arXiv:hep-th/9304011 v1 Apr 5 1993
arXiv:hep-th/9304011 v1 Apr 5 1993
arXiv:hep-th/9304011 v1 Apr 5 1993
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
YCTP-P23-92<br />
LA-UR-92-3479<br />
<strong>hep</strong>-<strong>th</strong>/<strong>9304011</strong><br />
<strong>arXiv</strong>:<strong>hep</strong>-<strong>th</strong>/<strong>9304011</strong> <strong>v1</strong> <strong>Apr</strong> 5 <strong>1993</strong><br />
P. Ginsparg<br />
ginsparg@xxx.lanl.gov<br />
MS-B285<br />
Los Alamos National Laboratory<br />
Los Alamos, NM 87545<br />
Lectures on 2D Gravity<br />
and<br />
2D String Theory<br />
and<br />
Gregory Moore<br />
moore@castalia.physics.yale.edu<br />
Dept. of Physics<br />
Yale University<br />
New Haven, CT 06511<br />
These notes are based on lectures delivered at <strong>th</strong>e 1992 Tasi summer school. They constitute<br />
<strong>th</strong>e preliminary version of a book which will include many corrections and much more<br />
useful information. Constructive comments are welcome.<br />
Lectures given June 11–19, 1992 at TASI summer school, Boulder, CO<br />
1992/<strong>1993</strong>
Contents<br />
0. Introduction, Overview, and Purpose . . . . . . . . . . . . . . . . . . . . . . . 3<br />
0.1. Philosophy and Diatribe . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
0.2. 2D Gravity and 2D String <strong>th</strong>eory . . . . . . . . . . . . . . . . . . . . . . . 5<br />
0.3. Review of reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
1. Loops and States in Conformal Field Theory . . . . . . . . . . . . . . . . . . . 8<br />
1.1. Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
1.2. Hamiltonian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
1.3. Equivalence of states and operators . . . . . . . . . . . . . . . . . . . . . . 10<br />
1.4. Gaussian Field wi<strong>th</strong> a Background Charge . . . . . . . . . . . . . . . . . . . 12<br />
2. 2D Euclidean Quantum Gravity I: Pa<strong>th</strong> Integral Approach . . . . . . . . . . . . . 13<br />
2.1. 2D Gravity and Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 13<br />
2.2. Pa<strong>th</strong> integral approach to 2D Euclidean Quantum Gravity . . . . . . . . . . . . 14<br />
3. Brief Review of <strong>th</strong>e Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 22<br />
3.1. Classical Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 22<br />
3.2. Classical Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . 24<br />
3.3. Quantum Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
3.4. Spectrum of Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . 28<br />
3.5. Semiclassical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
3.6. Seiberg bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
3.7. Semiclassical Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 35<br />
3.8. Operator Products in Liouville Theory . . . . . . . . . . . . . . . . . . . . 39<br />
3.9. Liouville Correlators from Analytic Continuation . . . . . . . . . . . . . . . . 40<br />
3.10. Quantum Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . 41<br />
3.11. Surfaces wi<strong>th</strong> boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 48<br />
4. 2D Euclidean Quantum Gravity II: Canonical Approach . . . . . . . . . . . . . . 50<br />
4.1. Canonical Quantization of Gravitational Theories . . . . . . . . . . . . . . . 50<br />
4.2. Canonical Quantization of 2D Euclidean Quantum Gravity . . . . . . . . . . . 51<br />
4.3. KPZ states in 2D Quantum Gravity . . . . . . . . . . . . . . . . . . . . . 52<br />
4.4. LZ states in 2D Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . 53<br />
4.5. States in 2D Gravity Coupled to a Gaussian Field: more BRST . . . . . . . . . 54<br />
5. 2D Critical String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61<br />
5.1. Particles in D Dimensions: QFT as 1D Euclidean Quantum Gravity. . . . . . . . 62<br />
5.2. Strings in D Dimensions: String Theory as 2D Euclidean Quantum Gravity . . . . 64<br />
5.3. 2D String Theory: Euclidean Signature . . . . . . . . . . . . . . . . . . . . 66<br />
5.4. 2D String Theory: Minkowskian Signature . . . . . . . . . . . . . . . . . . . 68<br />
5.5. Heterodox remarks regarding <strong>th</strong>e “special states” . . . . . . . . . . . . . . . . 69<br />
5.6. Bosonic String Amplitudes and <strong>th</strong>e “c > 1 problem” . . . . . . . . . . . . . . 72<br />
6. Discretized surfaces, matrix models, and <strong>th</strong>e continuum limit . . . . . . . . . . . . 75<br />
6.1. Discretized surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
6.2. Matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />
6.3. The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />
6.4. A first look at <strong>th</strong>e double scaling limit . . . . . . . . . . . . . . . . . . . . 83<br />
7. Matrix Model Technology I: Me<strong>th</strong>od of Or<strong>th</strong>ogonal Polynomials . . . . . . . . . . . 84<br />
1
7.1. Or<strong>th</strong>ogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84<br />
7.2. The genus zero partition function . . . . . . . . . . . . . . . . . . . . . . 86<br />
7.3. The all genus partition function . . . . . . . . . . . . . . . . . . . . . . . 88<br />
7.4. The Douglas Equations and <strong>th</strong>e KdV hierarchy . . . . . . . . . . . . . . . . 90<br />
7.5. Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92<br />
7.6. Multi-Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94<br />
7.7. Continuum Solution of <strong>th</strong>e Matrix Chains . . . . . . . . . . . . . . . . . . . 95<br />
8. Matrix Model Technology II: Loops on <strong>th</strong>e Lattice . . . . . . . . . . . . . . . . . 99<br />
8.1. Lattice Loop Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />
8.2. Precise definition of <strong>th</strong>e continuum limit . . . . . . . . . . . . . . . . . . 101<br />
8.3. The Loop Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 103<br />
9. Matrix Model Technology III: Free Fermions from <strong>th</strong>e Lattice . . . . . . . . . . . 105<br />
9.1. Lattice Fermi Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 105<br />
9.2. Eigenvalue distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 106<br />
9.3. Double–Scaled Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . 109<br />
10. Loops and States in Matrix Model Quantum Gravity . . . . . . . . . . . . . . 113<br />
10.1. Computation of Macroscopic Loops . . . . . . . . . . . . . . . . . . . . 113<br />
10.2. Loops to Local Operators . . . . . . . . . . . . . . . . . . . . . . . . 116<br />
10.3. Wavefunctions and Propagators from <strong>th</strong>e Matrix Model . . . . . . . . . . . 117<br />
10.4. Redundant operators, singular geometries and contact terms . . . . . . . . . 120<br />
11. Loops and States in <strong>th</strong>e c = 1 Matrix Model . . . . . . . . . . . . . . . . . . 120<br />
11.1. Definition of <strong>th</strong>e c = 1 Matrix Model . . . . . . . . . . . . . . . . . . . 120<br />
11.2. Matrix Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 122<br />
11.3. Double-Scaled Fermi Field Theory . . . . . . . . . . . . . . . . . . . . . 127<br />
11.4. Macroscopic Loops at c = 1 . . . . . . . . . . . . . . . . . . . . . . . . 129<br />
11.5. Wavefunctions and Wheeler–DeWitt Equations . . . . . . . . . . . . . . . 134<br />
11.6. Macroscopic Loop Field Theory and c = 1 scaling . . . . . . . . . . . . . . 134<br />
11.7. Correlation functions of Vertex Operators . . . . . . . . . . . . . . . . . 136<br />
12. Fermi Sea Dynamics and Collective Field Theory . . . . . . . . . . . . . . . . 139<br />
12.1. Time dependent Fermi Sea . . . . . . . . . . . . . . . . . . . . . . . . 139<br />
12.2. Collective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 140<br />
12.3. Relation to 1+1 dimensional relativistic field <strong>th</strong>eory . . . . . . . . . . . . . 142<br />
12.4. τ-space and φ-space . . . . . . . . . . . . . . . . . . . . . . . . . . . 143<br />
12.5. The w ∞ Symmetry of <strong>th</strong>e Harmonic Oscillator . . . . . . . . . . . . . . . 146<br />
12.6. The w ∞ Symmetry of Free Field Theory . . . . . . . . . . . . . . . . . . 148<br />
12.7. w ∞ symmetry of Classical Collective Field Theory . . . . . . . . . . . . . . 149<br />
13. String scattering in two spacetime dimensions . . . . . . . . . . . . . . . . . 151<br />
13.1. Definitions of <strong>th</strong>e S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . 151<br />
13.2. On <strong>th</strong>e Violation of Folklore . . . . . . . . . . . . . . . . . . . . . . . 155<br />
13.3. Classical scattering in collective field <strong>th</strong>eory . . . . . . . . . . . . . . . . 156<br />
13.4. Tree-Level Collective Field Theory S-Matrix . . . . . . . . . . . . . . . . 158<br />
13.5. Nonperturbative S-matrices . . . . . . . . . . . . . . . . . . . . . . . 159<br />
13.6. Properties of S-Matrix Elements . . . . . . . . . . . . . . . . . . . . . 163<br />
13.7. Unitarity of <strong>th</strong>e S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . 165<br />
2
13.8. Generating functional for S-matrix elements . . . . . . . . . . . . . . . . 167<br />
13.9. Tachyon recursion relations . . . . . . . . . . . . . . . . . . . . . . . . 169<br />
13.10. The many faces of c = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 171<br />
14. Vertex Operator Calculations and Continuum Me<strong>th</strong>ods . . . . . . . . . . . . . 172<br />
14.1. Review of <strong>th</strong>e Shapiro-Virasoro Amplitude . . . . . . . . . . . . . . . . . 172<br />
14.2. Resonant Amplitudes and <strong>th</strong>e “Bulk S-Matrix” . . . . . . . . . . . . . . . 174<br />
14.3. Wall vs. Bulk Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 177<br />
14.4. Algebraic Structures of <strong>th</strong>e 2D String: Chiral Cohomology . . . . . . . . . . 179<br />
14.5. Algebraic Structures of <strong>th</strong>e 2D String: Closed String Cohomology . . . . . . . 183<br />
15. Achievements, Disappointments, Future Prospects . . . . . . . . . . . . . . . 184<br />
15.1. Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185<br />
15.2. Disappointments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186<br />
15.3. Future prospects and Open Problems . . . . . . . . . . . . . . . . . . . 187<br />
Appendix A. Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . 188<br />
A.1. Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . 188<br />
A.2. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189<br />
0. Introduction, Overview, and Purpose<br />
0.1. Philosophy and Diatribe<br />
Following <strong>th</strong>e discovery of spacetime anomaly cancellation in 1984 [1], string <strong>th</strong>eory<br />
has undergone rapid development in several directions. The early hope of making<br />
direct contact wi<strong>th</strong> conventional particle physics phenomenology has however long since<br />
dissipated, and <strong>th</strong>ere is as yet no experimental program for finding even indirect manifestations<br />
of underlying string degrees of freedom in nature. The question of whe<strong>th</strong>er string<br />
<strong>th</strong>eory is “correct” in <strong>th</strong>e physical sense <strong>th</strong>us remains impossible to answer for <strong>th</strong>e foreseeable<br />
future. Particle/string <strong>th</strong>eorists none<strong>th</strong>eless continue to be tantalized by <strong>th</strong>e richness<br />
of <strong>th</strong>e <strong>th</strong>eory and by its natural ability to provide a consistent microscopic underpinning<br />
for bo<strong>th</strong> gauge <strong>th</strong>eory and gravity.<br />
A prime obstacle to our understanding of string <strong>th</strong>eory has been an inability to penetrate<br />
beyond its perturbative expansion. Our understanding of gauge <strong>th</strong>eory is enormously<br />
enhanced by having a fundamental formulation based on <strong>th</strong>e principle of local gauge invariance<br />
from which <strong>th</strong>e perturbative expansion can be derived. Symmetry breaking and<br />
nonperturbative effects such as instantons admit a clean and intuitive presentation. In<br />
string <strong>th</strong>eory, our lack of a fundamental formulation is compounded by our ignorance of<br />
<strong>th</strong>e true ground state of <strong>th</strong>e <strong>th</strong>eory. Beginning in 1989, <strong>th</strong>ere was some progress towards<br />
extracting such nonperturbative information from string <strong>th</strong>eory, at least in some simple<br />
3
contexts. The aim of <strong>th</strong>ese lectures is to provide <strong>th</strong>e conceptual background for <strong>th</strong>is work,<br />
and to describe some of its immediate consequences.<br />
In string <strong>th</strong>eory we wish to perform an integral over two dimensional geometries and<br />
a sum over two dimensional topologies,<br />
Z ∼<br />
∑ ∫<br />
Dg DX e −S , (0.1)<br />
topologies<br />
where <strong>th</strong>e spacetime physics (in <strong>th</strong>e case of <strong>th</strong>e bosonic string) resides in <strong>th</strong>e conformally<br />
invariant action<br />
∫<br />
S ∝ d 2 ξ √ g g ab ∂ a X µ ∂ b X ν G µν (X) . (0.2)<br />
Here µ, ν run from 1, . . . , D where D is <strong>th</strong>e number of spacetime dimensions, G µν (X) is <strong>th</strong>e<br />
spacetime metric, and <strong>th</strong>e integral Dg is over worldsheet metrics. Typically we “gauge-fix”<br />
<strong>th</strong>e worldsheet metric to g ab<br />
= e ϕ δ ab , where ϕ is known as <strong>th</strong>e Liouville field. Following<br />
<strong>th</strong>e formulation of string <strong>th</strong>eory in <strong>th</strong>is form (and in particular following <strong>th</strong>e appearance of<br />
<strong>th</strong>e work of Polyakov [2]), <strong>th</strong>ere was much work to develop <strong>th</strong>e quantum Liouville <strong>th</strong>eory<br />
(some of which is reviewed in chapt. 2 here), and conformal field <strong>th</strong>eory itself has been<br />
characterized as “an unsuccessful attempt to solve <strong>th</strong>e Liouville <strong>th</strong>eory” [3]. It has been<br />
recognized <strong>th</strong>at evaluation of <strong>th</strong>e partition function Z in (0.1) wi<strong>th</strong>out taking into account<br />
<strong>th</strong>e integral over geometry does not solve <strong>th</strong>e problem of interest, and moreover does not<br />
provide a systematic basis for a perturbation series in any known parameter.<br />
The program initiated in [4–6] relies on a discretization of <strong>th</strong>e string worldsheet to<br />
provide a me<strong>th</strong>od of taking <strong>th</strong>e continuum limit which incorporates simultaneously <strong>th</strong>e<br />
contribution of 2d surfaces wi<strong>th</strong> any number of handles. In one seemingly giant step, it is<br />
<strong>th</strong>us possible not only to integrate over all possible deformations of a given genus surface<br />
(<strong>th</strong>e analog of <strong>th</strong>e integral over Feynman parameters for a given loop diagram), but also<br />
to sum over all genus (<strong>th</strong>e analog of <strong>th</strong>e sum over all loop diagrams). This would in<br />
principle free us from <strong>th</strong>e ma<strong>th</strong>ematically fascinating but physically irrelevant problems<br />
of calculating conformal field <strong>th</strong>eory correlation functions on surfaces of fixed genus wi<strong>th</strong><br />
fixed moduli (objects which we never knew how to integrate over moduli or sum over<br />
genus anyway). The progress, however, is limited in <strong>th</strong>e sense <strong>th</strong>at <strong>th</strong>ese me<strong>th</strong>ods only<br />
apply currently for non-critical strings embedded in dimensions D ≤ 1 (or critical strings<br />
embedded in D ≤ 2), and <strong>th</strong>e nonperturbative information even in <strong>th</strong>is restricted context<br />
has proven incomplete. Due to familiar problems wi<strong>th</strong> lattice realizations of supersymmetry<br />
and chiral fermions, <strong>th</strong>ese me<strong>th</strong>ods have also resisted extension to <strong>th</strong>e supersymmetric case.<br />
4
The developments we shall describe here none<strong>th</strong>eless provide at least a half-step in<br />
<strong>th</strong>e correct direction, if only to organize <strong>th</strong>e perturbative expansion in a most concise<br />
way. They have also prompted much useful evolution of related continuum me<strong>th</strong>ods.<br />
Our point of view here is <strong>th</strong>at string <strong>th</strong>eories embedded in D ≤ 1 dimensions provide a<br />
simple context for testing ideas and me<strong>th</strong>ods of calculation. Just as we would encounter<br />
much difficulty calculating infinite dimensional functional integrals wi<strong>th</strong>out some prior<br />
experience wi<strong>th</strong> <strong>th</strong>eir finite dimensional analogs, progress in string <strong>th</strong>eory should be aided<br />
by experimentation wi<strong>th</strong> systems possessing a restricted number of degrees of freedom.<br />
While it is occasionally stated <strong>th</strong>at exactly solvable models are too special to provide<br />
useful lessons for physics, at least one striking historical example suggests <strong>th</strong>e opposite:<br />
Onsager’s exact solution of <strong>th</strong>e Ising model led to many fundamental ideas in quantum field<br />
<strong>th</strong>eory. In particular, ideas associated wi<strong>th</strong> <strong>th</strong>e renormalization group, phase transitions,<br />
non-mean field exponents, and <strong>th</strong>e operator product expansion all had <strong>th</strong>eir origin in <strong>th</strong>e<br />
solution of <strong>th</strong>e Ising model. We hope <strong>th</strong>at <strong>th</strong>is historical example will serve as well as <strong>th</strong>e<br />
paradigm for applications of exactly soluble spacetime solutions in string <strong>th</strong>eory.<br />
0.2. 2D Gravity and 2D String <strong>th</strong>eory<br />
We begin wi<strong>th</strong> a quick tour and overview of 2D gravity and 2D string <strong>th</strong>eory, emphasizing<br />
<strong>th</strong>e main physical ideas and lessons we have learned from recent progress in <strong>th</strong>e<br />
subject, so <strong>th</strong>at <strong>th</strong>ey are not lost in <strong>th</strong>e leng<strong>th</strong>y discussion <strong>th</strong>at follows. See also chapt. 15<br />
for ano<strong>th</strong>er appraisal when we are done.<br />
We have learned distinct lessons for 2D gravity and 2D string <strong>th</strong>eory due to <strong>th</strong>e twofold<br />
interpretation of <strong>th</strong>e models we discuss:<br />
Hartle-Hawking<br />
tadpole<br />
propagator<br />
Interaction Vertex<br />
= Topology Change<br />
Fig. 1: Correlation functions 〈 W (l) 〉 , 〈 W (l)W (l) 〉 , and 〈 W (l)W (l)W (l) 〉 .<br />
5
1) As Quantum Gravity<br />
In <strong>th</strong>is guise, we study a “field <strong>th</strong>eory of universes.” We will introduce an operator (<strong>th</strong>e<br />
macroscopic loop operator) W (l, . . .) <strong>th</strong>at creates universes of size l (1D and circular, in <strong>th</strong>e<br />
case of a 2D target space). The matrix model allows us to compute correlation functions<br />
〈<br />
W (l)<br />
〉<br />
,<br />
〈<br />
W (l) W (l)<br />
〉<br />
,<br />
〈<br />
W (l) W (l) W (l)<br />
〉<br />
, . . ., associated respectively wi<strong>th</strong> <strong>th</strong>e Hartle–<br />
Hawking wavefunction, wi<strong>th</strong> universe propagation, and wi<strong>th</strong> topology change (processes<br />
depicted in fig. 1). The matrix model even allows bo<strong>th</strong> calculation of <strong>th</strong>e effect of more<br />
drastic topology changes and summation over topology–changing amplitudes.<br />
2) As Critical String Theory<br />
We will consider in detail <strong>th</strong>e case for which (0.2) defines a flat Euclidean two dimensional<br />
target spacetime wi<strong>th</strong> coordinates φ, X. Physical interpretations of <strong>th</strong>e spacetime<br />
<strong>th</strong>eory require an analytic continuation to a spacetime of Minkowskian signature. There<br />
are two choices for <strong>th</strong>is continuation, but we shall argue <strong>th</strong>at <strong>th</strong>e clearest lessons for string<br />
<strong>th</strong>eory come from <strong>th</strong>e c = 1 model <strong>th</strong>at we will construct, wi<strong>th</strong> φ taken as <strong>th</strong>e Liouville<br />
coordinate, and t = −iX as <strong>th</strong>e Minkowskian time coordinate.<br />
t<br />
Free<br />
Strings<br />
1<br />
+log( )<br />
µ<br />
φ<br />
Strongly<br />
Coupled<br />
Strings<br />
Wall<br />
Fig. 2: Free, strong, and wall regions of <strong>th</strong>e tachyon potential. At left, particles<br />
are produced as tachyons scatter.<br />
6
This exactly solvable spacetime background is a strange world. Since <strong>th</strong>ere are only<br />
two spacetime dimensions, <strong>th</strong>ere are no transverse degrees of freedom. The only on-shell 1<br />
field <strong>th</strong>eoretic degree of freedom in <strong>th</strong>e <strong>th</strong>eory is <strong>th</strong>at of a massless bosonic field T (t, φ),<br />
called <strong>th</strong>e “massless tachyon” (because <strong>th</strong>e field T is ma<strong>th</strong>ematically analogous to a field<br />
which is tachyonic for any string propagating in more <strong>th</strong>an 2 spacetime dimensions, in<br />
particular for <strong>th</strong>e 26-dimensional bosonic string). Moreover, <strong>th</strong>e world is spatially inhomogeneous.<br />
We shall find (see eq. (5.14)) <strong>th</strong>at <strong>th</strong>e spacetime dilaton field has expectation<br />
value 〈D〉 = Qφ/2, so <strong>th</strong>at <strong>th</strong>e string coupling varies wi<strong>th</strong> <strong>th</strong>e spatial coordinate φ as<br />
κ eff<br />
= κ 0 e 1 2 Qφ . At φ = −∞, strings are free while at φ = +∞ strings are infinitely<br />
strongly coupled. Finally, <strong>th</strong>ere is a cosmological constant term ∼ µe φ in <strong>th</strong>e Liouville<br />
pa<strong>th</strong> integral formulation of <strong>th</strong>is <strong>th</strong>eory <strong>th</strong>at strongly suppresses contributions to pa<strong>th</strong><br />
integrals from large positive values of φ (we are assuming <strong>th</strong>roughout <strong>th</strong>ese notes <strong>th</strong>at <strong>th</strong>e<br />
cosmological constant satisfies µ > 0 , unless specified o<strong>th</strong>erwise). Since <strong>th</strong>e interaction<br />
turns on exponentially <strong>th</strong>ere is effectively a wall located at φ = log 1/µ, often called “<strong>th</strong>e<br />
Liouville wall”, and <strong>th</strong>e world looks as depicted in fig. 2. The most obvious physical experiment<br />
we can perform in <strong>th</strong>is world is to bounce our massless bosons off <strong>th</strong>e wall. In<br />
chapt. 13, we will show how to compute exactly <strong>th</strong>e S-matrix for such scattering processes.<br />
We are also able to compute <strong>th</strong>e “flows” <strong>th</strong>at relate physics in different (time-dependent)<br />
backgrounds.<br />
0.3. Review of reviews<br />
Several reviews overlap different portions of <strong>th</strong>e subject matter covered here. The<br />
Liouville <strong>th</strong>eory is reviewed in [7,8] and references <strong>th</strong>erein. The matrix model technology<br />
is reviewed in [9–14] (note <strong>th</strong>at some sections here are adapted directly from [11]), <strong>th</strong>e<br />
c = 1 matrix model is reviewed in [15,16], and spacetime properties are emphasized in [17].<br />
O<strong>th</strong>er recent reviews are [18].<br />
After treating <strong>th</strong>e necessary preliminaries, our viewpoint here is largely complementary<br />
to <strong>th</strong>e o<strong>th</strong>er treatments, placing an emphasis on <strong>th</strong>e properties of macroscopic loops,<br />
<strong>th</strong>e Wheeler–DeWitt equation, and <strong>th</strong>e scattering <strong>th</strong>eory in D = 2 target space dimensions.<br />
In chapters 1–5, we give an overview of <strong>th</strong>e continuum ideas and formalism we shall<br />
use for treating loops and states in conformal field <strong>th</strong>eory, for understanding <strong>th</strong>e classical,<br />
1 The graviton and dilaton in 2 spacetime target dimensions have no on-shell degrees of freedom,<br />
i.e. are not physical propagating particles.<br />
7
semiclassical, and quantum Liouville <strong>th</strong>eories, and for implementing <strong>th</strong>e pa<strong>th</strong> integral and<br />
canonical treatments of 2D Euclidean quantum gravity. In chapters 6–9, we focus on <strong>th</strong>e<br />
discretized approach to 2d quantum gravity and 2d string <strong>th</strong>eory, and explain various features<br />
of <strong>th</strong>e matrix model approach. In chapters 10–14, we employ <strong>th</strong>e techniques provided<br />
by <strong>th</strong>e discretized approach to calculate many of <strong>th</strong>e continuum quantities introduced earlier.<br />
In chapter 15, we assess our prospects for <strong>th</strong>e future. Appendix A contains some<br />
useful definitions and facts about some of <strong>th</strong>e special functions used in <strong>th</strong>ese lectures.<br />
Due to lack of spacetime, we will not be reviewing many o<strong>th</strong>er important works on<br />
<strong>th</strong>e subject of 2D gravity. These notes are a preliminary version of a book [19], 2 <strong>th</strong>at<br />
will contain much interesting material omitted from <strong>th</strong>ese lecture notes. We welcome<br />
constructive comments concerning typos, inconsistent conventions, inconsistent references,<br />
sign errors and conceptual errors in <strong>th</strong>ese notes.<br />
1. Loops and States in Conformal Field Theory<br />
We begin here wi<strong>th</strong> a review of how a loop in <strong>th</strong>e context of conformal field <strong>th</strong>eory<br />
can be replaced by a sum of local operators. For simplicity we will restrict attention to <strong>th</strong>e<br />
Gaussian model. The intuition from <strong>th</strong>is example will be essential to our later extraction<br />
of correlation functions from macroscopic loop amplitudes.<br />
1.1. Lagrangian formalism<br />
For simplicity, consider <strong>th</strong>e standard c = 1 Gaussian model,<br />
∫<br />
S = ∂X ∂X , (1.1)<br />
Σ<br />
where Σ is a surface perhaps wi<strong>th</strong> handles and boundaries. The objects of interest are <strong>th</strong>e<br />
pa<strong>th</strong> integrals<br />
∫<br />
DX(z, ¯z) e −S O i (p i ) , (1.2)<br />
where we integrate over maps X : Σ → IR.<br />
The space of local operators is spanned by expressions of <strong>th</strong>e form<br />
∏<br />
i<br />
O ∼ P(∂X, ∂ 2 X, . . . ; ∂X, ∂ 2 X, . . .) e ikX(z,¯z) ,<br />
(1.3)<br />
2 <strong>th</strong>e modest expenditure for which will be amply rewarded by <strong>th</strong>e substantial fur<strong>th</strong>er enlightenment<br />
contained <strong>th</strong>erein.<br />
8
where P is a polynomial and <strong>th</strong>e expression is suitably normal-ordered. When X is a<br />
periodic variable, i.e. a map X : Σ → S 1 , <strong>th</strong>en additional considerations apply: k is<br />
quantized and <strong>th</strong>ere are winding modes which allow <strong>th</strong>e definition of a more subtle <strong>th</strong>eory<br />
wi<strong>th</strong> an extra zero mode, leading to “magnetic” and “electric” quantum numbers. See [20]<br />
for more details.<br />
1.2. Hamiltonian formalism<br />
In <strong>th</strong>e radial Hamiltonian formalism (for a review, see e.g. [20]) we decompose <strong>th</strong>e<br />
fields into modes,<br />
i∂X = ∑ α n z −n−1 ,<br />
[α n , α m ] = n δ n+m<br />
i∂X = ∑ ᾱ n ¯z −n−1 , [ᾱ n , ᾱ m ] = n δ n+m .<br />
(1.4)<br />
The stress-energy tensor T = − 1 2 (∂X)2 defines a Virasoro algebra and radial propagation<br />
is generated by <strong>th</strong>e Hamiltonian L 0 + ¯L 0 .<br />
In terms of <strong>th</strong>e Fock space<br />
{∏<br />
}<br />
F p = Span (α −n ) m n<br />
|p〉 n > 0, m n ≥ 0<br />
n<br />
(1.5)<br />
α 0 |p〉 = p |p〉 α n |p〉 = 0 (n > 0) ,<br />
we see <strong>th</strong>at <strong>th</strong>e states of <strong>th</strong>e <strong>th</strong>eory lie in a Hilbert Space of <strong>th</strong>e form<br />
H = ⊕ p,¯p N p,¯p F p ⊗ F¯p . (1.6)<br />
Alternatively, we can <strong>th</strong>ink of <strong>th</strong>e Hilbert space as <strong>th</strong>e space of wavefunctionals Ψ[X(σ)].<br />
(More precisely, we may introduce <strong>th</strong>e loop space LIR and, wi<strong>th</strong> an appropriate measure,<br />
<strong>th</strong>e Hilbert space is simply <strong>th</strong>e space H = L 2 (LIR) of square integrable maps of loops into<br />
IR.)<br />
Exercise. Coherent states<br />
If we decompose<br />
∑ ∑<br />
X(σ) = x 0 + e inσ x n + e inσ ¯x n , (1.7)<br />
n>0<br />
n 0; α n ↔ ∂/∂x n ,<br />
for n < 0; and similarly for ᾱ and ¯x, translate <strong>th</strong>e Fock space states (1.5) into wavefunctionals<br />
Ψ[X(σ)].<br />
9
The Lagrangian and Hamiltonian formalisms are related by <strong>th</strong>e so-called “operator<br />
formalism”. The basic idea is <strong>th</strong>at <strong>th</strong>e neighborhood of any point on <strong>th</strong>e surface Σ locally<br />
looks like <strong>th</strong>e complex plane and information about <strong>th</strong>e rest of <strong>th</strong>e surface may be<br />
summarized in a state at infinity.<br />
1.3. Equivalence of states and operators<br />
One of <strong>th</strong>e basic properties of conformal field <strong>th</strong>eory is <strong>th</strong>e one-to-one correspondence<br />
between operators O and states |O〉.<br />
To map operators → states, we associate <strong>th</strong>e state |O〉 to <strong>th</strong>e operator O(z, ¯z) according<br />
to<br />
O ↦→ |O〉 ≡ lim<br />
z,¯z→0 O(z, ¯z)|0〉 . (1.8)<br />
Equivalently, we can create a state by performing a pa<strong>th</strong> integral on a hemisphere D.<br />
To evaluate such a pa<strong>th</strong> integral, <strong>th</strong>e boundary conditions for <strong>th</strong>e field X(z, ¯z) must be<br />
specified on <strong>th</strong>e equator, parametrized here by σ, and <strong>th</strong>e value of <strong>th</strong>e pa<strong>th</strong> integral defines<br />
<strong>th</strong>e wavefunction Ψ[X(σ)] (i.e. <strong>th</strong>e wavefunction for <strong>th</strong>e identity operator). Insertion of an<br />
operator O on <strong>th</strong>e hemisphere D gives <strong>th</strong>e wavefunction Ψ O for <strong>th</strong>e operator O,<br />
[ ]<br />
Ψ O X(σ) =<br />
(1.9)<br />
Using <strong>th</strong>e equivalence between <strong>th</strong>e Fock space and wavefunctional descriptions of <strong>th</strong>e states,<br />
<strong>th</strong>e two descriptions (1.8) and (1.9) of <strong>th</strong>e operator ↦→ state maps are easily seen to be<br />
O<br />
equivalent: namely 〈 X(σ) ∣ ∣ O<br />
〉<br />
= ΨO<br />
[<br />
X(σ)<br />
]<br />
(where<br />
〈<br />
X(σ)<br />
∣ ∣ is a basis of eigenstates of <strong>th</strong>e<br />
operator X(σ) in <strong>th</strong>e Fock space representation).<br />
Exercise. Wavefunctions from pa<strong>th</strong> integrals<br />
Consider a disk of radius r in <strong>th</strong>e complex plane, centered at z = 0, and consider<br />
<strong>th</strong>e pa<strong>th</strong> integral (1.9) wi<strong>th</strong> boundary condition<br />
∑ ∑<br />
X(σ) = x 0 + e inσ x n + e inσ ¯x n (1.10)<br />
n>0<br />
n
where <strong>th</strong>e constant C is determined by imposing some normalization condition.<br />
b) Why can <strong>th</strong>is be identified wi<strong>th</strong> <strong>th</strong>e state |0〉 ?<br />
c) Repeat <strong>th</strong>is exercise to calculate <strong>th</strong>e wavefunction for some o<strong>th</strong>er simple operators.<br />
d) Describe <strong>th</strong>e wavefunction associated to states created by local operators of <strong>th</strong>e<br />
form (1.3).<br />
Expansion of loops in terms of local operators<br />
Having described <strong>th</strong>e operator ↦→ state mapping, now we wish to consider <strong>th</strong>e inverse<br />
state ↦→ operator mapping. To “insert a state” into <strong>th</strong>e pa<strong>th</strong> integral, we cut a hole of<br />
radius r out of <strong>th</strong>e surface Σ, and insert a state wi<strong>th</strong> some wavefunction Ψ [ X(σ) ] on<br />
<strong>th</strong>e boundary of <strong>th</strong>e hole (i.e. use Ψ [ X(σ) ] as <strong>th</strong>e weight factor in <strong>th</strong>e functional integral<br />
over X(σ)).<br />
We shall see <strong>th</strong>at <strong>th</strong>e new pa<strong>th</strong> integral is equivalent to <strong>th</strong>at obtained by<br />
inserting a local operator into (1.2), <strong>th</strong>us providing a states → operators mapping. The<br />
mapping Ψ O ↦→ O is clearly linear and one-to-one, but to show <strong>th</strong>at <strong>th</strong>ere is moreover an<br />
isomorphism between states and operators, we need to show <strong>th</strong>at every state is equivalently<br />
represented by an operator insertion. The basic idea, physically, is to take a state inserted<br />
on a hole and by conformal invariance shrink <strong>th</strong>e hole to arbitrarily small size. Since an<br />
infinitesimally small hole has <strong>th</strong>e same effect as a local operator, <strong>th</strong>e insertion of a state on<br />
<strong>th</strong>e boundary of a hole in <strong>th</strong>e pa<strong>th</strong> integral is equivalent to <strong>th</strong>e insertion of some operator.<br />
In formulae, <strong>th</strong>e states ↦→ operators mapping equates <strong>th</strong>e insertion of a little hole of<br />
radius r on any surface Σ in state |Ψ〉 wi<strong>th</strong> <strong>th</strong>e insertion of a sum of operators at <strong>th</strong>e center<br />
of <strong>th</strong>e hole,<br />
W Ψ (r) ≡ ∑ i<br />
〈O i |Ψ〉 r ∆ i+ ¯∆ i<br />
O i . (1.12)<br />
Here O i (z, ¯z) is a basis of local operators diagonalizing L 0 + L 0 , and W Ψ (r) is an operator<br />
<strong>th</strong>at inserts a macroscopic loop of size r wi<strong>th</strong> wavefunction Ψ [ X(σ) ] .<br />
Example: Annular pa<strong>th</strong> integral.<br />
Consider <strong>th</strong>e pa<strong>th</strong> integral on a sphere wi<strong>th</strong> two holes, or, after a conformal transfor-<br />
11
mation, on an annulus. This is given in <strong>th</strong>e operator formalism by<br />
r=1<br />
Ψ 2<br />
(1.13)<br />
where ∣ ∣W Ψ1 (r) 〉 is <strong>th</strong>e state associated to <strong>th</strong>e local operator W Ψ1 (r). For more details on<br />
<strong>th</strong>e operator formalism, see e.g. [21–23].<br />
Our goal is to calculate macroscopic loop amplitudes in 2D gravity. Current matrix<br />
model technology will allow <strong>th</strong>is only for some very specific states |Ψ〉, but <strong>th</strong>ese states will<br />
have overlaps wi<strong>th</strong> sufficiently many interesting operators <strong>th</strong>at much useful information can<br />
be extracted. To provide a physical framework for interpreting <strong>th</strong>e matrix model results,<br />
we shall first investigate in <strong>th</strong>e next two chapters <strong>th</strong>e spectrum of Liouville <strong>th</strong>eory and of<br />
2D gravity.<br />
r<br />
= 〈Ψ<br />
Ψ 1<br />
= ∑<br />
= ∑<br />
= 〈<br />
1.4. Gaussian Field wi<strong>th</strong> a Background Charge<br />
For later comparison wi<strong>th</strong> <strong>th</strong>e Liouville results, we give here a brief overview of gaussian<br />
conformal field <strong>th</strong>eory in <strong>th</strong>e presence of a background charge, also known as Chodos–<br />
Thorn/Feigin–Fuks (CTFF) <strong>th</strong>eory. We consider <strong>th</strong>e action<br />
∫<br />
S CTFF =<br />
The additional term leads to <strong>th</strong>e modified stress-energy<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇φ) 2 + iα )<br />
0<br />
4π φR(ĝ) . (1.14)<br />
T = − 1 2 ∂φ∂φ + iα 0 ∂ 2 φ , (1.15)<br />
which generates a Virasoro algebra wi<strong>th</strong> central charge<br />
c = 1 − 12α 2 0 .<br />
We see <strong>th</strong>at <strong>th</strong>e effect of <strong>th</strong>e extra term in (1.15) is to shift c < 1 for α 0 real. Since <strong>th</strong>e<br />
stress-energy tensor in (1.15) has an imaginary part, <strong>th</strong>e <strong>th</strong>eory it defines is not unitary<br />
12
for arbitrary α 0 . For particular values of α 0 , it turns out to contain a consistent unitary<br />
subspace.<br />
Taking <strong>th</strong>e operator product wi<strong>th</strong> <strong>th</strong>e modified T of (1.15), we find <strong>th</strong>at <strong>th</strong>e conformal<br />
weight of e βφ is shifted to ∆ = 1 2 β(β−2α 0). The same conformal weights would be inferred<br />
from two-point functions calculated in <strong>th</strong>e presence of a ‘background charge’ − √ 2α 0 at<br />
infinity, so <strong>th</strong>e modification of T (z) in (1.15) is interpreted as <strong>th</strong>e presence of such a<br />
background charge. This formalism was originally used by Chodos and Thorn in [24], and<br />
was more recently revived by Feigen and Fuks in a form used in [25] to calculate correlation<br />
functions of <strong>th</strong>e c < 1 conformal field <strong>th</strong>eories.<br />
Exercise. Momentum Shift from Background Charge<br />
Consider a Gaussian field wi<strong>th</strong> background charge Q = 2iα 0 . Derive <strong>th</strong>e relation<br />
ip φ = α − Q 2 , (1.16)<br />
for <strong>th</strong>e momentum p φ of <strong>th</strong>e state created by <strong>th</strong>e vertex operator exp(αφ) by considering<br />
<strong>th</strong>e state created by <strong>th</strong>e pa<strong>th</strong> integral on <strong>th</strong>e disk, analogous to <strong>th</strong>e example of <strong>th</strong>e<br />
Gaussian model in (1.9).<br />
2. 2D Euclidean Quantum Gravity I: Pa<strong>th</strong> Integral Approach<br />
In <strong>th</strong>is chapter, we shall consider <strong>th</strong>e implementation of quantum gravity as a <strong>th</strong>eory<br />
which makes precise and well-defined sense out of a pa<strong>th</strong> integral over metrics on some<br />
spacetime.<br />
2.1. 2D Gravity and Liouville Theory<br />
Using <strong>th</strong>e principle of general covariance, any quantum field <strong>th</strong>eory S matter [X i ] in any<br />
number of dimensions may be coupled to gravity, resulting in an action S[g, X i ], where X i<br />
refer to “matter” fields in <strong>th</strong>e <strong>th</strong>eory and g is <strong>th</strong>e metric.<br />
Classically, <strong>th</strong>e <strong>th</strong>eory S[g, X i ] wi<strong>th</strong> gravity coupling in two dimensions is always a<br />
conformal field <strong>th</strong>eory. To see <strong>th</strong>is, recall <strong>th</strong>at <strong>th</strong>e stress energy tensor of <strong>th</strong>e <strong>th</strong>eory is<br />
δ<br />
δg αβ S = T αβ . Defining <strong>th</strong>e Liouville mode as <strong>th</strong>e overall scale of <strong>th</strong>e metric, g = e γφ ĝ,<br />
we see <strong>th</strong>at <strong>th</strong>e Liouville equation of motion is T α α = 0. This defines a classical conformal<br />
field <strong>th</strong>eory. In two dimensions, we may pass to local complex coordinates and write <strong>th</strong>is<br />
equation as T z¯z = 0. Conservation of energy-momentum <strong>th</strong>en shows <strong>th</strong>at <strong>th</strong>e <strong>th</strong>eory has a<br />
holomorphic energy momentum tensor T zz = T (z).<br />
13
Quantum mechanically, we try to understand <strong>th</strong>e (Euclidean) quantum gravitational<br />
pa<strong>th</strong> integrals<br />
〈O 1 . . . O n 〉 = 1 Z<br />
∫<br />
∫<br />
Dg DX<br />
vol(Diff ) e−κ R−µA−S[X] O 1 . . . O n , (2.1)<br />
where O i are generally covariant operators. By fixing a conformal gauge g ab<br />
= e φ δ ab ,<br />
Polyakov [2] observed <strong>th</strong>at <strong>th</strong>e matter/gravity <strong>th</strong>eory could be written as a coupled tensor<br />
product of Liouville <strong>th</strong>eory and <strong>th</strong>e “matter” <strong>th</strong>eory S matter . The passage to conformal<br />
gauge necessarily introduces <strong>th</strong>e Faddeev–Popov reparametrization ghosts. At a formal<br />
level, <strong>th</strong>e gauge invariance of <strong>th</strong>e <strong>th</strong>eory, expressed as <strong>th</strong>e independence of <strong>th</strong>e integral<br />
(2.1) to Weyl transformations of <strong>th</strong>e gauge slice, implies <strong>th</strong>at <strong>th</strong>e coupling of Liouville and<br />
matter <strong>th</strong>eories is itself a conformal field <strong>th</strong>eory. If <strong>th</strong>e original matter <strong>th</strong>eory was not<br />
conformal, however, <strong>th</strong>en <strong>th</strong>e resulting <strong>th</strong>eory will not be a simple tensor product.<br />
Example 1: Coupling <strong>th</strong>e massive Ising model<br />
∫<br />
S =<br />
¯ψ∂ψ + m ¯ψψ (2.2)<br />
to gravity results in <strong>th</strong>e lagrangian<br />
∫ ∫<br />
√g ( )<br />
S = ¯ψDψ + m ¯ψψ +<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇φ) 2 + µ<br />
8πγ 2 eγφ + Q )<br />
8π φR(ĝ) , (2.3)<br />
where D is <strong>th</strong>e covariant derivative (i.e. includes <strong>th</strong>e spin connection).<br />
Example 2: Coupling <strong>th</strong>e massive sine-Gordon model<br />
to gravity results in <strong>th</strong>e lagrangian<br />
∫<br />
S =<br />
+<br />
∫<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇X) 2 + m cos(pX/ √ )<br />
2)<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇X) 2 + me ξφ cos(pX/ √ )<br />
2)<br />
∫<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇φ) 2 + µ<br />
8πγ 2 eγφ + Q )<br />
8π φR(ĝ)<br />
.<br />
(2.4)<br />
(2.5)<br />
In bo<strong>th</strong> examples, we will see <strong>th</strong>at Q and γ are fixed by general covariance. The<br />
remarkable property of Liouville <strong>th</strong>eory <strong>th</strong>at allows it to associate an arbitrary quantum<br />
field <strong>th</strong>eory wi<strong>th</strong> some conformal field <strong>th</strong>eory deserves to be better understood.<br />
14
2.2. Pa<strong>th</strong> integral approach to 2D Euclidean Quantum Gravity<br />
The first success of <strong>th</strong>e discretized (matrix model) approach, <strong>th</strong>e focus of our later<br />
chapters here, was to reproduce <strong>th</strong>e critical exponents predicted from continuum (Liouville)<br />
me<strong>th</strong>ods. (In fact <strong>th</strong>e coincidence of results served to give post-facto verification of bo<strong>th</strong><br />
me<strong>th</strong>ods.) In <strong>th</strong>is section we review <strong>th</strong>e latter continuum me<strong>th</strong>ods from a fairly formal<br />
point of view. A more physical point of view will appear in <strong>th</strong>e next chapter.<br />
String susceptibility Γ str<br />
We consider <strong>th</strong>e continuum partition function<br />
∫<br />
Z =<br />
Dg DX<br />
vol(Diff ) e−S M(X; g) − µ 0<br />
8π<br />
∫<br />
d 2 ξ √ g<br />
, (2.6)<br />
where S M is some conformally invariant action for matter fields coupled to a two dimensional<br />
surface Σ wi<strong>th</strong> metric g, µ 0 is a bare cosmological constant, and we have symbolically<br />
divided <strong>th</strong>e measure by <strong>th</strong>e “volume” of <strong>th</strong>e diffeomorphism group (which acts as a local<br />
∫<br />
symmetry) of Σ . For <strong>th</strong>e free bosonic string, we take S M = 1<br />
8π d 2 ξ √ g g ab ∂ aX ⃗ · ∂bX<br />
⃗<br />
where <strong>th</strong>e X(ξ) ⃗ specify <strong>th</strong>e embedding of Σ into flat D-dimensional spacetime.<br />
To define (2.6), we need to specify <strong>th</strong>e measures for <strong>th</strong>e integrations over X and g<br />
(see, e.g. [26]). The measure DX is determined by requiring <strong>th</strong>at ∫ D g δX e −‖δX‖2 g = 1,<br />
where <strong>th</strong>e norm in <strong>th</strong>e gaussian functional integral is given by ‖δX‖ 2 g = ∫ d 2 ξ √ g δ ⃗ X · δ ⃗ X.<br />
Similarly, <strong>th</strong>e measure Dg is determined by normalizing ∫ D g δg e − 1 2 ‖δg‖2 g<br />
= 1, where<br />
‖δg‖ 2 g = ∫ d 2 ξ √ g (g ac g bd + 2g ab g cd ) δg ab<br />
δg cd<br />
, and δg represents a metric fluctuation at<br />
some point g ij in <strong>th</strong>e space of metrics on a genus h surface.<br />
The measures DX and Dg are invariant under <strong>th</strong>e group of diffeomorphisms of <strong>th</strong>e<br />
surface, but not necessarily under conformal transformations g ab<br />
→ e σ g ab<br />
. Indeed due to<br />
<strong>th</strong>e metric dependence in <strong>th</strong>e norm ‖δX‖ 2 g , it turns out <strong>th</strong>at<br />
D<br />
48π<br />
D e σ gX = e<br />
S L(σ)<br />
Dg X , (2.7)<br />
where<br />
∫<br />
S L (σ) =<br />
d 2 ξ √ (<br />
)<br />
1<br />
g<br />
2 gab ∂ a σ∂ b σ + Rσ + µe σ<br />
(2.8)<br />
is known as <strong>th</strong>e Liouville action. (This result may be derived diagrammatically, via <strong>th</strong>e<br />
Fujikawa me<strong>th</strong>od, or via an index <strong>th</strong>eorem; for a review see [27].)<br />
The metric measure Dg as well has an anomalous variation under conformal transformations.<br />
To express it in a form analogous to (2.7), we first need to recall some basic facts<br />
15
about <strong>th</strong>e domain of integration. The space of metrics on a compact topological surface Σ<br />
modulo diffeomorphisms and Weyl transformations is a finite dimensional compact space<br />
M h , known as moduli space. (It is 0-dimensional for genus h = 0; 2-dimensional for h = 1;<br />
and (6h − 6)-dimensional for h ≥ 2). If for each point τ ∈ M h , we choose a representative<br />
metric ĝ ij , <strong>th</strong>en <strong>th</strong>e orbits generated by <strong>th</strong>e diffeomorphism and Weyl groups acting on<br />
ĝ ij generate <strong>th</strong>e full space of metrics on Σ. Thus given <strong>th</strong>e slice ĝ(τ), any metric can be<br />
represented in <strong>th</strong>e form<br />
f ∗ g = e ϕ ĝ(τ) ,<br />
where f ∗ represents <strong>th</strong>e action of <strong>th</strong>e diffeomorphism f : Σ → Σ.<br />
Since <strong>th</strong>e integrand of (2.6) is diffeomorphism invariant, <strong>th</strong>e functional integral would<br />
be infinite unless we formally divide out by <strong>th</strong>e volume of orbit of <strong>th</strong>e diffeomorphism<br />
group. This is accomplished by gauge fixing to <strong>th</strong>e slice ĝ(τ); <strong>th</strong>e Jacobian <strong>th</strong>at enters can<br />
be represented in terms of Fadeev–Popov ghosts, as familiar from <strong>th</strong>e analogous procedure<br />
in gauge <strong>th</strong>eory. We parametrize an infinitesimal change in <strong>th</strong>e metric as<br />
δg zz = ∇ z ξ z ,<br />
δg¯z¯z = ∇¯z ξ¯z<br />
(where for convenience we employ complex coordinates, and recall <strong>th</strong>at <strong>th</strong>e components<br />
g = g¯zz z¯z<br />
are parametrized by e ϕ ). The measure Dg at ĝ(τ) splits into an integration<br />
[dτ] over moduli, an integration Dϕ over <strong>th</strong>e conformal factor, and an integration<br />
Dξ D ¯ξ over diffeomorphisms. The change of integration variables Dδg zz Dδg¯z¯z =<br />
(det ∇ z det ∇¯z ) Dξ D ¯ξ introduces <strong>th</strong>e Jacobian det ∇ z det ∇¯z for <strong>th</strong>e change from δg to ξ.<br />
The determinants in turn can be represented as<br />
∫<br />
det ∇ z det ∇¯z = Db Dc D¯b D¯c e − ∫ d 2 ξ √ g b zz ∇¯z c z − ∫ d 2 ξ √ g b¯z¯z ∇ z c¯z<br />
∫<br />
≡ D(gh) e −S gh(b, c, ¯b,<br />
(2.9)<br />
¯c) ,<br />
where D(gh) ≡ Db Dc D¯b D¯c is an abbreviation for <strong>th</strong>e measures associated to <strong>th</strong>e ghosts<br />
b, c, ¯b, ¯c; b zz is a holomorphic quadratic differential, and c z (c¯z ) is a holomorphic (antiholomorphic)<br />
vector.<br />
Finally, <strong>th</strong>e ghost measure D(gh) is not invariant under <strong>th</strong>e conformal transformation<br />
g → e σ g, instead we have [2,26,27]<br />
D eσ g(gh) = e − 26<br />
48π S L(σ, g)<br />
Dg (gh) , (2.10)<br />
16
where S L is again <strong>th</strong>e Liouville action (2.8). (In units in which <strong>th</strong>e contribution of a single<br />
scalar field to <strong>th</strong>e conformal anomaly is c = 1, and hence c = 1/2 for a single Majorana–<br />
Weyl fermion, <strong>th</strong>e conformal anomaly due to a spin j reparametrization ghost is given by<br />
c = (−1) F 2(1 + 6j(j − 1)). The contribution from a spin j = 2 reparametrization ghost is<br />
<strong>th</strong>us c = −26.)<br />
We have <strong>th</strong>us far succeeded to reexpress <strong>th</strong>e partition function (2.6) as<br />
∫<br />
Z =<br />
Choosing a metric slice g = e ϕ ĝ gives<br />
[dτ] D g ϕ D g (gh) D g X e −S M − S gh − µ ∫<br />
0<br />
2π d 2 ξ √ g<br />
.<br />
D eϕĝϕ D eϕĝ(gh) D eϕĝX = J(ϕ, ĝ) Dĝϕ Dĝ(gh) DĝX ,<br />
where <strong>th</strong>e Jacobian J(ϕ, ĝ) is easily calculated for <strong>th</strong>e matter and ghost sectors ( (2.7) and<br />
(2.10) ) but not for <strong>th</strong>e Liouville mode ϕ. The functional integral over ϕ is complicated by<br />
<strong>th</strong>e implicit metric dependence in <strong>th</strong>e norm<br />
∫<br />
‖δϕ‖ 2 g = d 2 ξ √ ∫<br />
g (δϕ) 2 =<br />
d 2 ξ √ ĝ e ϕ (δϕ) 2 ,<br />
since only if <strong>th</strong>e e ϕ factor were absent above would <strong>th</strong>e Dĝϕ measure reduce to <strong>th</strong>at of a<br />
free field.<br />
In [28], it is simply assumed 3 <strong>th</strong>at <strong>th</strong>e overall Jacobian J(ϕ, ĝ) takes <strong>th</strong>e form of an<br />
exponential of a local Liouville-like action ∫ d 2 ξ √ ĝ (ã ĝ ab ∂ a ϕ∂ b ϕ + ˜b ˆRϕ + µe˜cϕ ), where ã,<br />
˜b, and ˜c are constants <strong>th</strong>at will be determined by requiring overall conformal invariance (˜c<br />
is inserted in anticipation of rescaling of ϕ). Wi<strong>th</strong> <strong>th</strong>is assumption, <strong>th</strong>e partition function<br />
(2.6) takes <strong>th</strong>e form<br />
∫<br />
Z = [dτ] Dĝϕ Dĝ(gh) DĝX e −S M(X, ĝ) − S gh (b, c, ¯b, ¯c; ĝ)<br />
where <strong>th</strong>e ϕ measure is now <strong>th</strong>at of a free field.<br />
· e − ∫ d 2 ξ √ ĝ (ã ĝ ab ∂ a ϕ∂ b ϕ + ˜b ˆRϕ + µe˜cϕ )<br />
(2.11)<br />
The pa<strong>th</strong> integral (2.11) was defined to be reparametrization invariant, and should<br />
depend only on e ϕ ĝ = g (up to diffeomorphism), not on <strong>th</strong>e specific slice ĝ.<br />
Due to<br />
3 Some attempts to justify <strong>th</strong>is assumption may be found in [29]. In <strong>th</strong>e next two chapters, we<br />
shall see why <strong>th</strong>is result should be expected from <strong>th</strong>e canonical Hamiltonian point of view of [30].<br />
17
diffeomorphism invariance, (2.11) should <strong>th</strong>us be invariant under <strong>th</strong>e infinitesimal transformation<br />
δĝ = ε(ξ)ĝ , δϕ = −ε(ξ) , (2.12)<br />
and we can use <strong>th</strong>e known conformal anomalies (2.7) and (2.10) for ϕ, X, and <strong>th</strong>e ghosts to<br />
determine <strong>th</strong>e constants ã, ˜b, ˜c. Substituting <strong>th</strong>e variations (2.12) in (2.11), we find terms<br />
of <strong>th</strong>e form<br />
( D − 26 + 1<br />
48π<br />
˜b) ∫<br />
+ d 2 ξ √ ∫<br />
ĝ ˆR ε and (2ã − ˜b)<br />
d 2 ξ √ ĝ ε ϕ ,<br />
where <strong>th</strong>e D − 26 on <strong>th</strong>e left is <strong>th</strong>e contribution from <strong>th</strong>e matter and ghost measures DĝX<br />
and Dĝ(gh), and <strong>th</strong>e additional 1 comes from <strong>th</strong>e Dĝϕ measure. Invariance under (2.12)<br />
<strong>th</strong>us determines<br />
˜b =<br />
25 − D<br />
48π<br />
, ã = 1 2˜b . (2.13)<br />
(In general we would substitute here D → c matter , where c matter is <strong>th</strong>e contribution to <strong>th</strong>e<br />
central charge from <strong>th</strong>e “matter” sector of <strong>th</strong>e <strong>th</strong>eory.)<br />
Substituting <strong>th</strong>e values of ã, ˜b into <strong>th</strong>e Liouville action in (2.11) gives<br />
∫<br />
1<br />
8π<br />
d 2 ξ √ ( 25 − D ĝ ĝ ab ∂ a ϕ ∂ b ϕ + 25 − D<br />
12<br />
6<br />
)<br />
ˆR ϕ . (2.14)<br />
∫ √<br />
To obtain a conventionally normalized kinetic term 1<br />
8π (∂ϕ) 2 12<br />
, we rescale ϕ →<br />
25−D ϕ.<br />
(This normalization leads to <strong>th</strong>e leading short distance expansion ϕ(z) ϕ(w) ∼ − log(z −<br />
w).) In terms of <strong>th</strong>e rescaled ϕ, we write <strong>th</strong>e Liouville action as<br />
where<br />
∫<br />
1<br />
8π<br />
d 2 ξ √ (<br />
)<br />
ĝ ĝ ab ∂ a ϕ ∂ b ϕ + Q ˆR ϕ , (2.15)<br />
Q ≡<br />
√<br />
25 − D<br />
3<br />
. (2.16)<br />
The energy-momentum tensor T = − 1 2 ∂ϕ∂ϕ+ Q 2 ∂2 ϕ derived from (2.15) has leading short<br />
distance expansion T (z)T (w) ∼ 1 2 c Liouville /(z − w)4 + . . ., where c Liouville<br />
= 1 + 3Q 2 . Note<br />
<strong>th</strong>at if we substitute (2.16) into c Liouville<br />
and add an additional c = D −26 from <strong>th</strong>e matter<br />
and ghost sectors, we find <strong>th</strong>at <strong>th</strong>e total conformal anomaly vanishes,<br />
c matter + c Liouv + c ghost = D + (26 − D) − 26 = 0<br />
18
(consistent wi<strong>th</strong> <strong>th</strong>e required overall conformal invariance).<br />
It remains to determine <strong>th</strong>e coefficient ˜c in (2.11). We have since rescaled ϕ, so we<br />
write instead e γϕ and determine γ by <strong>th</strong>e requirement <strong>th</strong>at <strong>th</strong>e physical metric be g = ĝ e γϕ .<br />
Geometrically, <strong>th</strong>is means <strong>th</strong>at <strong>th</strong>e area of <strong>th</strong>e surface is represented by ∫ d 2 ξ √ ĝ e γϕ . γ<br />
is <strong>th</strong>ereby determined by <strong>th</strong>e requirement <strong>th</strong>at e γϕ behave as a (1,1) conformal field (so<br />
<strong>th</strong>at <strong>th</strong>e combination d 2 ξ e γϕ is conformally invariant). For <strong>th</strong>e energy-momentum tensor<br />
mentioned after (2.16), <strong>th</strong>e conformal weight 4 of e γϕ is<br />
∆(e γϕ ) = ∆(e γϕ ) = − 1 2γ(γ − Q) . (2.17)<br />
Requiring <strong>th</strong>at ∆(e γϕ ) = ∆(e γϕ ) = 1 determines <strong>th</strong>at<br />
Q = 2/γ + γ . (2.18)<br />
Using (2.16) and solving for γ <strong>th</strong>en gives 5<br />
γ = √ 1 (√ √ ) Q<br />
25 − D − 1 − D = 12 2 − 1 √<br />
Q<br />
2<br />
2 − 8 . (2.19)<br />
For spacetime embedding dimension d ≤ 1, we find from (2.16) and (2.19) <strong>th</strong>at Q and<br />
γ are bo<strong>th</strong> real (wi<strong>th</strong> γ ≤ Q/2). The D ≤ 1 domain is <strong>th</strong>us where <strong>th</strong>e Liouville <strong>th</strong>eory is<br />
well-defined and most easily interpreted. For D ≥ 25, on <strong>th</strong>e o<strong>th</strong>er hand, bo<strong>th</strong> γ and Q are<br />
imaginary. To define a real physical metric g = e γϕ ĝ, we need to Wick rotate ϕ → −iϕ.<br />
(This changes <strong>th</strong>e sign of <strong>th</strong>e kinetic term for ϕ. Precisely at D = 25 we can interpret<br />
X 0 = −iϕ as a free time coordinate. In o<strong>th</strong>er words, for a string naively embedded in 25<br />
flat euclidean dimensions, <strong>th</strong>e Liouville mode turns out to provide automatically a single<br />
timelike dimension, dynamically realizing a string embedded in 26 dimensional Minkowski<br />
spacetime. In general for D ≥ 25, we must have c Liouville<br />
< 1 and <strong>th</strong>e kinetic term of <strong>th</strong>e<br />
Liouville field changes sign. The conformal mode of <strong>th</strong>e metric in Euclidean space is a<br />
“wrong sign” scalar field analogous to <strong>th</strong>e conformal mode in four-dimensional Euclidean<br />
4 Recall <strong>th</strong>at ∆ is given by <strong>th</strong>e leading term in <strong>th</strong>e operator product expansion T (z) e γϕ(w) ∼<br />
∆ e γϕ /(z − w) 2 +. . . . Recall also <strong>th</strong>at for a conventional energy-momentum tensor T = − 1 2 ∂ϕ∂ϕ,<br />
<strong>th</strong>e conformal weight of e ipϕ is ∆ = ∆ = p 2 /2.<br />
5 One me<strong>th</strong>od for choosing <strong>th</strong>e root for γ is to make contact wi<strong>th</strong> <strong>th</strong>e classical limit of <strong>th</strong>e<br />
Liouville action. Note <strong>th</strong>at <strong>th</strong>e effective coupling in (2.14) goes as (25 − D) −1 so <strong>th</strong>e classical limit<br />
is given by D → −∞. In <strong>th</strong>is limit <strong>th</strong>e above choice of root has <strong>th</strong>e classical γ → 0 behavior. We<br />
shall discuss <strong>th</strong>is issue fur<strong>th</strong>er in <strong>th</strong>e next chapter.<br />
19
quantum gravity [31], so it would be useful to make sense of <strong>th</strong>is case (if possible). Finally,<br />
in <strong>th</strong>e regime 1 < D < 25, γ is complex, and Q is imaginary. As we shall see later in lurid<br />
detail, <strong>th</strong>is problem is equivalent to <strong>th</strong>e cosmological constant becoming a macroscopic<br />
state operator. Sadly, it is not yet known how to make sense of <strong>th</strong>e Liouville approach for<br />
<strong>th</strong>e regime of most physical interest.<br />
A useful critical exponent <strong>th</strong>at can be calculated in <strong>th</strong>is formalism is <strong>th</strong>e string susceptibility<br />
Γ str . We write <strong>th</strong>e partition function for fixed area A as<br />
∫<br />
Z(A) = Dϕ DX e −S ∫<br />
δ(<br />
d 2 ξ √ )<br />
ĝ e γϕ − A , (2.20)<br />
where for convenience we now group <strong>th</strong>e ghost determinant and integration over moduli<br />
into DX. We define a string susceptibility Γ str by<br />
and determine Γ str by a simple scaling argument.<br />
Z(A) ∼ A (Γ str−2)χ/2−1 , A → ∞ , (2.21)<br />
(Note <strong>th</strong>at for genus zero, we have<br />
Z(A) ∼ A Γ str−3 .) Under <strong>th</strong>e shift ϕ → ϕ + ρ/γ for ρ constant, <strong>th</strong>e measure in (2.20) does<br />
not change. The change in <strong>th</strong>e action (2.15) comes from <strong>th</strong>e term<br />
∫<br />
Q<br />
d 2 ξ √ ĝ<br />
8π<br />
ˆR ϕ → Q ∫<br />
d 2 ξ √ ĝ<br />
8π<br />
ˆR ϕ + Q ∫<br />
ρ<br />
d 2 ξ √ ĝ<br />
8π γ<br />
ˆR .<br />
∫<br />
Substituting in (2.20) and using <strong>th</strong>e Gauss-Bonnet formula 1<br />
4π d 2 ξ √ ĝ ˆR = χ toge<strong>th</strong>er<br />
wi<strong>th</strong> <strong>th</strong>e identity δ(λx) = δ(x)/|λ| gives Z(A) = e −Qρχ/2γ−ρ Z(e −ρ A). We may now<br />
choose e ρ = A, which results in<br />
Z(A) = A −Qχ/2γ−1 Z(1) = A (Γ str−2)χ/2−1 Z(1) ,<br />
and we confirm from (2.16) and (2.19) <strong>th</strong>at<br />
Γ str = 2 − Q γ = 1 12(<br />
D − 1 −<br />
√<br />
(D − 25)(D − 1)<br />
)<br />
. (2.22)<br />
In <strong>th</strong>e nomenclature of [21], so-called “minimal conformal field <strong>th</strong>eories” (<strong>th</strong>ose wi<strong>th</strong> a<br />
finite number of primary fields) are specified by a pair of relatively prime integers (p, q) and<br />
have central charge D = c p,q = 1 − 6(p − q) 2 /pq. The unitary discrete series, for example,<br />
is <strong>th</strong>e subset specified by (p, q) = (m + 1, m). After coupling to gravity, <strong>th</strong>e general (p, q)<br />
model has critical exponent Γ str = −2/(p + q − 1). Notice <strong>th</strong>at Γ str = −1/m for <strong>th</strong>e values<br />
D = 1 − 6/m(m + 1) ) of central charge in <strong>th</strong>e unitary discrete series. (In general, <strong>th</strong>e m <strong>th</strong><br />
20
order multicritical point of <strong>th</strong>e one-matrix model will turn out to describe <strong>th</strong>e (2m − 1, 2)<br />
model (in general non-unitary) coupled to gravity, so its critical exponent Γ str = −1/m<br />
happens to coincide wi<strong>th</strong> <strong>th</strong>at of <strong>th</strong>e m <strong>th</strong> member of <strong>th</strong>e unitary discrete series coupled<br />
to gravity.) Notice also <strong>th</strong>at (2.22) ceases to be sensible for D > 1, an indication of a<br />
“barrier” at D = 1 <strong>th</strong>at has already appeared and will reappear in various guises in what<br />
follows. 6<br />
Dressed operators / dimensions of fields<br />
Now we wish to determine <strong>th</strong>e effective dimension of fields after coupling to gravity.<br />
Suppose <strong>th</strong>at Φ 0 is some spinless primary field in a conformal field <strong>th</strong>eory wi<strong>th</strong> conformal<br />
weight ∆ 0 = ∆(Φ 0 ) = ∆(Φ 0 ) before coupling to gravity. The gravitational “dressing”<br />
can be viewed as a form of wave function renormalization <strong>th</strong>at allows Φ 0 to couple to<br />
gravity. The dressed operator Φ = e αϕ Φ 0 is required to have dimension (1,1) so <strong>th</strong>at it<br />
can be integrated over <strong>th</strong>e surface Σ wi<strong>th</strong>out breaking conformal invariance. (This is <strong>th</strong>e<br />
same argument used prior to (2.19) to determine γ). Recalling <strong>th</strong>e formula (2.17) for <strong>th</strong>e<br />
conformal weight of e αϕ , we find <strong>th</strong>at α is determined by <strong>th</strong>e condition<br />
∆ 0 − 1 α(α − Q) = 1 . (2.23)<br />
2<br />
We may now associate a critical exponent ∆ to <strong>th</strong>e behavior of <strong>th</strong>e one-point function<br />
of Φ at fixed area A,<br />
F Φ (A) ≡ 1 ∫<br />
Z(A)<br />
Dϕ DX e −S δ( ∫<br />
d 2 ξ √ ĝ e γϕ − A) ∫<br />
d 2 ξ √ ĝ e αϕ Φ 0 ∼ A 1−∆ . (2.24)<br />
This definition conforms to <strong>th</strong>e standard convention <strong>th</strong>at ∆ < 1 corresponds to a relevant<br />
operator, ∆ = 1 to a marginal operator, and ∆ > 1 to an irrelevant operator (and in<br />
particular <strong>th</strong>at relevant operators tend to dominate in <strong>th</strong>e infrared, i.e. large area, limit).<br />
6 The “barrier” occurs when coupling gravity to D = 1 matter in <strong>th</strong>e language of non-critical<br />
string <strong>th</strong>eory, or equivalently in <strong>th</strong>e case of d = 2 target space dimensions in <strong>th</strong>e language of<br />
critical string <strong>th</strong>eory. So-called non-critical strings (i.e. whose conformal anomaly is compensated<br />
by a Liouville mode) in D dimensions can always be reinterpreted as critical strings in d =<br />
D + 1 dimensions, where <strong>th</strong>e Liouville mode provides <strong>th</strong>e additional (interacting) dimension.<br />
(The converse, however, is not true since it is not always possible to gauge-fix a critical string and<br />
artificially disentangle <strong>th</strong>e Liouville mode (see e.g. [32]).)<br />
21
To determine ∆, we employ <strong>th</strong>e same scaling argument <strong>th</strong>at led to (2.22). We shift<br />
ϕ → ϕ + ρ/γ wi<strong>th</strong> e ρ = A on <strong>th</strong>e right hand side of (2.24), to find<br />
F Φ (A) = A−Qχ/2γ−1+α/γ<br />
A −Qχ/2γ−1 F Φ (1) = A α/γ F Φ (1) ,<br />
where <strong>th</strong>e additional factor of e ρα/γ = A α/γ comes from <strong>th</strong>e e αφ gravitational dressing of<br />
Φ 0 . The gravitational scaling dimension ∆ defined in (2.24) <strong>th</strong>us satisfies<br />
∆ = 1 − α/γ . (2.25)<br />
Solving (2.23) for α wi<strong>th</strong> <strong>th</strong>e same branch used in (2.19),<br />
α = 1 2 Q − √<br />
1<br />
4 Q2 − 2 + 2∆ 0 = 1 √<br />
12<br />
(√<br />
25 − D −<br />
√<br />
1 − D + 24∆0<br />
)<br />
(2.26)<br />
(for which α ≤ Q/2, and α → 0 as D → −∞). Finally we substitute <strong>th</strong>e above result for<br />
α and <strong>th</strong>e value (2.19) for γ into (2.25), and find 7<br />
∆ =<br />
√ 1 − D + 24∆0 − √ 1 − D<br />
√<br />
25 − D −<br />
√<br />
1 − D<br />
. (2.27)<br />
We can apply <strong>th</strong>ese results to <strong>th</strong>e (p, q) minimal models [21] mentioned after (2.22).<br />
These have a set of operators labelled by two integers r, s (satisfying 1 ≤ r ≤ q − 1, 1 ≤<br />
s ≤ p − 1) wi<strong>th</strong> bare conformal weights ∆ 0 = ( (pr − qs) 2 − (p − q) 2) /4pq (we take p > q).<br />
Coupled to gravity, <strong>th</strong>ese operators have dressed Liouville exponents<br />
1 − ∆ r,s = α r,s<br />
γ<br />
p + q − |pr − qs|<br />
=<br />
2q<br />
1 ≤ r ≤ q − 1, 1 ≤ s ≤ p − 1 (2.28)<br />
(p − q)2<br />
(note also <strong>th</strong>at c = 1 − 6<br />
pq<br />
=⇒ γ =<br />
√<br />
2q<br />
p ,<br />
√ √<br />
2p 2q<br />
Q = q + p<br />
)<br />
,<br />
in agreement wi<strong>th</strong> <strong>th</strong>e weights determined from <strong>th</strong>e (p, q) formalism (to be discussed in<br />
sections 7.4 and 7.7 ) for <strong>th</strong>e generalized KdV hierarchy (see e.g. [34,35]).<br />
7 We can also substitute α = γ(1 − ∆) from (2.25) into (2.23) and use − 1 γ(γ − Q) = 1 (from<br />
2<br />
before (2.19)) to rederive <strong>th</strong>e result ∆−∆ 0 = ∆(1−∆)γ 2 /2 for <strong>th</strong>e difference between <strong>th</strong>e “dressed<br />
weight” ∆ and <strong>th</strong>e bare weight ∆ 0 [33].<br />
22
3. Brief Review of <strong>th</strong>e Liouville Theory<br />
In <strong>th</strong>is chapter, we touch briefly on some of <strong>th</strong>e highlights of Liouville <strong>th</strong>eory from<br />
<strong>th</strong>e viewpoint advocated in [7,17,8,36]. For o<strong>th</strong>er points of view on <strong>th</strong>e Liouville <strong>th</strong>eory,<br />
see [37,38], and <strong>th</strong>e sequence of works [39]. The classical Liouville <strong>th</strong>eory was extensively<br />
studied at <strong>th</strong>e end of <strong>th</strong>e nineteen<strong>th</strong> century in connection wi<strong>th</strong> <strong>th</strong>e uniformization problem<br />
for Riemann surfaces. We will sketch some of <strong>th</strong>is in sections 3.2, 3.10 .<br />
3.1. Classical Liouville Theory<br />
We choose some reference metric ĝ on a surface Σ. The Liouville <strong>th</strong>eory is <strong>th</strong>e <strong>th</strong>eory<br />
of metrics g on Σ, and <strong>th</strong>e Liouville field φ is defined by<br />
The action is<br />
∫<br />
S Liouville =<br />
g = e γφ ĝ . (3.1)<br />
d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇φ) 2 + Q )<br />
8π φR(ĝ) + µ ∫<br />
8πγ 2<br />
d 2 z √ ĝ e γφ , (3.2)<br />
very similar to <strong>th</strong>e background–charge <strong>th</strong>eory (1.14) wi<strong>th</strong> a pure imaginary background<br />
charge Q = 2iα 0 . The interaction given by µ (<strong>th</strong>e “cosmological constant” term), while<br />
soft, will be seen to have profound effects on <strong>th</strong>e <strong>th</strong>eory. For <strong>th</strong>e particular choice<br />
Q = 2/γ , (3.3)<br />
<strong>th</strong>e action (3.2) defines a classical conformal field <strong>th</strong>eory, invariant under <strong>th</strong>e Weyl transformations<br />
ĝ → e 2ρ ĝ γφ → γφ − 2ρ . (3.4)<br />
Remark: The linear shift in φ under a conformal transformation shows <strong>th</strong>at φ can be<br />
interpreted as a Goldstone boson for broken Weyl invariance (broken by <strong>th</strong>e choice of ĝ).<br />
Exercise. Classical Liouville <strong>th</strong>eory<br />
a) Using <strong>th</strong>e transformation properties of <strong>th</strong>e Ricci scalar in two dimensions,<br />
R[e 2ρ ĝ] = e −2ρ( R[ĝ] − ˆ∇ 2 2ρ ) , (3.5)<br />
compute <strong>th</strong>e change in <strong>th</strong>e action (3.2) under (3.4), and show <strong>th</strong>at for Q = 2/γ <strong>th</strong>e<br />
change is independent of φ (so doesn’t affect <strong>th</strong>e classical equations of motion).<br />
b) Show <strong>th</strong>at <strong>th</strong>e classical equations of motion for (3.2) may be expressed as<br />
i.e. <strong>th</strong>ey describe a surface wi<strong>th</strong> constant negative curvature.<br />
R[g] = − 1 2 µ , (3.6)<br />
(We take µ positive.)<br />
Using again (3.5) (in <strong>th</strong>e form R[g] = e −γφ( R[ĝ] − ˆ∇ 2 γφ ) , wi<strong>th</strong> φ as in (3.1)), note <strong>th</strong>at<br />
(3.6) is explicitly invariant under (3.4).<br />
23
The stress-energy tensor following from (3.2), T µν<br />
Feigin–Fuks form<br />
T z¯z = 0<br />
T zz = − 1 2 (∂φ)2 + 1 2 Q∂2 φ<br />
= −2π δS<br />
δg µν , takes <strong>th</strong>e familiar<br />
(3.7)<br />
T¯z¯z = − 1 2 (∂φ)2 + 1 2 Q∂2 φ ,<br />
where we have used <strong>th</strong>e equations of motion in <strong>th</strong>e first line.<br />
Since φ is a component of a metric, its transformation law under conformal transformations<br />
z → w = f(z),<br />
φ → φ + 1 γ log ∣ ∣∣ dw<br />
dz<br />
∣ 2 , (3.8)<br />
is more complicated <strong>th</strong>an <strong>th</strong>at of an ordinary scalar field. In particular, <strong>th</strong>e U(1) current<br />
∂ z φ measuring <strong>th</strong>e Liouville momentum transforms as<br />
and <strong>th</strong>e stress tensor T zz transforms as<br />
∂ z φ → dw<br />
dz ∂ wφ + d 1<br />
∣ ∣∣<br />
dz γ log dw<br />
∣ , (3.9)<br />
dz<br />
T zz →<br />
( dw<br />
) 2Tww<br />
+ 1 S[w; z] . (3.10)<br />
dz γ2 The object S[w; z] is called <strong>th</strong>e “Schwartzian derivative” and has many equivalent definitions.<br />
8 Combining (3.7) and (3.9), we see <strong>th</strong>at<br />
S[w; z] = − 1 (<br />
∂ z log∣ dw<br />
) 2 ∣<br />
d2<br />
∣ ∣∣ + 2 dz dz log dw<br />
2 ∣<br />
dz<br />
= T zz<br />
(φ = log∣ dw<br />
)<br />
∣<br />
(3.11)<br />
dz<br />
= w′′′<br />
w ′ − 3 ( w<br />
′′ ) 2<br />
.<br />
2 w ′<br />
The unusual transformation laws (3.9) and (3.10) have counterparts in <strong>th</strong>e quantum <strong>th</strong>eory,<br />
where <strong>th</strong>ey result in shifted formulae for conformal charges and weights in Liouville <strong>th</strong>eory.<br />
8 It may be considered, for example, as <strong>th</strong>e integrated version of <strong>th</strong>e conformal anomaly, or it<br />
may be defined in terms of “projective connections.” For a discussion of <strong>th</strong>e latter concept see<br />
[40].<br />
24
3.2. Classical Uniformization<br />
The central <strong>th</strong>eorem in classical Liouville <strong>th</strong>eory is <strong>th</strong>e uniformization <strong>th</strong>eorem <strong>th</strong>at<br />
characterizes Riemann surfaces.<br />
Uniformization Theorem: Every Riemann surface Σ is conformally equivalent to<br />
1) CP 1 , <strong>th</strong>e Riemann sphere, or<br />
2) H, <strong>th</strong>e Poincaré upper half plane, or<br />
3) A quotient of H by a discrete subgroup Γ ⊂ SL(2, IR) acting as Möbius transformations.<br />
The first proofs of <strong>th</strong>e uniformization <strong>th</strong>eorem were based on <strong>th</strong>e existence of solutions<br />
to <strong>th</strong>e classical field equations (3.6); <strong>th</strong>e standard proofs use potential <strong>th</strong>eory and are<br />
nonconstructive (see [41]).<br />
The upper half plane supports <strong>th</strong>e standard solution of <strong>th</strong>e Liouville equation, namely<br />
<strong>th</strong>e Poincaré metric:<br />
ds 2 = e γφ |dz| 2 = 4 µ<br />
1<br />
(Im z) 2 |dz|2 , (3.12)<br />
a constant negative curvature solution to (3.6). This metric is invariant under <strong>th</strong>e Möbius<br />
transformations<br />
z → az + b , a, b, c, d ∈ IR, ad − bc = 1 (3.13)<br />
cz + d<br />
(i.e. <strong>th</strong>e group P SL(2, IR) = SL(2, IR)/ Z 2 ), and <strong>th</strong>us descends to a metric<br />
ds 2 = e γφ |dz| 2 = 4 µ<br />
∂A ∂B<br />
(<br />
A(z) − B(¯z) ) 2 |dz|2 (3.14)<br />
on <strong>th</strong>e Riemann surface X = H/Γ for some locally defined (anti-)holomorphic functions<br />
A(z) ( B(¯z) ) . In general, when we quotient H by <strong>th</strong>e action of a discrete group Γ to define<br />
a space X = H/Γ, <strong>th</strong>ere is a natural projection π : H → X and an “inverse” map<br />
known as <strong>th</strong>e “uniformizing map”.<br />
Exercise. Classical energy-momentum<br />
T zz = 0.<br />
f : X → H , (3.15)<br />
a) Evaluate <strong>th</strong>e energy-momentum tensor for <strong>th</strong>e Poincaré metric and show <strong>th</strong>at<br />
b) Let f : X → H be <strong>th</strong>e uniformizing map as in (3.15). Show <strong>th</strong>at <strong>th</strong>e solution<br />
(3.14) to <strong>th</strong>e field equations has energy-momentum<br />
wi<strong>th</strong> S[f; z] as in (3.11).<br />
T zz = 1 S[f; z] , (3.16)<br />
γ2 25
Note <strong>th</strong>at <strong>th</strong>e uniformizing map (3.15) is not well-defined. If we continue <strong>th</strong>e values of<br />
f from some coordinate patch around a nontrivial cycle, <strong>th</strong>en f will change by <strong>th</strong>e action<br />
of T ∈ Γ. The nature of <strong>th</strong>e surface near such a nontrivial curve depends on <strong>th</strong>e nature of<br />
<strong>th</strong>e conjugacy class of T , in turn classified by <strong>th</strong>e value of <strong>th</strong>e trace. There are <strong>th</strong>ree types<br />
of conjugacy classes in SL(2, IR):<br />
1) elliptic: |Tr T | < 2 (T conjugate to ( cos λ sin λ<br />
− sin λ cos λ)<br />
) ,<br />
2) parabolic: |Tr T | = 2 (T conjugate to ( 1 λ<br />
0 1)<br />
) , and<br />
3) hyperbolic: |Tr T | > 2 (T conjugate to ( cosh λ sinh λ<br />
sinh λ cosh λ)<br />
) .<br />
In cases 1,2), <strong>th</strong>e nontrivial curve surrounds a puncture on <strong>th</strong>e surface. In case 3), <strong>th</strong>e<br />
curve surrounds a handle. See [17] for fur<strong>th</strong>er discussion.<br />
Exercise.<br />
a) Show <strong>th</strong>at <strong>th</strong>e Schwartzian derivative is invariant under independent Möbius<br />
transformation of ei<strong>th</strong>er z or f.<br />
b) Show <strong>th</strong>at al<strong>th</strong>ough <strong>th</strong>e uniformizing map is not globally defined, <strong>th</strong>e energymomentum<br />
(3.16) is none<strong>th</strong>eless well-defined.<br />
3.3. Quantum Liouville Theory<br />
It is more subtle <strong>th</strong>at <strong>th</strong>e <strong>th</strong>eory (3.2) is also a quantum conformal field <strong>th</strong>eory.<br />
If µ were zero and φ were a free field, we would immediately conclude <strong>th</strong>at T defines<br />
a Virasoro algebra wi<strong>th</strong> central charge c = 1 + 3Q 2 , <strong>th</strong>at exponentials e αφ have conformal<br />
weight − 1 2α(α−Q), and <strong>th</strong>at <strong>th</strong>ese operators create states |α〉 on which we could construct<br />
Feigin–Fuks modules. 9 Since φ is not a free field, 10 we must be more careful.<br />
We proceed via canonical quantization, first passing from <strong>th</strong>e complex z-plane to<br />
cylindrical coordinates (t, σ) via z = e t+iσ , and expanding<br />
φ(σ, t) = φ 0 (t) + ∑ i<br />
(a n (t) e −inσ + b n (t) e inσ)<br />
n<br />
n≠0<br />
Π(σ, t) = p 0 (t) + ∑ 1<br />
)<br />
(a n (t) e −inσ + b n (t) e inσ<br />
4π<br />
n≠0<br />
.<br />
(3.17)<br />
9 A Feigin–Fuks module is a Fock space in which <strong>th</strong>e Virasoro algebra is represented by an<br />
energy-momentum tensor such as (3.7).<br />
10 We will see <strong>th</strong>at <strong>th</strong>e operator product of two exponentials is not given by <strong>th</strong>e free field<br />
expression.<br />
26
Here a † n = a −n , b † n = b −n , and we have <strong>th</strong>e equal-time canonical commutation relations<br />
[<br />
an (t), b m (t) ] = n δ n,m . (3.18)<br />
The energy-momentum tensor in canonical variables takes <strong>th</strong>e form (again using <strong>th</strong>e equations<br />
of motion)<br />
T +− = 0<br />
T ±± = 1 8 (4πΠ ± φ′ ) 2 − Q 4 (4πΠ ± φ′ ) ′ + µ<br />
8γ 2 eγφ + Q2<br />
8 . (3.19)<br />
The additive factor of Q 2 /8 in <strong>th</strong>e above arises from <strong>th</strong>e Schwartzian derivative in <strong>th</strong>e<br />
transformation properties (3.10) of T when mapping from <strong>th</strong>e plane to <strong>th</strong>e cylinder z →<br />
(t, σ).<br />
In [30], it was shown <strong>th</strong>at <strong>th</strong>e operators (3.19) satisfy a Virasoro algebra if<br />
Q = 2 γ + γ (3.20)<br />
(as we derived from ano<strong>th</strong>er point of view in (2.18)). Calculating <strong>th</strong>e [T, T ] commutator,<br />
one finds indeed a central charge<br />
c = 1 + 3Q 2 , (3.21)<br />
and calculating commutators wi<strong>th</strong> T shows <strong>th</strong>at exponentials e αφ have conformal weight<br />
∆(e αφ ) = − 1 2 (α − Q/2)2 + Q 2 /8 . (3.22)<br />
Note <strong>th</strong>at if we impose <strong>th</strong>e condition ∆(e αφ ) + ∆ 0 = 1, to dress an operator wi<strong>th</strong> bare<br />
weight ∆ 0 , we rederive <strong>th</strong>e KPZ equation (2.26).<br />
It is useful to have an intuitive understanding of eqns. (3.20–3.22).<br />
Note <strong>th</strong>at by<br />
rescaling φ → 1 γ<br />
φ in (3.2), we identify γ wi<strong>th</strong> <strong>th</strong>e coupling constant of <strong>th</strong>e <strong>th</strong>eory. The<br />
semiclassical <strong>th</strong>eory is <strong>th</strong>us defined by asymptotic expansion as γ → 0. We see <strong>th</strong>at<br />
(3.20) is <strong>th</strong>e quantum version of <strong>th</strong>e classical condition (3.3) for conformal invariance.<br />
Similarly, (3.21) consists of a classical part (∝ Q 2 ∝ 1/γ 2 ) already visible in <strong>th</strong>e classical<br />
transformation law (3.10), plus a quantum conformal anomaly (c = 1), familiar for a<br />
single scalar field. Finally, to understand (3.22) we note <strong>th</strong>at <strong>th</strong>e analog of (3.9) in <strong>th</strong>e full<br />
quantum <strong>th</strong>eory is<br />
∂ z φ → dw<br />
dz ∂ wφ + d Q<br />
∣ ∣∣<br />
dz 2 log dw<br />
∣ . (3.23)<br />
dz<br />
27
In particular, passing from <strong>th</strong>e plane to <strong>th</strong>e cylinder via <strong>th</strong>e conformal transformation<br />
w = log z we have ∂ z φ → (∂ w φ − Q/2)/z, so <strong>th</strong>e momentum “shifts” by Q/2. The vertex<br />
operators e αφ , inserted on <strong>th</strong>e z-plane, create states wi<strong>th</strong> Liouville momentum p φ<br />
given by<br />
ip φ = α − Q 2 . (3.24)<br />
(Note <strong>th</strong>at “states” refer to quantization on <strong>th</strong>e cylinder, and “Liouville momentum” refers<br />
to <strong>th</strong>e zero mode p 0 of Π in (3.17).) We see <strong>th</strong>at <strong>th</strong>e first term in (3.22) is simply 1 2 p2 φ ,<br />
and <strong>th</strong>e second term is <strong>th</strong>e shift in <strong>th</strong>e energy (relative to <strong>th</strong>e Gaussian case) due to <strong>th</strong>e<br />
“extra” central charge.<br />
The above formulae are valid for cosmological constant µ ≥ 0. It may seem curious<br />
<strong>th</strong>at <strong>th</strong>e quantum formulae for µ > 0 are identical to <strong>th</strong>ose obtained for a free field, i.e.<br />
wi<strong>th</strong> cosmological constant µ = 0 in (3.2). (N.B.: a translation of φ cannot transform<br />
one case into <strong>th</strong>e o<strong>th</strong>er.) Heuristically, we can understand <strong>th</strong>is by noting <strong>th</strong>at in <strong>th</strong>e<br />
worldsheet ultraviolet, φ → −∞, <strong>th</strong>e interaction term disappears: Quantities such as c<br />
and ∆, determined by <strong>th</strong>e singular terms in operator product expansion, depend only on<br />
<strong>th</strong>e ultraviolet behavior of <strong>th</strong>e <strong>th</strong>eory.<br />
3.4. Spectrum of Liouville Theory<br />
We now proceed to study <strong>th</strong>e Hilbert space of <strong>th</strong>e <strong>th</strong>eory, all of whose subtleties lie<br />
in <strong>th</strong>e zero modes φ 0 (t), p 0 (t). We can understand <strong>th</strong>e physics of <strong>th</strong>ese degrees of freedom<br />
by studying <strong>th</strong>e action (3.2) for field configurations independent of σ, i.e., we study <strong>th</strong>e<br />
Liouville quantum mechanics<br />
∫<br />
S =<br />
dt ( ˙φ 2 + µ γ 2 eγφ ) . (3.25)<br />
The Hamiltonian H 0 = 1 2<br />
∮<br />
(T++ + T −− ), after substituting (3.19), takes <strong>th</strong>e form<br />
H 0 = − 1 2<br />
( ∂<br />
∂φ 0<br />
) 2<br />
+<br />
µ<br />
8γ 2 eγφ 0 +<br />
1<br />
8 Q2 . (3.26)<br />
28
|ψ|<br />
V<br />
ϕ<br />
Fig. 3: Particle wavefunctions in <strong>th</strong>e exponential potential.<br />
The spectrum of H 0 is easily understood. In <strong>th</strong>e worldsheet ultraviolet, φ 0 ∼ −∞ (i.e.<br />
at short physical distances), <strong>th</strong>e potential disappears so <strong>th</strong>at normalizable states behave<br />
like plane waves, ψ E<br />
∼ sin Eφ 0 , wi<strong>th</strong> energy 1 2 E2 + 1 8 Q2 . The exponential grow<strong>th</strong> of <strong>th</strong>e<br />
potential prevents <strong>th</strong>e “particle” at φ 0 from penetrating too far to <strong>th</strong>e right, and hence<br />
gives total reflections of any incoming wave. Because of <strong>th</strong>e total reflection property <strong>th</strong>ere<br />
is no distinction between states wi<strong>th</strong> +E and −E and we can <strong>th</strong>erefore take E > 0. The<br />
wavefunctions look as in fig. 3.<br />
The circumference of <strong>th</strong>e 1D “universe” in physical units is measured by <strong>th</strong>e quantity<br />
1<br />
2<br />
l = e<br />
γφ 0<br />
. (3.27)<br />
Using l, it is moreover possible to give an exact description of <strong>th</strong>e eigenstates of H 0 in<br />
(<br />
terms of Bessel functions. Changing variables to l, <strong>th</strong>e eigenvalue equation H 0 ψ E<br />
=<br />
1<br />
2 E2 + 1 8<br />
)ψ Q2 E<br />
becomes<br />
(−(l ∂ ) (<br />
∂l )2 + 4ˆµl 2 + Q2<br />
Q<br />
2 )<br />
ψ<br />
γ 2 E (l) =<br />
γ + Ê2<br />
ψ 2 2 E , (3.28)<br />
where ˆµ = µ/(4γ 4 ) and Ê = √ 8E/γ.<br />
We recognize (3.28) as <strong>th</strong>e Bessel differential<br />
equation. Imposing <strong>th</strong>e boundary condition <strong>th</strong>at ψ decays for large universes, l → ∞,<br />
gives <strong>th</strong>e eigenfunction<br />
ψ E = 1 π<br />
√Ê sinh πÊ KiÊ (2√ˆµl) , (3.29)<br />
where K is <strong>th</strong>e modified Bessel function. We have chosen a δ-function normalization for<br />
<strong>th</strong>e states,<br />
∫ ∞<br />
0<br />
dl<br />
l ψ E(l) ψ E<br />
′ (l) = δ(E − E ′ ) . (3.30)<br />
29
Exercise. The wall analogy<br />
Show <strong>th</strong>at <strong>th</strong>e solutions ψ E behave asymptotically for φ 0 → −∞ as sin ( 1<br />
γEφ 2 0 +<br />
α(E) ) , where<br />
α(E) = arg Γ(1 + iE) = 1 Γ(1 + iE)<br />
log<br />
2 Γ(1 − iE) .<br />
For intuitive purposes, it is useful to replace <strong>th</strong>e exponentially growing Liouville potential<br />
by an infinite hard wall. Where should <strong>th</strong>is wall be located for energies of order<br />
E?<br />
Now we proceed from <strong>th</strong>e zero mode structure of <strong>th</strong>e <strong>th</strong>eory to construct <strong>th</strong>e full<br />
field <strong>th</strong>eory. Combining <strong>th</strong>e above discussion on zero modes wi<strong>th</strong> canonical quantization,<br />
one expects [30] <strong>th</strong>e Hilbert space (as a Vir ⊕ Vir representation space) of <strong>th</strong>e Liouville<br />
conformal field <strong>th</strong>eory to take <strong>th</strong>e form<br />
∫ ∞<br />
H = ⊕<br />
0<br />
dE F ∆(E) ⊗ F ∆(E) . (3.31)<br />
Here F is a Feigin–Fuks module wi<strong>th</strong> weight ∆(E) = 1 2 E2 + 1 8 Q2 and central charge<br />
c = 1 + 3Q 2 . Our notation on <strong>th</strong>e r.h.s. of (3.31) is meant to indicate a direct integral of<br />
Hilbert spaces [42]. Each Feigin–Fuks module is generated by adding oscillator excitations<br />
to some primary state. As usual, <strong>th</strong>is structure may be understood heuristically since in<br />
<strong>th</strong>e worldsheet ultraviolet (φ → −∞) <strong>th</strong>e <strong>th</strong>eory becomes free.<br />
In a conventional conformal field <strong>th</strong>eory we are able to associate each state in <strong>th</strong>e<br />
Hilbert space to an operator, and <strong>th</strong>en determine <strong>th</strong>e operator algebra of <strong>th</strong>e <strong>th</strong>eory. How<br />
do we construct <strong>th</strong>e vertex operators <strong>th</strong>at create states in <strong>th</strong>e space (3.31)? According to<br />
(3.24), we might expect <strong>th</strong>e primary fields of Liouville momentum p φ<br />
= E to have quantum<br />
vertex operators<br />
V E (z, ¯z) = e αφ = e iEφ e 1 2 Qφ . (3.32)<br />
At <strong>th</strong>is point, however, we begin to encounter some of <strong>th</strong>e confusing subtleties of Liouville<br />
<strong>th</strong>eory cum quantum gravity, as first emphasized in [7,8]: <strong>th</strong>e operators (3.32) by no means<br />
encompass all of <strong>th</strong>e quantities of interest in <strong>th</strong>e <strong>th</strong>eory. For example, a natural quantity<br />
in quantum gravity is <strong>th</strong>e “volume of <strong>th</strong>e universe,”<br />
∫<br />
A = e γφ√ ĝ , (3.33)<br />
Σ<br />
given by integrating <strong>th</strong>e area operator e γφ . But comparing wi<strong>th</strong> (3.32), we see <strong>th</strong>at <strong>th</strong>is<br />
operator has imaginary momentum and cannot correspond to a normalizable state (in<br />
<strong>th</strong>e sense of (3.30)). To obtain some insight into <strong>th</strong>is puzzling observation, we return to<br />
examine <strong>th</strong>e semiclassical <strong>th</strong>eory.<br />
30
3.5. Semiclassical States<br />
The semiclassical approximation is an important source of intuition for understanding<br />
Liouville <strong>th</strong>eory. Classical Liouville <strong>th</strong>eory describes <strong>th</strong>e geometry of negatively curved<br />
surfaces (eq. (3.6)). Earlier, we identified γ wi<strong>th</strong> <strong>th</strong>e coupling constant of <strong>th</strong>e <strong>th</strong>eory (by<br />
rescaling φ → φ/γ), so <strong>th</strong>e semiclassical limit is defined by γ → 0 asymptotics. In <strong>th</strong>is<br />
limit, we expect <strong>th</strong>e above quantum states to correspond to specific constant curvature<br />
surfaces, and <strong>th</strong>is has been verified in detail [7,17]. Briefly, <strong>th</strong>e σ-independent metric<br />
e γφ(t) (dt 2 + dσ 2 ) = 4 µ<br />
ε 2<br />
sin 2 (εt) (dt2 + dσ 2 )<br />
nπ<br />
ε<br />
< t <<br />
(n + 1)π<br />
ε<br />
(3.34a)<br />
(where z = e t+iσ and ε is a real number), is a solution to <strong>th</strong>e Liouville equation (3.6) <strong>th</strong>at<br />
looks some<strong>th</strong>ing like<br />
hyperbolic:<br />
(3.34b)<br />
Exercise. Classical field energy<br />
Use <strong>th</strong>e energy momentum tensor (3.7) to show <strong>th</strong>at <strong>th</strong>e Liouville field configuration<br />
(3.34) has classical energy 1 2 (ε2 /γ 2 ) + 1 8 Q2 .<br />
Recalling <strong>th</strong>at <strong>th</strong>e quantum state ψ E of (3.28) has energy 1 2 E2 + 1 8 Q2 , <strong>th</strong>is classical<br />
field energy allows us to identify (3.34) as <strong>th</strong>e semiclassical picture of <strong>th</strong>e quantum state<br />
ψ E for E = ε/γ.<br />
As noted in sec. 3.2, <strong>th</strong>ere are <strong>th</strong>ree kinds of local behavior of a solution to <strong>th</strong>e<br />
classical Liouville equation, classified by <strong>th</strong>e monodromy properties of A, B in (3.14). In<br />
<strong>th</strong>e solution (3.34a) above, we have A = z iε = B, <strong>th</strong>us giving an example of a hyperbolic<br />
class. From <strong>th</strong>e semiclassical point of view, we are naturally led to ask what quantum<br />
states correspond to <strong>th</strong>e o<strong>th</strong>er two classes, namely <strong>th</strong>e elliptic and parabolic solutions.<br />
These are given respectively by<br />
ν 2<br />
e γφ (dt 2 + dσ 2 ) = 4 µ sinh 2 (νt) (dt2 + dσ 2 ) t < 0<br />
e γφ (dt 2 + dσ 2 ) = 4 1<br />
µ t 2 (dt2 + dσ 2 ) t < 0 .<br />
(3.35a)<br />
31
Here ν is a real number. An important feature of <strong>th</strong>e solutions (3.35a) is <strong>th</strong>eir equivalence<br />
for bo<strong>th</strong> ±ν. These solutions look some<strong>th</strong>ing like<br />
elliptic, parabolic:<br />
(3.35b)<br />
and have energy − 1 2 (ν2 /γ 2 ) + 1 8 Q2 . Quantum mechanically, according to (3.28, 3.29) <strong>th</strong>ey<br />
<strong>th</strong>erefore correspond to states wi<strong>th</strong> imaginary momentum E = iν/γ. The corresponding<br />
wavefunctions are of type K ν (l), ν real. These wavefunctions blow up (as l −|ν| ) at short<br />
distances l → 0 and consequently might appear unphysical. On <strong>th</strong>e contrary, as we saw at<br />
<strong>th</strong>e end of <strong>th</strong>e previous section, operators corresponding to imaginary momentum states,<br />
including for example <strong>th</strong>e volume of <strong>th</strong>e universe, appear quite naturally in 2D quantum<br />
gravity and play an important role.<br />
The distinction between normalizable and non-normalizable states in Liouville <strong>th</strong>eory,<br />
and <strong>th</strong>e necessity to include <strong>th</strong>e non-normalizable states in <strong>th</strong>e <strong>th</strong>eory, was first emphasized<br />
by Seiberg in [7]. Motivated by <strong>th</strong>e geometries illustrated above, <strong>th</strong>e normalizable states<br />
were labelled “macroscopic states,” and <strong>th</strong>e non-normalizable states were labelled “microscopic<br />
states”. Semiclassically, <strong>th</strong>e macroscopic states do not have a well-defined insertion<br />
point in <strong>th</strong>e intrinsic geometry of <strong>th</strong>e surface. The microscopic states, on <strong>th</strong>e o<strong>th</strong>er hand,<br />
correspond semiclassically to <strong>th</strong>e elliptic geometry pictured in (3.35b), and <strong>th</strong>us to local<br />
operators — <strong>th</strong>e operator insertion in <strong>th</strong>is case is localized at <strong>th</strong>e tip of <strong>th</strong>e “funnel.” We<br />
discuss <strong>th</strong>ese issues fur<strong>th</strong>er in sec. 4.3 below.<br />
Exercise. Curvature Sources<br />
z-plane:<br />
a) Use <strong>th</strong>e exponential map z = e t+iσ to transform <strong>th</strong>e solutions (3.35) to <strong>th</strong>e<br />
e γφ dz d¯z = 16<br />
µ<br />
ν 2 (z¯z) ν dz d¯z<br />
(1 − (z¯z) ν ) 2 z¯z<br />
b) Show <strong>th</strong>at <strong>th</strong>e field φ solves <strong>th</strong>e Liouville equation wi<strong>th</strong> source,<br />
. (3.36)<br />
1<br />
4π ∆φ − µ<br />
8πγ eγφ + 1 − ν<br />
γ δ(2) (z) = 0 , (3.37)<br />
i.e., such solutions have a source of curvature at z = 0.<br />
32
c) Show <strong>th</strong>at <strong>th</strong>e solution corresponds to <strong>th</strong>e choice of functions (in (3.14)):<br />
A(z) = i zν + 1<br />
z ν − 1<br />
B(¯z) = −i ¯z ν + 1<br />
¯z ν − 1 . (3.38)<br />
d) Show <strong>th</strong>at <strong>th</strong>e monodromy when z circles around <strong>th</strong>e puncture is <strong>th</strong>e real elliptic<br />
Möbius transformation<br />
cos πνA(z) − sin πν<br />
A(z) →<br />
sin πνA(z) + cos πν . (3.39)<br />
e) Show <strong>th</strong>at a straight line <strong>th</strong>rough z = 0 is conformally mapped into an angle<br />
πν, and hence we must have ν ≥ 0 on geometric grounds.<br />
f) Repeat <strong>th</strong>e above for <strong>th</strong>e hyperbolic and parabolic cases.<br />
3.6. Seiberg bound<br />
Classically, <strong>th</strong>e metrics (3.34, 3.35) are invariant under ν → −ν, E → −E. Quantum<br />
mechanically, <strong>th</strong>e wavefunctions K ν , K iE share <strong>th</strong>is invariance, due to <strong>th</strong>e total reflection<br />
property of <strong>th</strong>e Liouville “wall.” Turning on <strong>th</strong>e wall by setting µ > 0 effectively halves<br />
<strong>th</strong>e states <strong>th</strong>at exist in <strong>th</strong>e (µ = 0) free spectrum.<br />
In <strong>th</strong>e DDK/KPZ formalism described in chapt. 2, on <strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>e choice of<br />
root in <strong>th</strong>e KPZ formula (2.27) affects <strong>th</strong>e scaling properties of <strong>th</strong>e operator e αφ Φ 0 . Since<br />
<strong>th</strong>e Liouville interaction truncates <strong>th</strong>e spectrum by half, we must choose a root. In [7,8],<br />
it is argued <strong>th</strong>at only <strong>th</strong>ose operators wi<strong>th</strong><br />
α ≤ 1 2 Q (3.40)<br />
can exist. This choice of root has many distinguishing properties, some of which will be<br />
noted in later sections:<br />
1) This root gives a smoo<strong>th</strong> semiclassical limit in quantum gravity, as we saw in sec. 2.2.<br />
2) The area element is integrable only for sources satisfying (3.40). A related fact in terms<br />
of deficit angles has appeared in part (e) in <strong>th</strong>e exercise above (following (3.39)).<br />
3) As we will see in <strong>th</strong>e penultimate paragraph of sec. 4.3 , <strong>th</strong>e Wheeler–DeWitt wavefunction<br />
for a local operator in quantum gravity, related to <strong>th</strong>e vertex operator V (φ)<br />
by ψ(φ) = e − 1 2 Qφ V (φ), must be concentrated on l → 0, <strong>th</strong>at is, where φ → −∞.<br />
4) A closely related question is <strong>th</strong>e nature of gravitational dressing (see sec. 2.2). Only<br />
wi<strong>th</strong> <strong>th</strong>e choice of root (3.40) do gravitationally dressed relevant/irrelevant operators<br />
grow/decay in <strong>th</strong>e worldsheet infrared.<br />
33
5) As will be seen in sec. 5.4 below, <strong>th</strong>e bound (3.40) has <strong>th</strong>e following spacetime interpretation:<br />
When scattering in a left half-space, incomers must be rightmovers and<br />
outgoers must be leftmovers.<br />
In addition, <strong>th</strong>ere is circumstantial evidence <strong>th</strong>at (3.40) is correct:<br />
1) In <strong>th</strong>e matrix model, we will see <strong>th</strong>at only scaling operators wi<strong>th</strong> scaling corresponding<br />
to α ≤ Q/2 appear.<br />
2) In <strong>th</strong>e semiclassical calculations of [30], <strong>th</strong>ere are difficulties constructing correlation<br />
functions of “wrong branch” operators.<br />
3) In <strong>th</strong>e SL(2, IR) quantum group approach pursued by Gervais and o<strong>th</strong>ers, inverse<br />
powers of <strong>th</strong>e metric e −jγφ , 2j ∈ Z + , are easily constructed, while <strong>th</strong>e positive powers<br />
have <strong>th</strong>us far eluded construction.<br />
The above reasoning is qualitative. While <strong>th</strong>e Seiberg bound (3.40) is undoubtedly correct,<br />
a precise ma<strong>th</strong>ematical understanding of <strong>th</strong>e statement would be useful.<br />
<br />
A common confusion<br />
There are two distinct novelties in <strong>th</strong>e description of <strong>th</strong>e Hilbert space of Liouville<br />
<strong>th</strong>eory: (1) Vertex operators wi<strong>th</strong> α > 1 2Q do not exist. (2) States wi<strong>th</strong> real momentum<br />
p φ<br />
= E, which formally correspond to vertex operators wi<strong>th</strong> α = 1 2Q + iE, do not have a<br />
correspondence wi<strong>th</strong> local operators. These two distinct points are often confused in <strong>th</strong>e<br />
literature. There is no (obvious) connection between <strong>th</strong>e Seiberg bound, which specifies<br />
<strong>th</strong>e operators <strong>th</strong>at exist, and <strong>th</strong>e difficulty of localizing operators <strong>th</strong>at correspond to states<br />
|E〉 of energy 1 2 E2 + 1 8 Q2 .<br />
<br />
An unresolved confusion<br />
There is some confusion in <strong>th</strong>e literature as to whe<strong>th</strong>er <strong>th</strong>e states |E〉 have a correspondence<br />
wi<strong>th</strong> vertex operators V E = e iEφ e 1 2 Qφ . Al<strong>th</strong>ough <strong>th</strong>e semiclassical pictures of<br />
states (e.g. (3.34b)) makes any correspondence wi<strong>th</strong> local operators seem unlikely [7], we<br />
shall find <strong>th</strong>ese operators necessary to formulate our scattering <strong>th</strong>eory in two Minkowskian<br />
dimensions. This problem is closely linked to <strong>th</strong>e problem of time in string <strong>th</strong>eory.<br />
Exercise. Seiberg bound and semiclassical limits<br />
Consider <strong>th</strong>e minimal conformal field <strong>th</strong>eories labelled by (2, 2m − 1) in <strong>th</strong>e BPZ<br />
classification, as mentioned after (2.22). Show <strong>th</strong>at γ = 2/ √ 2m − 1 so <strong>th</strong>at as m → ∞<br />
<strong>th</strong>e semiclassical approximation becomes valid. Note <strong>th</strong>at <strong>th</strong>e root chosen in sec. 2.2 on<br />
<strong>th</strong>e basis of <strong>th</strong>e semiclassical limit coincides wi<strong>th</strong> <strong>th</strong>at dictated by <strong>th</strong>e bound (3.40).<br />
34
3.7. Semiclassical Amplitudes<br />
In <strong>th</strong>e semiclassical approximation, we evaluate <strong>th</strong>e amplitudes<br />
〈∏ 〉 ∫<br />
e α iφ(z i )<br />
≡ [dφ] e ∏ −S Liouville[φ]<br />
e α iφ<br />
i<br />
i<br />
via <strong>th</strong>e saddle point approximation by first solving <strong>th</strong>e classical equation<br />
1<br />
4π ∆φ − µ<br />
8πγ eγφ + ∑ i<br />
(3.41)<br />
α i δ (2) (z − z i ) = 0 . (3.42)<br />
Integrating (3.42) over <strong>th</strong>e surface Σ we find, since µ > 0, <strong>th</strong>at a necessary condition for<br />
<strong>th</strong>e existence of a solution is<br />
Exercise.<br />
1<br />
( 1<br />
γ γ (2 − 2h) − ∑ )<br />
α i < 0 . (3.43)<br />
Derive (3.43) using <strong>th</strong>e Gauss-Bonnet <strong>th</strong>eorem, ∫ 1<br />
4π R√ g = χ = 2 − 2h.<br />
The particular combination<br />
s ≡ Q 2γ χ − ∑ α i<br />
γ , (3.44)<br />
known as <strong>th</strong>e KPZ exponent [33], plays an important role in <strong>th</strong>e <strong>th</strong>eory. Equation (3.43)<br />
says <strong>th</strong>at <strong>th</strong>e semiclassical KPZ exponent (Q → 2/γ) is negative.<br />
When s < 0, <strong>th</strong>ere is indeed a solution to (3.42), and we can expand around it to<br />
evaluate (3.41). Near z = z i , it follows from (3.42) <strong>th</strong>at<br />
φ cl ∼ −α i log |z − z i | 2 (3.45)<br />
for α i < 1/γ. For α = 1/γ we have instead<br />
φ cl ∼ − 1 (<br />
log |z − z i | 2 1<br />
)<br />
+ 2 log log . (3.46)<br />
γ<br />
|z − z i |<br />
In ei<strong>th</strong>er case, to write <strong>th</strong>e semiclassical amplitudes we must excise disks B(r, z i ) around<br />
z i of radius r (in <strong>th</strong>e ĝ metric), and define <strong>th</strong>e regularized action 11<br />
S[φ cl ] ≡ lim<br />
L −<br />
r→0<br />
(∫Σ−∪ ∑ i B(r,z i ) i<br />
)<br />
(∆ i + ∆ i ) log r<br />
(3.47)<br />
[7].<br />
11 The need for <strong>th</strong>ese subtractions reflects <strong>th</strong>e need for renormalization of <strong>th</strong>e vertex operators<br />
35
where ∆ i is <strong>th</strong>e conformal weight of e α iφ . In terms of S, <strong>th</strong>e leading γ → 0 asymptotics of<br />
<strong>th</strong>e correlator are given by<br />
〈∏<br />
e<br />
α i φ(z i )〉<br />
s.c.<br />
∼ e −S[φ cl ] . (3.48)<br />
There are also cases of interest wi<strong>th</strong> s ≥ 0, notably, for genus zero correlators of vertex<br />
operators wi<strong>th</strong> small total “Liouville charge” ∑ α i . In <strong>th</strong>ese cases, as described in [7], we<br />
can still perform semiclassical calculations by fixing <strong>th</strong>e total area of <strong>th</strong>e surface: we insert<br />
δ( ∫ e γφ − A) into <strong>th</strong>e pa<strong>th</strong> integral to ensure <strong>th</strong>at (3.42) has a solution. We obtain <strong>th</strong>e<br />
A dependence of fixed area correlators by a scaling argument (similar to <strong>th</strong>at used before<br />
(2.22) and (2.25)): shifting φ → φ + 1 γ<br />
log A gives<br />
〈∏ 〉 〈∏ 〉<br />
e α iφ(z i )<br />
= A −1−s e<br />
α i φ(z i )<br />
i<br />
A<br />
A=1<br />
, (3.49)<br />
wi<strong>th</strong> s as in (3.44). To Laplace transform to fixed cosmological constant, we integrate<br />
∫ ∞<br />
0<br />
dA<br />
A A−s e −µA = µ s Γ(−s) . (3.50)<br />
The UV divergence as A → 0 for s ≥ 0 (reflecting <strong>th</strong>e absence of a classical solution<br />
wi<strong>th</strong>out <strong>th</strong>e area constraint) plays an important role in speculations on <strong>th</strong>e relation of free<br />
field <strong>th</strong>eory to <strong>th</strong>e Liouville <strong>th</strong>eory described in sections 3.9, 14.2, 14.3 below.<br />
Using (3.49), <strong>th</strong>e problem of calculating correlators is reduced to <strong>th</strong>e case A = 1. The<br />
semiclassical formulae for genus zero correlators are obtained by averaging over <strong>th</strong>e space<br />
of classical solutions. These solutions are obtained for s ≥ 0 by applying complex Möbius<br />
transformations from <strong>th</strong>e standard round-sphere metric<br />
ds 2 = e γ ¯φ |dz| 2 = 16 µ<br />
|dz| 2<br />
(<br />
1 + |z|<br />
2 ) 2<br />
(3.51)<br />
(which, unlike (3.6), has positive curvature R = + 1 2<br />
µ). The result is<br />
〈∏ 〉 ∫<br />
s.c.<br />
e<br />
α i φ(z i )<br />
= d 2 a d 2 b d 2 c d 2 d δ (2) (ad − bc − 1) ∏ e α i ¯φ(z i )<br />
A=1<br />
i<br />
= ∏ ( 16<br />
) ∫ αi /γ ∞ ∫<br />
dλ λ<br />
µ<br />
i<br />
0 C d2 w ∏ (3.52)<br />
1<br />
(<br />
i |λzi + w| 2 + λ −2) , 2α i /γ<br />
where in <strong>th</strong>e second line we have parametrized SL(2,C) elements by a unitary matrix times<br />
an upper triangular matrix, and we have dropped <strong>th</strong>e volume of SU(2). See [7] for more<br />
details.<br />
36
Semiclassical Seiberg Bound [7,8]<br />
The semiclassical approach provides a key insight into <strong>th</strong>e Seiberg bound (3.40). Consider<br />
<strong>th</strong>e classical equation (3.42) in <strong>th</strong>e neighborhood of a vertex operator insertion. If<br />
we neglect <strong>th</strong>e cosmological constant term, <strong>th</strong>e solution must behave as in (3.45). To check<br />
if <strong>th</strong>is is self-consistent, we insert (3.45) back into (3.42) and note <strong>th</strong>at <strong>th</strong>e neglected term<br />
behaves as<br />
e γφ ∼<br />
1<br />
|z − z i | 2α iγ . (3.53)<br />
If α i γ > 1, <strong>th</strong>e cosmological constant operator is not integrable at z = 0 and we expect<br />
trouble.<br />
Indeed, <strong>th</strong>e careful considerations leading to <strong>th</strong>e classification of solutions in<br />
sec. 3.2 (following eq. (3.16)) show <strong>th</strong>at <strong>th</strong>ere is no solution for α i > 1/γ ∼ Q/2. The<br />
essential point is <strong>th</strong>at too much curvature cannot be localized at a single point.<br />
Here are two examples of semiclassical correlators:<br />
Example 1: Consider <strong>th</strong>e <strong>th</strong>ree-point function on <strong>th</strong>e sphere,<br />
〈<br />
e<br />
θ 1 φ/γ (z 1 , ¯z 1 ) e θ 2φ/γ (z 2 , ¯z 2 ) e θ 3φ/γ (z 3 , ¯z 3 ) 〉 , (3.54)<br />
where θ i < 1 are considered to be O(1) as γ → 0 and ∑ i θ i > 2, so s < 0. The classical<br />
solution is known in <strong>th</strong>is case and is Möbius invariant. It follows immediately from (3.47)<br />
and <strong>th</strong>e transformation properties of circles under Möbius transformations <strong>th</strong>at<br />
〈<br />
e<br />
θ 1 φ/γ (z 1 , ¯z 1 ) e θ2φ/γ (z 2 , ¯z 2 ) e θ3φ/γ (z 3 , ¯z 3 ) 〉 C[θ i ]<br />
∼ ∣<br />
∣z ∆ 123<br />
12<br />
z ∆ 132<br />
13<br />
z ∆ 231<br />
23<br />
∣ 2 , (3.55)<br />
where ∆ 123 = ∆ 1 + ∆ 2 − ∆ 3 , etc. The coefficient function C is generically nonzero. Sadly,<br />
<strong>th</strong>is example cannot be extended to higher point functions because <strong>th</strong>e classical solutions<br />
to Liouville <strong>th</strong>eory are not known in explicit form, except in special cases which have <strong>th</strong>e<br />
punctures symmetrically located [43].<br />
Example 2: Consider now <strong>th</strong>e <strong>th</strong>ree-point function on <strong>th</strong>e sphere, but wi<strong>th</strong> s ≥ 0 so we<br />
must fix <strong>th</strong>e area. Using <strong>th</strong>e Möbius invariance of (3.52) gives<br />
C[α i ] =<br />
〈<br />
e<br />
α 1 φ (z 1 , ¯z 1 ) e α2φ (z 2 , ¯z 2 ) e α3φ (z 3 , ¯z 3 ) 〉 s.c.<br />
∼ C[α i ]<br />
A=1<br />
|z ∆ 123<br />
∫ ∞<br />
∫<br />
dλ<br />
0 λ λ4(α 1+α 2 −α 3 )/γ<br />
1<br />
C d2 w (<br />
|w| 2 + 1 ) 2α 1 /γ<br />
= π 4<br />
Γ(j 1 − j 2 − j 3 ) Γ(j 2 − j 3 − j 1 ) Γ(j 3 − j 1 − j 2 )<br />
Γ(−2j 1 ) Γ(−2j 2 ) Γ(−2j 3 )<br />
12<br />
z ∆ 132<br />
13<br />
z ∆ 231<br />
23<br />
| 2<br />
1<br />
(<br />
|w + λ 2 | 2 + 1 ) 2α 2 /γ<br />
Γ(−j 1 − j 2 − j 3 − 1) ,<br />
(3.56)<br />
37
where j k ≡ −α k /γ. Strictly speaking, <strong>th</strong>e integrals above converge only for ranges of j k<br />
for which <strong>th</strong>e arguments of all <strong>th</strong>e Γ–functions in (3.56) are positive. As in ordinary string<br />
<strong>th</strong>eory, we define <strong>th</strong>e amplitude at o<strong>th</strong>er values of j by analytic continuation.<br />
Remarks:<br />
1) The formula (3.52) has an interesting group-<strong>th</strong>eoretical meaning. Defining j ≡ −α/γ,<br />
<strong>th</strong>e field e αφ transforms under SL(2,C) in <strong>th</strong>e (j, j) representation of SL(2,C), hence<br />
<strong>th</strong>e suggestive notation. Group <strong>th</strong>eoretically, <strong>th</strong>e integral (3.55) computes <strong>th</strong>e overlap<br />
between a product of vectors in infinite-dimensional highest weight representations<br />
and <strong>th</strong>e trivial representation.<br />
2) Analytically continuing j 3 to pure imaginary values, and taking <strong>th</strong>e limit j 3 → 0, gives<br />
<strong>th</strong>e two-point function<br />
〈e −j 1φ (z 1 , ¯z 1 ) e −j 2φ (z 2 , ¯z 2 )〉 = − iπ2<br />
4<br />
1<br />
2j 1 + 1 δ(j 1 − j 2 )<br />
1<br />
|z 12 | 4∆ 1 , (3.57)<br />
obtained in [7] by analytic continuation of <strong>th</strong>e integration over <strong>th</strong>e dilation subgroup<br />
IR ∗ + ⊂ SL(2,C). For a fur<strong>th</strong>er discussion of <strong>th</strong>e subtleties of one- and two-point<br />
functions, and <strong>th</strong>eir relations to <strong>th</strong>e regularized volumes of IR ∗ + and SL(2,C), see<br />
[7]. 12<br />
3) Note <strong>th</strong>at fields wi<strong>th</strong> j ∈ 1 2 Z + generically decouple. If all <strong>th</strong>ree fields satisfy j ∈ 1 2 Z + ,<br />
<strong>th</strong>en we recover <strong>th</strong>e SU(2) fusion rules.<br />
4) The two above examples make it clear <strong>th</strong>at <strong>th</strong>e Liouville correlators are entirely different<br />
from <strong>th</strong>e correlators of a Coulomb gas wi<strong>th</strong> a charge at infinity. In particular,<br />
<strong>th</strong>ere is no Liouville charge conservation.<br />
5) It is very unusual to have short distance singularities in <strong>th</strong>e correlators of <strong>th</strong>e classical<br />
<strong>th</strong>eory as we do in (3.56) and (3.57). This is due to <strong>th</strong>e average over <strong>th</strong>e noncompact<br />
group SL(2,C), so we see again <strong>th</strong>at <strong>th</strong>e geometrical origin of <strong>th</strong>e Liouville field<br />
distinguishes it from an ordinary scalar field. Note <strong>th</strong>at for <strong>th</strong>e n ≥ 4 -point functions,<br />
<strong>th</strong>e operator products are typically smoo<strong>th</strong> in <strong>th</strong>e semiclassical correlator.<br />
12 That Liouville two-point functions are diagonal in Liouville charges might have important<br />
implications for <strong>th</strong>e implementation of string field <strong>th</strong>eory identities [44].<br />
38
3.8. Operator Products in Liouville Theory<br />
The existence of <strong>th</strong>e Seiberg bound (3.40) dictates <strong>th</strong>at <strong>th</strong>e operator product expansion<br />
in Liouville <strong>th</strong>eory will be ra<strong>th</strong>er different from <strong>th</strong>at in free field <strong>th</strong>eory. Indeed it is not<br />
clear <strong>th</strong>at <strong>th</strong>e notion of <strong>th</strong>e operator product expansion is <strong>th</strong>e correct language to use when<br />
discussing Liouville correlators. Already in <strong>th</strong>e semiclassical <strong>th</strong>eory, we shall see <strong>th</strong>at <strong>th</strong>e<br />
putative short distance behavior of correlators depends on global properties of <strong>th</strong>e surface<br />
and <strong>th</strong>e “operator product expansion” appears to be nonlocal [7,8].<br />
Fig. 4: Left: Insertion of operators e αφ and e βφ in a surface wi<strong>th</strong> boundary. Right:<br />
<strong>th</strong>e same operation on a higher genus surface.<br />
Consider <strong>th</strong>e state arising from <strong>th</strong>e insertion of two vertex operators e αφ and e βφ on<br />
a surface wi<strong>th</strong> boundary C as in <strong>th</strong>e l.h.s. diagram of fig. 4, or, more generally, on a higher<br />
genus surface as in <strong>th</strong>e r.h.s. diagram of fig. 4. In <strong>th</strong>e semiclassical approximation, <strong>th</strong>e<br />
state created by <strong>th</strong>e surface on <strong>th</strong>e boundary C has a wavefunction <strong>th</strong>at depends on <strong>th</strong>e<br />
zero-mode φ 0 of φ as<br />
ψ(φ 0 ) ∼ e αφ 0 +βφ 0 − 1 2 Qχφ 0 , (3.58)<br />
where χ = 1 − 2h is <strong>th</strong>e Euler character for <strong>th</strong>e surface (wi<strong>th</strong> a single boundary).<br />
α + β < 1 2 Qχ, <strong>th</strong>en <strong>th</strong>e wavefunction (3.58) is a real exponential diverging at φ 0 → −∞<br />
(short distance). Then we may expect to replace <strong>th</strong>e holes in <strong>th</strong>e diagrams in fig. 4 each<br />
by a sum of local (“microscopic”) operators, as in ordinary conformal field <strong>th</strong>eory. Note<br />
<strong>th</strong>at since <strong>th</strong>e <strong>th</strong>ree-point functions are generically nonzero, we would naively expect a<br />
disastrous sum over operators wi<strong>th</strong> conformal dimensions unbounded from below.<br />
If α + β > 1 2Qχ, on <strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>en <strong>th</strong>e state is normalizable and we certainly<br />
cannot expand it in an operator product expansion of local operators. Instead <strong>th</strong>e state<br />
must be expanded in <strong>th</strong>e normalizable macroscopic state operators. Thus, by sewing, <strong>th</strong>e<br />
surface amplitude must have <strong>th</strong>e form<br />
〈<br />
e αφ(z,¯z) e βφ(z,¯z) · · ·〉 ∼<br />
∫ ∞<br />
0<br />
dE c αβE |z − w| 2(E2 /2+Q 2 /8−∆ α −∆ β ) 〈E|Σ〉 , (3.59)<br />
39<br />
If
where |Σ〉 is a state created by <strong>th</strong>e rest of <strong>th</strong>e surface (as is standard in discussions of<br />
<strong>th</strong>e “operator formalism” [21–23]). If we interpret <strong>th</strong>e integral in (3.59) as an OPE over<br />
macroscopic vertex operators V E , <strong>th</strong>en since we sum over operators wi<strong>th</strong> weights ∆ ≥<br />
Q 2 /8, we see <strong>th</strong>at <strong>th</strong>e OPE is much softer <strong>th</strong>an in ordinary CFT. This discussion can be<br />
generalized.<br />
The essential message is <strong>th</strong>at while we may insert microscopic operators on a surface we<br />
should only do so “externally.” We must factorize on macroscopic states. The factorization<br />
on macroscopic states also ameliorates <strong>th</strong>e disastrous sum noted above for α + β < 1 2 Qχ.<br />
The essentially non-free field nature of <strong>th</strong>e operator product expansion in Liouville<br />
<strong>th</strong>eory accounts for some unusual properties of <strong>th</strong>e <strong>th</strong>eory.<br />
As one example, note <strong>th</strong>at<br />
since <strong>th</strong>e Liouville <strong>th</strong>eory is conformal for all µ > 0, <strong>th</strong>e cosmological constant e γφ is<br />
an exactly marginal operator.<br />
This appears to conflict wi<strong>th</strong> <strong>th</strong>e fact <strong>th</strong>at its n-point<br />
correlation functions are nonvanishing, since <strong>th</strong>e standard obstruction to exact marginality<br />
is <strong>th</strong>e existence of a “potential” for such couplings. However, <strong>th</strong>e standard discussion of<br />
<strong>th</strong>e obstruction to exact marginality does not apply because of <strong>th</strong>e strange nature of <strong>th</strong>e<br />
operator product expansion. In later sections on string <strong>th</strong>eory (sec. 5.6 ) we will see <strong>th</strong>at<br />
<strong>th</strong>e unusual OPE of Liouville also has important consequences for <strong>th</strong>e finiteness of <strong>th</strong>e<br />
<strong>th</strong>eory and for <strong>th</strong>e existence of an infinite dimensional space of background deformations.<br />
3.9. Liouville Correlators from Analytic Continuation<br />
In <strong>th</strong>e past two years <strong>th</strong>ere has been very interesting progress in understanding Liouville<br />
correlation functions via “analytic continuation in <strong>th</strong>e number of operators.” The first<br />
step in <strong>th</strong>e calculation of continuum correlators was provided in [45], where <strong>th</strong>e free field<br />
formulation by zero mode integration of <strong>th</strong>e Liouville field was established. The essential<br />
idea is to treat <strong>th</strong>e Liouville pa<strong>th</strong> integral measure as a free field measure and separate out<br />
a zero-mode φ 0 via φ = φ 0 + ˆφ, so <strong>th</strong>at [dφ] = [d ˆφ] dφ 0 . The integral over <strong>th</strong>e zero mode is<br />
∫ ∞<br />
−∞<br />
s ≡ 1 γ<br />
∑<br />
dφ 0 e<br />
αi φ 0 e<br />
−Qχφ 0<br />
/2−Be γφ 0<br />
(<br />
1<br />
2 Qχ − ∑ α i<br />
)<br />
= 1 Γ(−s) Bs<br />
γ<br />
B ≡ µ ∫ √ĝ<br />
e<br />
γ ˆφ .<br />
8πφ 2<br />
(3.60)<br />
In references [46–49], it is proposed <strong>th</strong>at when s ∈ Z + (so <strong>th</strong>ere is no negative curvature<br />
solution) <strong>th</strong>e ˆφ integral can be done using free field techniques. One <strong>th</strong>en obtains a class<br />
40
of amplitudes as “functions of s,” manipulates <strong>th</strong>e s-dependence to reside solely in <strong>th</strong>e arguments<br />
of Γ-functions (<strong>th</strong>rough factorials) and <strong>th</strong>en “analytically continues” to all values<br />
of s using Γ(x + 1) as an analytic continuation of x! .<br />
This curious procedure has scored many impressive successes. In particular [46], <strong>th</strong>e<br />
incorporation of <strong>th</strong>e Liouville mode was shown to cancel <strong>th</strong>e ghastly assemblage of Γ-<br />
functions familiar from <strong>th</strong>e conformal field <strong>th</strong>eory result and reproduce <strong>th</strong>e relatively simple<br />
matrix model result for many continuum correlation functions. Additional genus zero<br />
correlation functions for D ≤ 1 were computed in [48]. The genus one partition function<br />
for <strong>th</strong>e AD series was calculated via KdV me<strong>th</strong>ods in [50], and was confirmed from <strong>th</strong>e<br />
continuum Liouville approach in [51].<br />
Attempts to justify <strong>th</strong>e technique on physical grounds are based on arguments <strong>th</strong>at<br />
for s ∈ Z + , <strong>th</strong>e coefficient of log µ in <strong>th</strong>e correlation function is dominated by <strong>th</strong>e regions<br />
where φ → −∞ (short distance). In <strong>th</strong>ese regions <strong>th</strong>e Liouville interaction is small and <strong>th</strong>e<br />
<strong>th</strong>eory can be treated as free field <strong>th</strong>eory. See [48] for more detailed discussion. Ano<strong>th</strong>er<br />
justification, using <strong>th</strong>e quantum group approach to Liouville <strong>th</strong>eory, has been proposed in<br />
[52].<br />
Never<strong>th</strong>eless, <strong>th</strong>ese results remain to be better understood. The proper and complete<br />
calculation of correlation functions in Liouville <strong>th</strong>eory remains <strong>th</strong>e most important open<br />
problem in <strong>th</strong>e subject.<br />
3.10. Quantum Uniformization<br />
The most ambitious approach to <strong>th</strong>e evaluation of Liouville correlators proceeds by<br />
attempting to generalize <strong>th</strong>e original uniformization program of Klein and Poincaré (described<br />
in sec. 3.2) to quantum field <strong>th</strong>eory. This program was one of <strong>th</strong>e central motivations<br />
<strong>th</strong>at led Belavin, Polyakov, and Zamolodchikov to <strong>th</strong>e study of <strong>th</strong>e minimal models<br />
of conformal field <strong>th</strong>eory [21]. This program has also been pursued in a series of papers<br />
by Gervais and collaborators [39,52] using <strong>th</strong>e operator formalism. In <strong>th</strong>is section we shall<br />
try to clarify <strong>th</strong>e relation of <strong>th</strong>e original uniformization program to <strong>th</strong>e quantum Liouville<br />
<strong>th</strong>eory, and in particular elucidate <strong>th</strong>e role played by what we now interpret as quantum<br />
Liouville correlators.<br />
We begin by recalling <strong>th</strong>e classical <strong>th</strong>eory of Poincaré [53]. 13 We restrict attention to<br />
<strong>th</strong>e n-punctured sphere X = Ĉ − {z 1, . . . z n }. Let us try to solve <strong>th</strong>e classical Liouville<br />
13 We do not adhere here to <strong>th</strong>e historical development.<br />
41
equation wi<strong>th</strong> sources, (3.42). We set θ i ≡ γα i and consider <strong>th</strong>e case wi<strong>th</strong> sγ 2 = 2− ∑ θ i <<br />
0. The metric e γφ |dz| 2 on X will be obtained as a pullback of <strong>th</strong>e Poincaré metric on <strong>th</strong>e<br />
unit disk D,<br />
ds 2 = 16 µ<br />
|dw| 2<br />
(1 − |w| 2 ) 2 , (3.61)<br />
via a uniformization map w = f(z), where f : X → D. (The metric (3.61) is related to<br />
<strong>th</strong>e metric (3.12) on <strong>th</strong>e upper half plane via <strong>th</strong>e Cayley map w = (z − i)/(z + i) from <strong>th</strong>e<br />
upper half plane to <strong>th</strong>e disk.)<br />
The main observation is <strong>th</strong>at f(z) can be obtained as a ratio of solutions of a linear<br />
differential equation of Fuchsian type (i.e. having only regular singular points). To see<br />
<strong>th</strong>is, recall <strong>th</strong>e result (3.16),<br />
T (z)(dz) 2 = 1<br />
γ 2 S[ f(z); z ] (dz) 2 , (3.62)<br />
where S is <strong>th</strong>e Schwartzian derivative. T (z) is analytic and has second order poles at <strong>th</strong>e<br />
sources of curvature so we may write a partial fraction decomposition,<br />
ω X ≡ γ2<br />
2 T (z) = ∑ i<br />
(<br />
h i<br />
(z − z i ) + c )<br />
i<br />
, (3.63)<br />
2 z − z i<br />
where<br />
and <strong>th</strong>e c i are constants known as accessory parameters.<br />
equation<br />
h i = 1 4(<br />
1 − (1 − θi ) 2) , (3.64)<br />
In order to find <strong>th</strong>e map f, one might first turn to solve <strong>th</strong>e nonlinear differential<br />
S[f; z] = 2 ω X (z) . (3.65)<br />
This problem may be linearized by considering <strong>th</strong>e Fuchsian differential equation<br />
d 2 y<br />
dz 2 + ω X y = 0 (3.66)<br />
since, if y 1 , y 2 are any two linearly independent solutions of (3.66) <strong>th</strong>en f(z) = y 1 /y 2<br />
satisfies (3.65).<br />
Exercise.<br />
Check (3.65) for f = y 1 /y 2 . Note <strong>th</strong>at <strong>th</strong>e transformation properties of S[f; z]<br />
under <strong>th</strong>e Möbius group insure <strong>th</strong>at we can take any two solutions y 1 , y 2 .<br />
42
The differential equation (3.66) has regular singular points at z = z i , so if z is continued<br />
around z i <strong>th</strong>e solutions y 1 , y 2 will have monodromy<br />
y 1 → M 11 y 1 + M 12 y 2<br />
y 2 → M 21 y 1 + M 22 y 2 ,<br />
(3.67)<br />
inducing a Möbius transformation on f. Thus, if X = D/Γ where D is <strong>th</strong>e Poincaré disk<br />
and Γ is a discrete subgroup of SU(1, 1)/ Z 2<br />
∼ = P SL(2, IR), <strong>th</strong>en <strong>th</strong>ere exist y1 , y 2 such<br />
<strong>th</strong>at f : X → D is a multivalued mapping, inverting locally <strong>th</strong>e projection D → X. The<br />
different values of f(z) are obtained by applying Möbius transformations in Γ. Thus <strong>th</strong>e<br />
n-punctured sphere X and its accompanying uniformization map f have led to a Fuchsian<br />
differential equation (3.66) wi<strong>th</strong> <strong>th</strong>e property <strong>th</strong>at <strong>th</strong>e monodromy around regular singular<br />
points forms a discrete subgroup Γ ⊂ SU(1, 1)/ Z 2 .<br />
Klein and Poincaré tried to show <strong>th</strong>e converse of <strong>th</strong>e above chain of logic. Namely,<br />
if <strong>th</strong>e parameters c i in (3.63) are appropriately chosen, <strong>th</strong>en <strong>th</strong>e monodromy group of <strong>th</strong>e<br />
differential equation (3.66) would be a discrete subgroup Γ of SU(1, 1), and f = y 1 /y 2<br />
could be normalized so <strong>th</strong>at <strong>th</strong>e images of f(X) under Γ tesselate D. (In general <strong>th</strong>e c i<br />
have to be highly non-trivial functions of <strong>th</strong>e z i and h i to result in discrete monodromy<br />
and hence in reasonable surfaces X = D/Γ.) By suitable choice of <strong>th</strong>e c i , in principle any<br />
surface X could be obtained. This original approach to uniformization foundered on <strong>th</strong>e<br />
inability to calculate, or even show existence of, appropriate parameters c i . As we shall<br />
see, Klein and Poincaré got stuck on <strong>th</strong>e problem of computing Liouville correlators.<br />
In <strong>th</strong>e cases where f is a uniformizing map, we may obtain a solution to <strong>th</strong>e Liouville<br />
equation simply by pulling back <strong>th</strong>e Poincaré metric,<br />
e γφ |dz| 2 = f ∗ ( 16<br />
µ<br />
|dw| 2<br />
(1 − |w| 2 ) 2 )<br />
= 16 µ<br />
|C| 2 |dz| 2<br />
(<br />
|y2 | 2 − |y 1 | 2) 2 , (3.68)<br />
which yields<br />
e − 1 2 γφ =<br />
√ µ (<br />
|y2 | 2 − |y<br />
4|C|<br />
1 | 2) , (3.69)<br />
where <strong>th</strong>e constant C is <strong>th</strong>e Wronskian of y 1 , y 2 . Note <strong>th</strong>at al<strong>th</strong>ough y 1 , y 2 have monodromy<br />
(3.67), e − 1 2 γφ is single-valued. Finally, we combine (3.62, 3.63, 3.66, 3.69) to obtain<br />
<strong>th</strong>e important result<br />
∂ 2 ze − 1 2 γφ + γ2<br />
2 T (z) e− 1 2 γφ = 0 . (3.70)<br />
43
Exercise.<br />
Show <strong>th</strong>at (3.70) is an identity by working out <strong>th</strong>e second derivative of <strong>th</strong>e exponential<br />
and using <strong>th</strong>e formula for T in terms of φ.<br />
Example: The triangle functions.<br />
The uniformization of <strong>th</strong>e <strong>th</strong>ree-punctured sphere is explicitly known [54]. In <strong>th</strong>is<br />
case, (3.66) has <strong>th</strong>ree regular singular points and can <strong>th</strong>erefore be transformed to <strong>th</strong>e<br />
Gauss hypergeometric equation. The mapping is given by<br />
f(z) = N F 2(x)<br />
F 1 (x) ,<br />
(3.71a)<br />
where<br />
(<br />
F 2 (x) = x 1−θ 1<br />
2 F 1 2 −<br />
1<br />
2 (θ 1 + θ 2 + θ 3 ), 1 + 1 2 (−θ 1 + θ 2 − θ 3 ); 2 − θ 1 ; x )<br />
(<br />
F 1 (x) = 2 F 1 1 +<br />
1<br />
2 (θ 1<br />
1 − θ 2 − θ 3 ),<br />
2 (θ 1 + θ 2 − θ 3 ); θ 1 ; x )<br />
N 2 = (1 − θ 1 ) 2( ∆(θ 1 − 1) ) 2 ∆ ( 2 − 1 2 (θ 1 + θ 2 + θ 3 ) ∆ ( 1<br />
2 (−θ 1 + θ 2 + θ 3 ) )<br />
∆ ( 1<br />
2 (θ 1 + θ 2 − θ 3 ) ) ∆ ( 1<br />
2 (θ 1 − θ 2 + θ 3 ) )<br />
x = z − z 1<br />
z − z 2<br />
z 32<br />
z 31<br />
∆(y) = Γ(y)<br />
Γ(1 − y) . (3.71b)<br />
Fig. 5: A tesselation of <strong>th</strong>e Poincaré disk: A copy of an adjacent white and black<br />
triangle maps to <strong>th</strong>e <strong>th</strong>rice-punctured sphere. The images of <strong>th</strong>e triangles under<br />
<strong>th</strong>e monodromy group of <strong>th</strong>e associated Fuchsian differential equation tesselate <strong>th</strong>e<br />
Poincaré disk.<br />
44
The mapping (3.71) carries a circle <strong>th</strong>rough <strong>th</strong>e points z 1 , z 2 , z 3 to a curvilinear triangle<br />
in D wi<strong>th</strong> opening angles π(1 − θ i ). Note <strong>th</strong>e geometrical conditions<br />
θ i ≤ 1 , (3.72)<br />
since <strong>th</strong>e opening angles are ≥ 0 (recall eqs. (3.36–3.39)), and<br />
θ 1 + θ 2 + θ 3 − 2 ≥ 0 (3.73)<br />
since a hyperbolic triangle must have its sum of interior angles less <strong>th</strong>an or equal to π. If<br />
<strong>th</strong>e θ i are reciprocals of integers, <strong>th</strong>en <strong>th</strong>e triangles tesselate <strong>th</strong>e disk D as in fig. 5.<br />
Finally we write <strong>th</strong>e classical answer for <strong>th</strong>e monodromy-invariant solution to (3.70).<br />
Combining (3.69) and (3.71) we obtain<br />
e − 1 2 γφ =<br />
√ ∣ µ 1<br />
z − z 2 z 31 ∣∣∣<br />
θ 3<br />
∣ ∣ ∣∣∣ z − z 1 z 32 ∣∣∣<br />
θ 1<br />
∣ ∣ ∣∣∣ (z − z 3 ) 2 z 21 ∣∣∣<br />
2 (1 − θ 1 )|N| ∣z − z 3 z 21 z − z 2 z 31 z 31 z 23<br />
)<br />
·<br />
(|F 1 | 2 − N 2 |F 2 | 2 .<br />
(3.74)<br />
Using <strong>th</strong>e transformation properties of hypergeometric functions and identities on Γ–<br />
functions, it can be shown <strong>th</strong>at (3.74) is fully symmetric in (z 1 , θ 1 ), (z 2 , θ 2 ), and (z 3 , θ 3 ).<br />
We now interpret <strong>th</strong>e above equations in terms of conformal field <strong>th</strong>eory. The vertex<br />
operator Ψ = e − 1 2 γφ has conformal weight ∆ = − 1 2 − 3 8 γ2 . The central charge is c = 1+3Q 2<br />
and <strong>th</strong>erefore we have<br />
∆ = 1 16<br />
(<br />
c − 5 + √ )<br />
(c − 1)(c − 25) . (3.75)<br />
It immediately follows, as discussed in [21] (see also [20]), <strong>th</strong>at ( L 2 −1 − 2(2∆+1)<br />
3<br />
L −2<br />
)<br />
|∆〉 is<br />
a singular vector in <strong>th</strong>e Verma module built on |∆〉, and <strong>th</strong>erefore<br />
∂ 2 ze − 1 2 γφ + γ2<br />
2 : T (z) e− 1 2 γφ : (3.76)<br />
is a null field (where we use conformal normal-ordering in <strong>th</strong>e second term). Now, if <strong>th</strong>e<br />
null field decouples in correlation functions, 14 we may put<br />
〈 (∂<br />
2<br />
z e − 1 2 γφ + γ2<br />
2 T (z)e− 1 γφ) ∏<br />
〉<br />
2 e θiφ/γ (z i , ¯z i ) = 0 . (3.77)<br />
14 The Liouville <strong>th</strong>eory is sufficiently subtle <strong>th</strong>at <strong>th</strong>is is an open question.<br />
i<br />
45
In view of <strong>th</strong>ese observations, <strong>th</strong>e classical uniformization <strong>th</strong>eory takes on new meaning:<br />
<strong>th</strong>e classical solution in <strong>th</strong>e presence of sources e − 1 2 γφ cl corresponds to <strong>th</strong>e semiclassical<br />
correlator 〈e − 1 2 γφ ∏ e θ i<br />
γ φ 〉 s.c. . The classical equation (3.70) is <strong>th</strong>e null-vector decoupling<br />
equation, while (3.69) becomes <strong>th</strong>e decomposition of <strong>th</strong>e correlation function into “conformal<br />
blocks” y 1 , y 2 . These blocks are assembled into monodromy–invariant combinations.<br />
The geometrical conditions (3.72) and (3.73) become respectively <strong>th</strong>e Seiberg bound and<br />
<strong>th</strong>e condition for <strong>th</strong>e existence of a classical solution. Finally, <strong>th</strong>e partial fraction decomposition<br />
(3.63) is <strong>th</strong>e familiar Ward identity for <strong>th</strong>e insertion of an energy-momentum tensor<br />
in a correlator of primary fields:<br />
〈<br />
∏<br />
〉<br />
ω X = 1 2 γ2 T zz e<br />
θ i /γφ(z i )<br />
s.c.<br />
. (3.78)<br />
In particular, <strong>th</strong>e accessory parameters c i are given by<br />
c i = 1 ∂ 〈 ∏<br />
2 γ2 log e<br />
θ i /γφ(z i )〉<br />
∂z i<br />
s.c.<br />
(3.79)<br />
When combined wi<strong>th</strong> (3.48), <strong>th</strong>is last formula for <strong>th</strong>e accessory parameters makes sense<br />
independently of <strong>th</strong>e existence of a quantum Liouville <strong>th</strong>eory and has been rigorously<br />
proven recently by Takhtadjan and Zograf [38].<br />
<br />
Some four-point functions.<br />
Let us assume <strong>th</strong>at <strong>th</strong>e null-field decouples as in (3.77).<br />
Then, it follows directly<br />
from <strong>th</strong>e SL(2,C) Ward identities <strong>th</strong>at (3.77) reduces to an ODE related to Riemann’s<br />
differential equation (as in [21]). A straightforward calculation shows <strong>th</strong>at<br />
〈<br />
e − 1 2 γφ (z, ¯z)<br />
3∏<br />
i=1<br />
〉<br />
e θ i<br />
φ γ<br />
(z i , ¯z i )<br />
= ( µ ) 1<br />
−∑ 2<br />
θ i /γ 2 1<br />
4<br />
(1 − ˆθ 1 )| ˆN| |z∆ 123<br />
12<br />
z ∆ 132<br />
13<br />
z ∆ 231<br />
23<br />
| −2<br />
· ∣ z − z 2 z<br />
∣ ∣<br />
31 ∣∣ˆθ3<br />
∣∣ z − z 1 z<br />
∣ ∣<br />
32 ∣∣ˆθ1<br />
∣∣ (z − z 3 ) 2 z<br />
∣∣ 21 ∣∣ ∣∣ (z − z 1 ) 2 (z − z 2 ) 2 (z − z 3 ) 2 ∣ ∣∣<br />
γ 2 /4<br />
z − z 3 z 21 z − z 2 z 31 z 31 z 23 z 21 z 31 z<br />
) 23<br />
·<br />
(| ˆF 1 (x)| 2 − ˆN 2 | ˆF 2 (x)| 2<br />
(3.80)<br />
(<br />
ˆF 1 (x) = 2 F 1 1+<br />
1<br />
2 (ˆθ 1 − ˆθ 2 − ˆθ 3 ), 1 2 (ˆθ 1 + ˆθ 2 − ˆθ 3 ); ˆθ 1 ; x)<br />
ˆF 2 (x) = x 1−ˆθ 1<br />
(<br />
2 F 1 2 −<br />
1<br />
2 (ˆθ 1 +ˆθ 2 + ˆθ 3 ), 1 + 1 2 (−ˆθ 1 + ˆθ 2 − ˆθ 3 ); 2 − ˆθ 1 ; x) ,<br />
where <strong>th</strong>e quantum and classical expressions are related by <strong>th</strong>e simple shift ˆθ = θ −<br />
1<br />
2 γ2 . Of course, conformal invariance only determines <strong>th</strong>e correlator up to an overall<br />
function n(θ 1 , θ 2 , θ 3 ) which is totally symmetric in <strong>th</strong>e θ i . The prefactor ( (1 − ˆθ 1 )| ˆN| ) −1<br />
46
in (3.80) is obtained by comparing wi<strong>th</strong> <strong>th</strong>e semiclassical answer (3.74), where <strong>th</strong>e overall<br />
normalization is determined.<br />
Since <strong>th</strong>e rest of <strong>th</strong>e terms in <strong>th</strong>e expression satisfy <strong>th</strong>e<br />
substitution rule θ → ˆθ relating classical and quantum expressions, it is a fair guess <strong>th</strong>at<br />
<strong>th</strong>e prefactor ( (1 − ˆθ 1 )| ˆN| ) −1<br />
in (3.80) is exact.<br />
The fully quantum correlator (3.80) is a new result. As opposed to <strong>th</strong>e matrix model<br />
results we will describe in later chapters, (3.80) gives <strong>th</strong>e Liouville correlator as a function<br />
of <strong>th</strong>e moduli of <strong>th</strong>e 4-punctured sphere — if properly understood, (3.80) could be integrated<br />
over <strong>th</strong>e positions of <strong>th</strong>e punctures to derive <strong>th</strong>e (already automatically integrated)<br />
matrix model results for pure gravity. The correlator (3.80) has many strange properties<br />
possibly illustrating <strong>th</strong>e strange nature of <strong>th</strong>e OPE in Liouville <strong>th</strong>eory. Of particular note<br />
is <strong>th</strong>e case where some of <strong>th</strong>e operators saturate <strong>th</strong>e Seiberg bound α = Q/2, which, classically,<br />
corresponds to sources producing triangles wi<strong>th</strong> corner angle = 0. For example,<br />
if all <strong>th</strong>ree θ i /γ = Q/2 <strong>th</strong>en <strong>th</strong>e prefactor in (3.80) develops a pole and <strong>th</strong>e difference of<br />
hypergeometric functions vanishes. A short calculation shows <strong>th</strong>at <strong>th</strong>e limit α i → Q/2 is<br />
smoo<strong>th</strong> and<br />
〈e − 1 2 γφ 3∏<br />
i=1<br />
〉<br />
e 1 2 Qφ (z i , ¯z i )<br />
= ( µ ) 1<br />
2 −3Q/2γ |z ∆ 123<br />
12 z ∆ 132<br />
13 z ∆ 231<br />
23 | −2<br />
4<br />
∣<br />
∣∣ ·<br />
(z − z 1 )(z − z 3 ) ∣∣∣ ∣∣∣ (z − z 1 )(z − z 2 ) 1/2 (z 23 ) 1/4 γ 2<br />
∣ z 1 − z 3 z 1/4<br />
∣<br />
31 z3/4 21<br />
(<br />
· π F (1 − x) ¯F (x) + F (x) ¯F (1 − x)<br />
)<br />
,<br />
(3.81)<br />
where F (x) = F ( 1 2 , 1 2<br />
; 1, x) is an elliptic integral of <strong>th</strong>e first kind. In particular, F has<br />
logari<strong>th</strong>mic singularities F (x) ∼ 1 π log( 1<br />
1−x ) as x → 1− , resulting in logari<strong>th</strong>mic short–<br />
distance singularities in <strong>th</strong>e correlator (3.81). 15<br />
Remarks:<br />
1) The formula (3.80) probably only applies when s ≤ 0, o<strong>th</strong>erwise <strong>th</strong>ere are paradoxes.<br />
2) The operator e − 1 2 γφ is by no means <strong>th</strong>e only null vector in <strong>th</strong>e Liouville <strong>th</strong>eory. Using<br />
<strong>th</strong>e Kac determinant formula, one may ask for <strong>th</strong>e set of all operators e αφ = e −jγφ<br />
which weights ∆ = ∆ p,q (c) where p, q ≥ 1 are integer. The result is<br />
j p,q = 1 2<br />
1<br />
(p − 1) + (q − 1) .<br />
γ2 In principle <strong>th</strong>is allows one to extend <strong>th</strong>e above example to an infinite set of correlators.<br />
15 It is sometimes suggested in <strong>th</strong>e literature <strong>th</strong>at <strong>th</strong>e subleading logari<strong>th</strong>ms indicate <strong>th</strong>at <strong>th</strong>e<br />
correct vertex operator is φ e (Q/2)φ = ∂<br />
∂α eαφ | α=Q/2 , wi<strong>th</strong> <strong>th</strong>e derivative corresponding to <strong>th</strong>e<br />
limiting procedure needed above.<br />
47
3.11. Surfaces wi<strong>th</strong> boundaries<br />
The final me<strong>th</strong>od for extracting Liouville correlators, and <strong>th</strong>e one which is most closely<br />
connected to matrix model me<strong>th</strong>ods, is <strong>th</strong>e computation of macroscopic loop amplitudes.<br />
These are amplitudes in Liouville <strong>th</strong>eory for manifolds wi<strong>th</strong> boundary, for which <strong>th</strong>e Liouville<br />
action picks up <strong>th</strong>e extra boundary contribution<br />
S → S Bulk + Q ∮<br />
dŝ φ<br />
8π<br />
ˆk +<br />
ρ<br />
∂Σ<br />
4πγ 2 ∮∂Σ<br />
dŝ e 1 2 γϕ , (3.82)<br />
where ˆk is <strong>th</strong>e extrinsic curvature of <strong>th</strong>e boundary, dŝ is <strong>th</strong>e reference line element, and ρ<br />
is <strong>th</strong>e boundary cosmological constant.<br />
We have a well-defined variational principle if we choose Dirichlet boundary conditions<br />
(δϕ| ∂Σ = 0), or Neumann boundary conditions:<br />
∂(γϕ)<br />
∂n + ˆk + ρ 2 e 1 2 γϕ = 0 , (3.83)<br />
where <strong>th</strong>e first term is <strong>th</strong>e normal derivative.<br />
Just as we can introduce amplitudes at fixed area using <strong>th</strong>e operator (3.33), when using<br />
Neumann boundary conditions we can introduce amplitudes at fixed leng<strong>th</strong> by introducing<br />
<strong>th</strong>e leng<strong>th</strong> operator of a boundary loop C, given by<br />
∮<br />
l = dŝ e 1 2 γφ . (3.84)<br />
C<br />
While <strong>th</strong>is is obvious in <strong>th</strong>e classical <strong>th</strong>eory, surprisingly it continues to hold exactly in <strong>th</strong>e<br />
quantum <strong>th</strong>eory [55].<br />
Exercise. Boundary operators<br />
a) Assuming φ has free field Neumann short distance singularities near <strong>th</strong>e boundary,<br />
〈 φ(z) φ(w)<br />
〉 ∼ − log |z − w| 2 − log |¯z − ¯w| 2<br />
(where we <strong>th</strong>ink of <strong>th</strong>e boundary as <strong>th</strong>e x–axis for <strong>th</strong>e upper half plane), show <strong>th</strong>at <strong>th</strong>e<br />
vertex operator e αφ , when inserted on <strong>th</strong>e boundary has boundary conformal weight<br />
∆ b = −2α 2 + Qα and <strong>th</strong>us (3.84) is well-defined. A discussion of boundary operators<br />
in conformal field <strong>th</strong>eory may be found in [56].<br />
b) Show <strong>th</strong>at <strong>th</strong>e argument analogous to (3.53) suggests <strong>th</strong>e bound<br />
α ≤ Q 4<br />
(3.85)<br />
for boundary operators. 16<br />
16 In <strong>th</strong>e dense phase of <strong>th</strong>e O(n) model coupled to gravity, Kostov and Staudacher [57] have<br />
given examples of loop operator exponents which appear to give counterexamples to <strong>th</strong>e bound<br />
(3.85).<br />
48
From our experience wi<strong>th</strong> conformal field <strong>th</strong>eories in chapt. 1, we may expect <strong>th</strong>at if<br />
we insert a “macroscopic loop operator”<br />
W C (l) = δ<br />
( ∮<br />
l − dŝ e 1 γφ) 2 (3.86)<br />
C<br />
in <strong>th</strong>e Liouville pa<strong>th</strong> integral and <strong>th</strong>en shrink <strong>th</strong>e circumference to zero, <strong>th</strong>en W C (l) may<br />
be replaced by an infinite sum<br />
W (l) ∼ ∑ j≥0<br />
l x j<br />
σ j , (3.87)<br />
where σ j are local operators which can couple to <strong>th</strong>e boundary state created by (3.86).<br />
Exercise. Exponents in Loop Expansion<br />
Suppose <strong>th</strong>at <strong>th</strong>ere is an expansion like (3.87) in which <strong>th</strong>e operators σ j<br />
Liouville charge α j , i.e.<br />
σ j ∼ P(∂ ∗ φ, ∂ ∗ φ) e α j φ ,<br />
have<br />
where P is a polynomial. Show <strong>th</strong>at <strong>th</strong>e exponents x j of (3.87) can <strong>th</strong>en be found by<br />
a variant of <strong>th</strong>e simple scaling argument we have used in (3.49) and earlier. Consider<br />
a Liouville pa<strong>th</strong> integral wi<strong>th</strong> <strong>th</strong>e operator W (l) inserted, and shift φ, remembering to<br />
take into account <strong>th</strong>e change in Euler character from shrinking <strong>th</strong>e hole. Show <strong>th</strong>at <strong>th</strong>e<br />
pa<strong>th</strong> integral scales as<br />
e − 1 2 Q δφ+α j δφ+ 1 2 γ x j δφ ,<br />
from which follows<br />
x j = Q/γ − 2α j /γ = 2 γ ( 1 2 Q − α j) . (3.88)<br />
Note <strong>th</strong>at x j ≥ 0.<br />
It turns out <strong>th</strong>at, because of <strong>th</strong>e geometrical nature of Liouville <strong>th</strong>eory, <strong>th</strong>e expansion<br />
(3.87) is only valid under certain circumstances. This may be seen by a semiclassical study<br />
of amplitudes wi<strong>th</strong> loops [36], analogous to <strong>th</strong>e semiclassical considerations above. The<br />
main results of <strong>th</strong>is study are <strong>th</strong>e following:<br />
1) Let −sγ = ∑ α i − 1 2Qχ, where χ = 2 − 2h − B on a surface wi<strong>th</strong> h handles and B<br />
boundaries. If −sγ > 0, <strong>th</strong>e l → 0 behavior of W (l) is equivalent to a sum of local<br />
operators. In particular, <strong>th</strong>is is always <strong>th</strong>e case if <strong>th</strong>ere are two or more loops on <strong>th</strong>e<br />
surface (including <strong>th</strong>e one <strong>th</strong>at shrinks).<br />
2) As noted in <strong>th</strong>e above exercise, x j ≥ 0 for local operators. Coefficients of negative<br />
powers of l as l → 0 arise from small area divergences and are analytic in µ (and o<strong>th</strong>er<br />
coupling constants, in <strong>th</strong>e context of 2D gravity). Therefore <strong>th</strong>ey are interpreted as<br />
arising from infinitesimal size surfaces, and such terms are classified as non-universal<br />
contributions when comparing wi<strong>th</strong> matrix model answers.<br />
49
4. 2D Euclidean Quantum Gravity II: Canonical Approach<br />
It will be useful to work wi<strong>th</strong> <strong>th</strong>e canonical approach to two dimensional gravity. In<br />
<strong>th</strong>is chapter, we are led to introduce in particular some of <strong>th</strong>e details of <strong>th</strong>e algebraic<br />
(BRST) point of view (to be pursued fur<strong>th</strong>er in secs. 14.4, 14.5 ). We hope <strong>th</strong>at providing<br />
a common language will help bridge <strong>th</strong>e schism between <strong>th</strong>e algebraicists and <strong>th</strong>e matrix<br />
model <strong>th</strong>eorists, who are after all employing two complementary approaches to study <strong>th</strong>e<br />
same subject.<br />
4.1. Canonical Quantization of Gravitational Theories<br />
For a review of <strong>th</strong>e canonical approach to Einsteinian general relativity, see [58,59].<br />
Since gravitational <strong>th</strong>eories are gauge <strong>th</strong>eories, we are immediately led to study constrained<br />
dynamics.<br />
The canonical approach applies to spacetimes M which admit a space-time foliation:<br />
we assume <strong>th</strong>ere is a diffeomorphism Σ × IR → M, where Σ is a D-dimensional spacelike<br />
manifold. Choosing a unit normal n µ to <strong>th</strong>e surface Σ, we may project <strong>th</strong>e metric onto<br />
components parallel and perpendicular to <strong>th</strong>e surface. The restriction (D) g of <strong>th</strong>e metric g<br />
to <strong>th</strong>e spatial surface defines <strong>th</strong>e canonical coordinates, while <strong>th</strong>e time–time and time–space<br />
components of <strong>th</strong>e metric are expressed in terms of Lagrange multipliers for constraints<br />
of <strong>th</strong>e <strong>th</strong>eory (<strong>th</strong>e “lapse” and “shift”). In Einsteinian general relativity, <strong>th</strong>e constraints<br />
associated wi<strong>th</strong> <strong>th</strong>e lapse and shift are <strong>th</strong>e time–time and time–space components of <strong>th</strong>e<br />
Einstein equations: G 00 = 0, G 0i = 0. In <strong>th</strong>e canonical <strong>th</strong>eory, <strong>th</strong>ese are <strong>th</strong>e generators of<br />
time and space diffeomorphisms.<br />
In <strong>th</strong>e canonical quantization of gravity, wavefunctions are functions of <strong>th</strong>e spatial<br />
metric (and o<strong>th</strong>er fields in <strong>th</strong>e <strong>th</strong>eory): Ψ = Ψ[ (D) g, matter]. The requirement of gauge<br />
invariance states <strong>th</strong>at wavefunctions are required to obey operator versions of <strong>th</strong>e space<br />
and time diffeomorphism constraints. The Wheeler–DeWitt equation is <strong>th</strong>e equation expressing<br />
<strong>th</strong>e invariance under <strong>th</strong>e generator of time-diffeomorphisms [60–62] and plays a<br />
fundamental role in <strong>th</strong>e <strong>th</strong>eory.<br />
In non-Einsteinian metric <strong>th</strong>eories of gravity, including for example gravity in one and<br />
two spacetime dimensions, a completely analogous formulation may still be obtained by<br />
performing constrained quantization of a <strong>th</strong>eory wi<strong>th</strong> diffeomorphism invariance.<br />
50
4.2. Canonical Quantization of 2D Euclidean Quantum Gravity<br />
Diffeomorphisms are generated by <strong>th</strong>e energy-momentum tensor T αβ . In <strong>th</strong>e canonical<br />
approach <strong>th</strong>e diffeomorphism constraints of quantum gravity become <strong>th</strong>e statements <strong>th</strong>at<br />
<strong>th</strong>e tensor product <strong>th</strong>eory Liouville ⊗ matter is a conformal field <strong>th</strong>eory of central charge<br />
c = 0 (including <strong>th</strong>e ghosts) wi<strong>th</strong> a BRST complex, and moreover <strong>th</strong>e states in <strong>th</strong>e <strong>th</strong>eory<br />
lie in <strong>th</strong>e BRST cohomology of <strong>th</strong>e <strong>th</strong>eory.<br />
If massive matter is coupled to gravity, <strong>th</strong>en <strong>th</strong>e realization of <strong>th</strong>e Virasoro algebra<br />
on <strong>th</strong>e full Hilbert space is far from obvious. In <strong>th</strong>e special case where <strong>th</strong>e Hilbert space<br />
is a tensor product L ⊗ M of Liouville and matter M conformal field <strong>th</strong>eories (e.g., M =<br />
M(p, q) minimal conformal field <strong>th</strong>eory, is frequently considered), however, <strong>th</strong>e situation<br />
simplifies dramatically. Naively <strong>th</strong>e wavefunctions are now functions of <strong>th</strong>e spatial metric,<br />
parametrized by φ(σ) and <strong>th</strong>e matter degrees of freedom. When formulating 2D quantum<br />
gravity in <strong>th</strong>e context of conformal field <strong>th</strong>eory, however, <strong>th</strong>e diffeomorphism constraints<br />
are properly enforced <strong>th</strong>rough <strong>th</strong>e calculation of BRST cohomology wi<strong>th</strong> respect to <strong>th</strong>e<br />
Virasoro algebra. The condition <strong>th</strong>at nontrivial cohomology exists immediately implies<br />
<strong>th</strong>at <strong>th</strong>e total central charge is zero (see comment after (2.16)) so <strong>th</strong>at<br />
c + 1 + 3Q 2 − 26 = 0 =⇒ Q 2 = 25 − c , (4.1)<br />
3<br />
where c is <strong>th</strong>e central charge of M.<br />
Remarks:<br />
1) There are <strong>th</strong>ree kinds of cohomology problems we can study, depending on how we<br />
treat <strong>th</strong>e zero modes of b(z), ¯b(¯z). In “relative cohomology” we require b 0 = ¯b 0 = 0 on<br />
states and gauge parameters. In “semi-relative cohomology” we impose <strong>th</strong>e condition<br />
b − 0 = b 0 − ¯b 0 = 0 on states and gauge parameters. In “absolute cohomology” we<br />
impose no conditions pertaining to <strong>th</strong>e b, ¯b zero modes.<br />
2) In 2D gravity (and string <strong>th</strong>eory) <strong>th</strong>ere is an important duality on <strong>th</strong>e cohomology<br />
spaces. If Φ a forms a basis for <strong>th</strong>e semi-relative cohomology <strong>th</strong>en <strong>th</strong>ere will be a<br />
dual basis defined such <strong>th</strong>at <strong>th</strong>e BPZ inner product 〈Φ b , Φ a 〉 ≡ 〈0|Φ b (∞)Φ a (0)|0〉 is<br />
diagonalized. If Φ a is in <strong>th</strong>e semi-relative cohomology <strong>th</strong>en Φ a will not be in <strong>th</strong>e semirelative<br />
cohomology. One can define ˜Φ b ≡ b − 0 Φb which will be in <strong>th</strong>e semi-relative<br />
cohomology. This conjugation Φ b → ˜Φ b which exchanges states of ghost number G<br />
and 5 − G plays a crucial role in string field <strong>th</strong>eory, and will be important in <strong>th</strong>e<br />
considerations of chapt. 14.<br />
3) In <strong>th</strong>e literature, not much attention is devoted to defining precisely <strong>th</strong>e boundary<br />
conditions on field space (i.e., spacetime) for <strong>th</strong>e cohomology problem. However, such<br />
boundary conditions are very important physically, as we shall see.<br />
51
4.3. KPZ states in 2D Quantum Gravity<br />
KPZ states refer to a special class of BRST cohomology classes associated to <strong>th</strong>e<br />
primary fields of <strong>th</strong>e (p, q) minimal models which result when <strong>th</strong>ese <strong>th</strong>eories are coupled<br />
to gravity.<br />
For states wi<strong>th</strong> a trivial ghost structure <strong>th</strong>e Wheeler–DeWitt constraint, implementing<br />
invariance under time diffeomorphisms, becomes<br />
(<br />
L0 + L 0 − 2 ) Ψ = 0 (4.2)<br />
where L n are <strong>th</strong>e modes of <strong>th</strong>e total stress energy tensor for L⊗M, and <strong>th</strong>e −2 is <strong>th</strong>e ghost<br />
contribution. For M = M(p, q), <strong>th</strong>e KPZ operators O = e αφ Φ 0 , where Φ 0 is a primary in<br />
M(p, q) (as described in sec. 2.2), <strong>th</strong>e wavefunction Ψ O fur<strong>th</strong>er factorizes<br />
Ψ O = Ψ matter<br />
O<br />
⊗ Ψ gravity<br />
O<br />
, (4.3)<br />
and is separately an eigenstate of (L 0 +L 0 ) matter . In <strong>th</strong>is case <strong>th</strong>e WdW equation becomes<br />
(<br />
)<br />
(L 0 + L 0 ) Liouv + ∆ X + ∆ X − 2 Ψ gravity = 0 . (4.4)<br />
ϕ<br />
K ν(e )<br />
V<br />
Fig. 6: Solution to minisuperspace Wheeler–DeWitt equation decaying at large<br />
leng<strong>th</strong>s.<br />
ϕ<br />
If matter boundary conditions are separately diffeomorphism invariant, we expect Ψ<br />
to depend on only <strong>th</strong>e diffeomorphism invariant information in φ(σ), namely, on <strong>th</strong>e leng<strong>th</strong><br />
l = ∮ e 1 2 γφ(σ) . In any case, in <strong>th</strong>e minisuperspace approximation we replace<br />
1<br />
2 (L 0 + L 0 ) Liouv →<br />
(−(l γ2 ∂ )<br />
4 ∂l )2 + 4µl 2 + 1 8 Q2 . (4.5)<br />
52
Using <strong>th</strong>e KPZ formula (2.26) written as<br />
∆ X − 1 + 1 8 Q2 = 1 2(<br />
α −<br />
1<br />
2 Q) 2<br />
(4.6)<br />
(as suggested after (3.22)), we obtain <strong>th</strong>e minisuperspace Wheeler–DeWitt equation<br />
(<br />
−(l ∂ ∂l )2 + 4µl 2 + ν 2) ψ(l) = 0 ,<br />
ν = ± 2 γ<br />
(<br />
α −<br />
1<br />
2 Q) . (4.7)<br />
The solution decaying at large leng<strong>th</strong>s is <strong>th</strong>e non-normalizable wavefunction<br />
ψ O (l) ∝ K ν (2 √ µl) , (4.8)<br />
illustrated in fig. 6.<br />
As promised in sec. 3.5, <strong>th</strong>e wavefunctions corresponding to geometries (3.35) appear<br />
naturally in <strong>th</strong>e <strong>th</strong>eory. From <strong>th</strong>e geometrical picture of chapt. 3, it is natural to associate<br />
<strong>th</strong>is geometry wi<strong>th</strong> <strong>th</strong>e insertion of a local operator at t = −∞. In [7], Seiberg has<br />
fur<strong>th</strong>er interpreted <strong>th</strong>e blowup of <strong>th</strong>e wavefunction at short distances as being physically<br />
appropriate. The idea is <strong>th</strong>at <strong>th</strong>e wavefunctions associated to local operators in quantum<br />
gravity should have support on metrics which are infinitesimally small in <strong>th</strong>e physical<br />
metric e γφ ĝ (because <strong>th</strong>ey are local).<br />
Remark: In (4.5) we appear to have made an approximation. Astonishingly, matrix<br />
model calculations (for example eq. 10.19 below) confirm <strong>th</strong>at (4.8) is exact. It is not<br />
understood why <strong>th</strong>is should be so.<br />
4.4. LZ states in 2D Quantum Gravity<br />
So far we have discussed only <strong>th</strong>e KPZ states in which <strong>th</strong>e ghost modes are not excited.<br />
These form only part of <strong>th</strong>e full spectrum of <strong>th</strong>e <strong>th</strong>eory, as demonstrated in <strong>th</strong>e continuum<br />
formulation in <strong>th</strong>e work of Lian–Zuckerman [63,64]. Treating <strong>th</strong>e Liouville field as free,<br />
<strong>th</strong>ey calculated <strong>th</strong>e semi-infinite (BRST) cohomology of L ⊗ M(p, q), and found <strong>th</strong>at <strong>th</strong>e<br />
cohomology is spanned by operators of <strong>th</strong>e form<br />
O n e α nφ ,<br />
α n<br />
γ = p + q − n<br />
2q<br />
n ≥ 1, ≠ 0 mod p, ≠ 0 mod q , (4.9)<br />
and γ is determined as in (2.19). The operator O n is made of ghosts, matter, and derivatives<br />
of φ. The ghost number of O n depends linearly on n.<br />
53
In <strong>th</strong>e KP formalism of <strong>th</strong>e matrix model to be described in sec. 7.7 , on <strong>th</strong>e o<strong>th</strong>er<br />
hand, scaling operators formed from fractional powers of Lax operators (which have known<br />
lattice analogs) will be constructed and scale like Liouville operators of <strong>th</strong>e form<br />
O n e α nφ ,<br />
α n<br />
γ<br />
= p + q − n<br />
2q<br />
n ≥ 1, ≠ 0 mod q , (4.10)<br />
where q < p (but <strong>th</strong>e n ≠ 0 mod p restriction is lifted). In sec. 7.7 , we will see how <strong>th</strong>ese<br />
operators arise in <strong>th</strong>e matrix model formulation.<br />
Let us now consider <strong>th</strong>e discrepancies between <strong>th</strong>e calculations.<br />
First, in <strong>th</strong>e LZ<br />
computation <strong>th</strong>ere is no reason to restrict attention to states satisfying α ≤ Q/2. This is<br />
quite appropriate, since <strong>th</strong>e computation applies equally well when µ = 0, in which case<br />
<strong>th</strong>ere is no wall to induce total reflection of <strong>th</strong>e wavefunctions and hence identify states wi<strong>th</strong><br />
±E or ±ν. There is a fur<strong>th</strong>er discrepancy of operators wi<strong>th</strong> n = 0 mod p. This has been<br />
partially explained wi<strong>th</strong> boundary operators [55]. Apart from <strong>th</strong>is, <strong>th</strong>e two calculations<br />
are in remarkable agreement. Never<strong>th</strong>eless it is an important open problem to understand<br />
better <strong>th</strong>e physical meaning of <strong>th</strong>e Lian–Zuckerman states and <strong>th</strong>eir relationship, if any,<br />
to <strong>th</strong>e infinite tower of scaling operators in <strong>th</strong>e matrix model.<br />
In <strong>th</strong>e case of <strong>th</strong>e one-matrix model, <strong>th</strong>e infinite tower of operators corresponding to<br />
K j− 1 2 | + (in <strong>th</strong>e notation of sec. 7.7 ) are denoted by σ j , and will be studied in more detail<br />
in sec. 10.2 below.<br />
4.5. States in 2D Gravity Coupled to a Gaussian Field: more BRST<br />
Consider now <strong>th</strong>e coupling of Euclidean gravity to a massless Euclidean scalar field in<br />
two dimensions:<br />
∫<br />
S = d 2 z √ ( 1 ĝ<br />
8π ( ˆ∇φ) 2 + Q 8π φR(ĝ) + µ ) ∫<br />
8πγ 2 eγφ +<br />
d 2 z √ ĝ 1<br />
8π ( ˆ∇X) 2 , (4.11)<br />
where X is <strong>th</strong>e real massless boson. The KPZ equations (2.16) and (2.19) for D = 1 imply<br />
<strong>th</strong>at Q = √ 8 and γ = √ 2.<br />
<br />
Cosmological constant operator at c = 1<br />
According to some au<strong>th</strong>ors, <strong>th</strong>e correct quantum effective action must have a cosmological<br />
constant term given by φ e γφ . Many confusing issues related to <strong>th</strong>is point are not<br />
well understood (as of Sep. 92). The argument in favor of <strong>th</strong>is identification is <strong>th</strong>at <strong>th</strong>e<br />
usual relation between <strong>th</strong>e wavefunction and vertex operator, toge<strong>th</strong>er wi<strong>th</strong> <strong>th</strong>e wavefunction<br />
behavior K 0 (l) ∼ log l, suggests an extra factor of φ. A second argument is based on<br />
54
<strong>th</strong>e p → 0 behavior of amplitudes studied in sec. 13.6 below, and a <strong>th</strong>ird is based on <strong>th</strong>e<br />
relation between bare and renormalized cosmological constants at c = 1 given in sec. 11.6 .<br />
We find none of <strong>th</strong>ese arguments entirely convincing.<br />
<br />
Spectrum of µ = 0 versus µ > 0<br />
The BRST cohomology of <strong>th</strong>e <strong>th</strong>eory (4.11) was calculated in <strong>th</strong>e µ = 0 <strong>th</strong>eory by<br />
Lian and Zuckerman. Their results were simplified and extended in [65,66,67]. In <strong>th</strong>is<br />
section we describe some of <strong>th</strong>ese results. The following argument, based on <strong>th</strong>e string<strong>th</strong>eoretic/spacetime<br />
interpretation of <strong>th</strong>ese <strong>th</strong>eories described in chapt. 5, suggests <strong>th</strong>at,<br />
except for <strong>th</strong>e imposition of <strong>th</strong>e Seiberg bound, <strong>th</strong>e physical states should be <strong>th</strong>e same:<br />
<strong>th</strong>e Liouville interaction disappears for φ → −∞. Thus, states <strong>th</strong>at have wavefunctions<br />
concentrated in <strong>th</strong>is region must behave like states in <strong>th</strong>e free <strong>th</strong>eory, in particular, <strong>th</strong>e<br />
interaction is arbitrarily weak in <strong>th</strong>is region and “ought not” create or destroy extra states.<br />
This is not true of states concentrated at φ → +∞, which is why we must impose <strong>th</strong>e<br />
Seiberg bound. This argument is surely correct for <strong>th</strong>e tachyon cohomology classes, but is<br />
not obviously correct for <strong>th</strong>e global modes associated to <strong>th</strong>e discrete states.<br />
The nature of <strong>th</strong>e cohomology depends strongly on <strong>th</strong>e value of q, <strong>th</strong>e X-field momentum<br />
as measured by √ 2 ∂X. For generic q /∈ Z <strong>th</strong>ere are states in <strong>th</strong>e BRST cohomology<br />
of ghost number G = 2 and dimension zero. 17 These are <strong>th</strong>e gravitationally dressed vertex<br />
operators<br />
V q = c¯c e iqX/√2 √<br />
e 2(1−<br />
1<br />
2 |q|)φ<br />
V q = c¯c e iqX/√2 e<br />
√<br />
2(1+<br />
1<br />
2 |q|)φ .<br />
(4.12)<br />
The operators V q violate <strong>th</strong>e condition α ≤ Q/2 discussed in sec. 3.6. We will confirm<br />
below <strong>th</strong>at <strong>th</strong>ey do not appear in <strong>th</strong>e matrix model computations. As in sec. 4.3, we expect<br />
2D gravity wavefunctions associated to <strong>th</strong>e operators V q to be µ |q|/2 K q (2 √ µl). We will<br />
confirm <strong>th</strong>is in chapt. 11. As discussed in sec. 4.2 above we should distinguish between<br />
absolute, relative, and semi-relative cohomology.<br />
cohomology, we must introduce <strong>th</strong>e operator [66]<br />
If we are working wi<strong>th</strong> <strong>th</strong>e absolute<br />
a = [Q, φ] = c ∂φ + √ 2 ∂c , (4.13)<br />
and its holomorphic conjugate. Then we have extra states: aV q , āV q , aāV q . In <strong>th</strong>e semirelative<br />
cohomology, we must include <strong>th</strong>e extra state (a + ā) V q .<br />
17 Ghost number G always refers to <strong>th</strong>e total left+right moving ghost number in <strong>th</strong>e closed<br />
string case.<br />
55
In fact, <strong>th</strong>e c = 1 model has much more cohomology. First of all, <strong>th</strong>ere are many more<br />
primary fields in <strong>th</strong>e <strong>th</strong>eory which may be gravitationally dressed by Liouville exponentials.<br />
This is most elegantly seen by considering <strong>th</strong>e chiral SU(2) current algebra <strong>th</strong>at arises when<br />
a Gaussian field X is compactified on a self-dual radius [20]. The currents are given by<br />
J (±) (z) = e ±i√ 2X<br />
J (3) (z) =<br />
i √<br />
2<br />
∂X . (4.14)<br />
is√2X<br />
Then, for s = 0, 1/2, 1, . . ., we have highest weight fields ψ s,s = e for<br />
SU(2). We can <strong>th</strong>us make chiral weight (1, 0) Virasoro highest weight fields from<br />
√<br />
(j + m)!<br />
ψ j,m (z) =<br />
(j − m)! (2j)!<br />
where m ∈ {−j, −j + 1, . . . , j − 1, j}.<br />
<strong>th</strong>e global<br />
(∮ dz<br />
) j−m<br />
2πi e−i√ 2X ψj,j . (4.15)<br />
Exercise. Characters of Fock modules<br />
When q ∈ Z, <strong>th</strong>e Fock module has a highest weight vector wi<strong>th</strong> Virasoro weight<br />
∆ = q 2 /4. In <strong>th</strong>is case it is known from Virasoro representation <strong>th</strong>eory <strong>th</strong>at <strong>th</strong>e Fock<br />
space F q becomes infinitely reducible, i.e., <strong>th</strong>at F q contains infinitely many Virasoro<br />
primaries.<br />
The characters of <strong>th</strong>e irreducible c = 1 representations of <strong>th</strong>e Virasoro algebra wi<strong>th</strong><br />
weight ∆ are [20,68]:<br />
χ ∆ = q∆ η<br />
√<br />
4∆ /∈ Z<br />
χ ∆n = qn2 /4 − q (n+2)2 /4<br />
η<br />
∆ = n 2 /4, n ∈ Z .<br />
(4.16)<br />
a) Using <strong>th</strong>ese characters and <strong>th</strong>e fact <strong>th</strong>at F q contains no singular vectors, show<br />
<strong>th</strong>at when q ∈ Z <strong>th</strong>e Fock module can be written as<br />
F n/<br />
√<br />
2<br />
= ⊕ ∞ r=0L ( c = 1, ∆ =<br />
(n + 2r)2 ) , (4.17)<br />
4<br />
where L is <strong>th</strong>e irreducible representation wi<strong>th</strong> highest weight ∆.<br />
b) Show <strong>th</strong>at <strong>th</strong>e state α −1 α −1 |0〉 corresponding to ∂X∂X is an example of a<br />
nontrivial Virasoro primary in <strong>th</strong>e Fock module wi<strong>th</strong> q = 0.<br />
56
Therefore <strong>th</strong>e chiral cohomology contains <strong>th</strong>e fields<br />
Y +<br />
√<br />
j,m (z) = c ψ j,m(z) e 2(1−j)φ<br />
= c P n,r (∂X) e i 1 √<br />
2 nX e<br />
√<br />
2<br />
2 (2−(n+2r))φ (4.18)<br />
wi<strong>th</strong> ghost number G = 1 and dimension zero. In <strong>th</strong>e second line of (4.18), we set n = 2m<br />
and we have emphasized <strong>th</strong>e description of <strong>th</strong>e exercise: <strong>th</strong>e highest weight in <strong>th</strong>e r <strong>th</strong> term<br />
of (4.17) is generated by <strong>th</strong>e highest weight state P n,r (∂X) e i 1 √<br />
2 nX , where P n,r (∂X) is a<br />
polynomial in derivatives of X of dimension nr + r 2 , and s = r + n/2. This state has<br />
Liouville momentum −ip φ / √ 2 = −α/γ + Q/2γ = (n + 2r)/2 = s.<br />
Al<strong>th</strong>ough we have constructed <strong>th</strong>ese states by appealing to <strong>th</strong>e symmetry structure<br />
at <strong>th</strong>e self-dual radius, <strong>th</strong>ey will give rise to BRST cohomology classes at o<strong>th</strong>er radii by<br />
combining left– and right–movers. In particular, at infinite radius we may form <strong>th</strong>e states<br />
S j,m = Y +<br />
j,m Y + j,m (4.19)<br />
wi<strong>th</strong> ghost number G = 2 and dimension zero. In <strong>th</strong>e absolute cohomology we must include<br />
<strong>th</strong>e states aY +<br />
j,m , and so on. .<br />
.<br />
Y + 3 ,<br />
2<br />
3<br />
−<br />
2<br />
.<br />
.<br />
.<br />
+<br />
Y<br />
1 − 1<br />
3<br />
− 2 − − 1 −<br />
2<br />
,<br />
Y + 3 ,<br />
2<br />
Y + ,<br />
1<br />
2<br />
.<br />
1<br />
−<br />
2<br />
−<br />
.<br />
1<br />
2<br />
1<br />
2<br />
2<br />
3<br />
2<br />
1<br />
1<br />
2<br />
.<br />
.<br />
.<br />
− i p ϕ<br />
2<br />
Y + ,<br />
1 0<br />
.<br />
1<br />
2<br />
Y +<br />
3<br />
2<br />
,<br />
Y +<br />
1<br />
2<br />
1<br />
2<br />
,<br />
1<br />
2<br />
.<br />
.<br />
1<br />
Y + 1,<br />
1<br />
.<br />
3<br />
2<br />
Y +<br />
3<br />
2<br />
,<br />
3<br />
2<br />
.<br />
2<br />
p x 2<br />
Fig. 7: A plot of <strong>th</strong>e quantum numbers of <strong>th</strong>e special states in <strong>th</strong>e (p X , −ip φ )/ √ 2<br />
plane. The special states intersecting <strong>th</strong>e tachyon dispersion line at |m| = j are<br />
called “special tachyons.” Note <strong>th</strong>at if one works at µ = 0, <strong>th</strong>e Seiberg bound<br />
does not hold and one should include <strong>th</strong>e o<strong>th</strong>er states Y −<br />
j,m<br />
. These constitute an<br />
identical plot obtained by reflecting p φ → −p φ .<br />
57
We may plot <strong>th</strong>e quantum numbers of <strong>th</strong>ese states as in fig. 7.<br />
The big surprise,<br />
discovered by Lian and Zuckerman, is <strong>th</strong>at at <strong>th</strong>e points in fig. 7 interior to <strong>th</strong>e wedge<br />
<strong>th</strong>ere are extra cohomology classes. The above-mentioned classes only account for half<br />
of <strong>th</strong>e BRST cohomology. For every class Y +<br />
j,m<br />
wi<strong>th</strong> j = 1, 2, ..., and |m| < j <strong>th</strong>ere is<br />
a corresponding class O j−1,m wi<strong>th</strong> <strong>th</strong>e same X, φ momenta but wi<strong>th</strong> ghost number zero.<br />
The first <strong>th</strong>ree examples are<br />
O 0,0 = 1<br />
(<br />
O 1/2,1/2 = bc − √ 1<br />
)<br />
(∂φ + i∂X)<br />
2<br />
O 1/2,−1/2 =<br />
e −(φ−iX)/√ 2<br />
(<br />
bc − √ 1<br />
)<br />
(∂φ − i∂X) e −(φ+iX)/√2 .<br />
2<br />
(4.20)<br />
.<br />
.<br />
.<br />
O<br />
1 − 1<br />
.<br />
3<br />
− 2 − 1 −<br />
2<br />
,<br />
O 1<br />
2 ,<br />
.<br />
−<br />
.<br />
1<br />
2<br />
1<br />
2<br />
2<br />
3<br />
2<br />
1<br />
1<br />
2<br />
.<br />
.<br />
.<br />
− i p ϕ<br />
2<br />
O 0,<br />
0<br />
.<br />
.<br />
1<br />
2<br />
O 1<br />
2<br />
,<br />
1<br />
2<br />
.<br />
O 1,<br />
0 O ,<br />
.<br />
1<br />
1 1<br />
.<br />
3<br />
2<br />
.<br />
2<br />
p x 2<br />
Fig. 8: The wedge of fig. 7, wi<strong>th</strong> <strong>th</strong>e chiral ground ring states enumerated.<br />
A plot of <strong>th</strong>ese “ground ring” states is shown in fig. 8. Lian and Zuckerman show<br />
<strong>th</strong>at <strong>th</strong>ere are no o<strong>th</strong>er chiral cohomology classes. The full closed string cohomology is<br />
formed by combining <strong>th</strong>e above classes subject to constraints on left– and right–moving<br />
momenta. Since we do not compactify <strong>th</strong>e Liouville field, we must impose p L φ = pR φ . The<br />
conditions on p X<br />
depend on <strong>th</strong>e radius of compactification [20]. For <strong>th</strong>e X-field wi<strong>th</strong><br />
infinite radius R = ∞ (our usual case), we have p R X = pL X<br />
. When X is compactified at<br />
special radii, e.g. <strong>th</strong>e self-dual radius, <strong>th</strong>is condition may be relaxed and <strong>th</strong>ere will be more<br />
BRST cohomology classes.<br />
58
Remark: In general <strong>th</strong>ere will be “special states” when <strong>th</strong>e X-field is compactified<br />
on a circle of radius r = √ 1 p<br />
2 q<br />
where p, q are relatively prime integers. The special states<br />
must have <strong>th</strong>e (p L X , pR X )√ 2 = (kq + lp, kq − lp) where k, l are arbitrary integers. Note <strong>th</strong>at<br />
<strong>th</strong>e zero momentum special states are present at every radius.<br />
The appearance and disappearance of special states as <strong>th</strong>e radius is varied is a puzzling<br />
phenomenon. It has been discussed in [69].<br />
Thus we may finally summarize <strong>th</strong>e relative closed string cohomology at R = ∞: In<br />
addition to <strong>th</strong>e tachyon states (4.12) we have four states at ghost numbers G = 0, 1, 2 in<br />
<strong>th</strong>e relative cohomology:<br />
G = 0 : R j,m = O j,m Ō j,m j = 0, 1/2, . . . ; |m| ≤ j<br />
G = 1 : J j,m = Y +<br />
j,m Ōj−1,m<br />
¯ J j,m = O j−1,m Ȳ +<br />
j,m<br />
G = 2 : S j,m = Y +<br />
j,m Ȳ +<br />
j,m<br />
j = 1, 3/2, . . . ; |m| < j .<br />
j = 1, 3/2, . . . ; |m| < j<br />
(4.21)<br />
As pointed out in [66], <strong>th</strong>e semi-relative cohomology is more appropriate for comparison<br />
wi<strong>th</strong> closed string field <strong>th</strong>eory (see [44]). The semi-relative cohomology has 4 more states<br />
at ghost numbers 1,2,3 obtained by multiplying <strong>th</strong>e above operators by a + ā. Explicit<br />
formulae for special state representatives, as well as an alternative proof of <strong>th</strong>e Lian–<br />
Zuckerman <strong>th</strong>eorem has been given in [65].<br />
Remark: Conjugate States 18<br />
The tilde conjugation Φ s → ˜Φ s described in sec. 4.2 above is important for understanding<br />
<strong>th</strong>e factorization properties of amplitudes. The behavior of <strong>th</strong>is conjugation is<br />
ra<strong>th</strong>er different at µ = 0 and µ > 0. At µ = 0 we have standard free-field formulae. In<br />
particular Φ s → ˜Φ s exchanges states wi<strong>th</strong> ghost number G and 5 − G. It also exchanges<br />
(+)–states wi<strong>th</strong> (−)–states. At µ > 0, <strong>th</strong>ere are no (−) states and it might appear <strong>th</strong>at a<br />
fundamental axiom for constructing string field <strong>th</strong>eory has broken down. This is not <strong>th</strong>e<br />
case, since <strong>th</strong>e Liouville 2-point function has a geometrical divergence coming from <strong>th</strong>e<br />
volume of <strong>th</strong>e dilation group IR ∗ + (see sec. 3.7). This divergent numerator is precisely what<br />
is needed to cancel <strong>th</strong>e division by <strong>th</strong>e volume of <strong>th</strong>e conformal Killing group <strong>th</strong>at results<br />
if we only insert 4 out of 6 c, ¯c –zero modes. Thus we can have a nonzero 2-point function:<br />
〈<br />
c¯c e ip 1X/ √2 √ 1<br />
e 2(1−<br />
2 |p1|)φ c¯c e ip 2X/ √2 √ 1<br />
〉<br />
e 2(1−<br />
2 |p 2|)φ<br />
∼ δ(p 1 + p 2 ) . (4.22)<br />
18 We <strong>th</strong>ank N. Seiberg for clarifying <strong>th</strong>is point.<br />
59
On <strong>th</strong>e RHS we have one, ra<strong>th</strong>er <strong>th</strong>an two, δ–functions in <strong>th</strong>e momenta of <strong>th</strong>e problem.<br />
In general, we see <strong>th</strong>at <strong>th</strong>e conjugation Φ s → ˜Φ s at µ > 0 exchanges ghost numbers G and<br />
4 − G, and preserves <strong>th</strong>e (+)–states satisfying <strong>th</strong>e Seiberg bound.<br />
As emphasized in [67], <strong>th</strong>e existence of <strong>th</strong>e ghost number one BRST classes implies<br />
<strong>th</strong>e existence of a large symmetry algebra. Indeed, quite generally, given a dimension zero<br />
BRST class Ω (0) we may associate wi<strong>th</strong> it a descent multiplet (Ω (0) , Ω (1) , Ω (2) ) consisting<br />
of 0, 1, 2 forms defined by <strong>th</strong>e descent equations:<br />
0 = {Q, Ω (0) }<br />
dΩ (0) = {Q, Ω (1) }<br />
(4.23)<br />
dΩ (1) = {Q, Ω (2) }<br />
Exercise. Descent Equations<br />
a) Using {Q, b −1 } = L −1 , show <strong>th</strong>at in terms of states associated to <strong>th</strong>e operators<br />
<strong>th</strong>e descent equations read:<br />
|Ω (1)<br />
z 〉 = b −1 |Ω (0) 〉<br />
|Ω (1)<br />
¯z 〉 = ¯b −1 |Ω (0) 〉<br />
|Ω (2)<br />
z¯z 〉 = b −1¯b −1 |Ω (0) 〉 .<br />
(4.24)<br />
The significance of <strong>th</strong>e descent multiplet is <strong>th</strong>at to any BRST invariant dimension<br />
zero operator Ω (0) , we may associate 1) a corresponding charge<br />
∮<br />
A(Ω (0) ) ≡ Ω (1) , (4.25)<br />
conserved up to BRST exact operators, and 2) a corresponding modulus, by which we can<br />
deform <strong>th</strong>e action,<br />
while preserving BRST symmetry.<br />
Exercise. Tachyon descent multiplet<br />
∫<br />
∆S =<br />
Σ<br />
Ω (2) , (4.26)<br />
Show <strong>th</strong>at <strong>th</strong>e descent multiplet for <strong>th</strong>e tachyon vertex operator is<br />
G = 2 : V (0)<br />
p<br />
= cc e ipX/√2 e (√ 2(1− 1 2 |p|)φ<br />
G = 1 : V (1)<br />
p<br />
= (dz c − dz c) e ipX/√2 e (√ 2(1− 1 2 |p|)φ<br />
(4.27)<br />
G = 0 : V (2)<br />
p = dz ∧ dz e ipX/√2 e (√ 2(1− 1 2 |p|)φ .<br />
60
.<br />
.<br />
A<br />
.<br />
.<br />
3<br />
− 2 − 1 −<br />
2<br />
,<br />
2 − 1<br />
A 3<br />
2 ,<br />
.<br />
−<br />
.<br />
1<br />
2<br />
1<br />
2<br />
2<br />
3<br />
2<br />
1<br />
1<br />
2<br />
.<br />
.<br />
.<br />
− i p ϕ<br />
2<br />
A 1,<br />
0<br />
.<br />
.<br />
1<br />
2<br />
A 3<br />
2<br />
,<br />
1<br />
2<br />
.<br />
A 2,<br />
0 A ,<br />
.<br />
1<br />
2 1<br />
.<br />
3<br />
2<br />
.<br />
2<br />
p x 2<br />
Fig. 9: Closed string symmetry charges A j,m ≡ ∮ dz Ω (1)<br />
j,m<br />
. Note <strong>th</strong>ere are also<br />
conjugate charges Ā j,m at <strong>th</strong>e same values of (p x , p ϕ ).<br />
The descent multiplet turns out to be nontrivial for <strong>th</strong>e ghost number G = 1 states<br />
in (4.21): J (0)<br />
j,m = Y +<br />
j,mŌj−1,m and its holomorphic conjugate. Therefore, <strong>th</strong>ere are corresponding<br />
currents Ω (1)<br />
j,m<br />
, conserved up to BRST exact operators, which produce “discrete<br />
charges”<br />
∮<br />
A j,m ≡<br />
dz Ω (1)<br />
j,m , (4.28)<br />
and <strong>th</strong>eir holomorphic conjugates Āj,m, which are conserved up to BRST exact operators.<br />
As described in [66] and in chapt. 14 below, <strong>th</strong>e existence of <strong>th</strong>ese charges have nontrivial<br />
consequences for correlation functions computed in <strong>th</strong>e µ = 0 <strong>th</strong>eory. The quantum<br />
numbers of <strong>th</strong>e charges are plotted in fig. 9.<br />
As at c < 1, an important open problem is to understand better <strong>th</strong>e role of <strong>th</strong>ese<br />
states in quantum gravity. Moreover, an important open problem is to find matrix model<br />
techniques for investigating <strong>th</strong>e O u,n .<br />
5. 2D Critical String Theory<br />
Fur<strong>th</strong>er insight into <strong>th</strong>e spectrum of 2D gravity is obtained when we consider <strong>th</strong>e<br />
string–<strong>th</strong>eory/target space point of view, in which we regard φ as a spacetime coordinate.<br />
The KPZ formula is now interpreted as <strong>th</strong>e on-shell condition for Euclidean target space.<br />
61
5.1. Particles in D Dimensions: QFT as 1D Euclidean Quantum Gravity.<br />
In chapters 2 and 4, we have discussed 2D Euclidean quantum gravity. In <strong>th</strong>is section,<br />
we apply <strong>th</strong>e same techniques to 1D Euclidean Quantum Gravity. While <strong>th</strong>e <strong>th</strong>eory is<br />
trivial as a <strong>th</strong>eory of quantum gravity, it has an important and obvious reinterpretation in<br />
terms of target space Euclidean quantum field <strong>th</strong>eory.<br />
Pa<strong>th</strong> Integral Approach<br />
An example which will illuminate our later considerations is <strong>th</strong>at of a particle moving<br />
<strong>th</strong>rough Euclidean spacetime. This may be <strong>th</strong>ought of as 1D quantum gravity since <strong>th</strong>e<br />
system is described by <strong>th</strong>e action<br />
S = 1 2<br />
We consider <strong>th</strong>e pa<strong>th</strong> integral<br />
∫<br />
dτ √ g(τ)<br />
(g ττ ( dX µ ) ) 2<br />
− m<br />
2<br />
. (5.1)<br />
dτ<br />
A(X I , X f ) =<br />
∫ dg dX<br />
Diff eS , (5.2)<br />
wi<strong>th</strong> boundary conditions X µ i , Xµ f on Xµ . We can fix <strong>th</strong>e gauge by transforming <strong>th</strong>e<br />
einbein to a constant, f ∗ e = s, where s is <strong>th</strong>e single coordinate invariant quantity (i.e.<br />
modulus), namely <strong>th</strong>e leng<strong>th</strong>. The pa<strong>th</strong> integral becomes<br />
A(X i , X f ) =<br />
∝<br />
∫ ∞<br />
0<br />
∫ ∞<br />
since <strong>th</strong>e determinant is proportional to s.<br />
Canonical Approach<br />
0<br />
ds (<br />
det ′ (−s −2 ∂ 2<br />
s 1/2 t ) ) (1−D)/2<br />
e<br />
−(∆X) 2 /2s−m 2 s/2<br />
∫<br />
ds<br />
/2s−m 2 s/2 d D p<br />
∝<br />
s D/2 e−(∆X)2 (2π) D<br />
e ip∆X (5.3)<br />
p 2 + m , 2<br />
Turning to <strong>th</strong>e canonical approach, <strong>th</strong>e action (5.1) has a gauge invariance:<br />
δX = ɛ(τ)X ′ δe(τ) = ɛ ′ (τ)e(τ) + ɛ(τ)e ′ (τ) . (5.4)<br />
We can fix <strong>th</strong>e gauge by putting e = 1 at <strong>th</strong>e price of imposing a constraint. The Wheeler–<br />
DeWitt operator, which generates τ diffeomorphisms, is simply H = p 2 + m 2 , where p µ (τ)<br />
is <strong>th</strong>e field canonically conjugate to x µ (τ). The Wheeler–DeWitt equation is <strong>th</strong>e Euclidean<br />
Klein-Gordon equation:<br />
Hψ(x) =<br />
(− ∂2<br />
∂x 2 + m2 )<br />
ψ = 0 . (5.5)<br />
62
If we isolate one Euclidean coordinate, call it φ, as a special coordinate, <strong>th</strong>en we can<br />
√p<br />
write <strong>th</strong>e Euclidean on-shell wavefunctions as e ipx e ± 2 +m 2φ . As long as <strong>th</strong>ere are no<br />
tachyons in <strong>th</strong>e <strong>th</strong>eory, <strong>th</strong>ese wavefunctions have exponential grow<strong>th</strong> and are not normalizable.<br />
Conversely, <strong>th</strong>e existence of Euclidean on-shell normalizable wavefunctions is a<br />
signal of tachyons in <strong>th</strong>e <strong>th</strong>eory.<br />
In order to describe off-shell physics, we introduce <strong>th</strong>e normalizable states which<br />
diagonalize <strong>th</strong>e Wheeler–DeWitt operator: e ipx+iEφ , wi<strong>th</strong> eigenvalue E 2 + p 2 + m 2 . For<br />
example, in a mixed position-space/momentum-space representation where we Fourier<br />
transform wi<strong>th</strong> respect to all o<strong>th</strong>er coordinates, we may describe <strong>th</strong>e propagator as<br />
∫ ∞<br />
G(ϕ 1 , p; ϕ 2 , −p) = dE e−iEϕ 1<br />
e iEϕ 2<br />
−∞ E 2 + ⃗p 2 + m 2<br />
1<br />
√⃗p<br />
= θ(φ 1 − φ 2 ) √<br />
⃗p 2 + m 2 e− 2 +m 2 |ϕ 1 −ϕ 2 | + [1 ↔ 2] .<br />
(5.6)<br />
Exercise. Back to <strong>th</strong>e wall<br />
What happens if φ is restricted to be semi-infinite? Put a boundary condition <strong>th</strong>at<br />
<strong>th</strong>e wavefunctions vanish at φ = log µ and calculate <strong>th</strong>e analog of (5.6).<br />
Interactions and Topology-Change<br />
One-dimensional quantum gravity from <strong>th</strong>e target space viewpoint provides a useful insight<br />
into <strong>th</strong>e origin of <strong>th</strong>e violation of <strong>th</strong>e Wheeler–DeWitt constraint in topology-changing<br />
processes. In <strong>th</strong>is case, a topology-changing process corresponds to one 0-dimensional space<br />
splitting into two as in<br />
p 1<br />
.<br />
p<br />
2<br />
p = p + p<br />
3 1 2<br />
(5.7)<br />
The “violation of <strong>th</strong>e Wheeler–DeWitt constraint” is simply <strong>th</strong>e familiar fact <strong>th</strong>at if p 2 1 =<br />
p 2 2 = −m 2 are on-shell momenta <strong>th</strong>en in general p 2 3 = (p 1 +p 2 ) 2 ≠ −m 2 will not be on-shell.<br />
This above basic phenomenon can also be realized as <strong>th</strong>e result of a contact term<br />
arising from a singularity at <strong>th</strong>e boundary of “moduli space.” Consider <strong>th</strong>e wavefunction<br />
of a particle <strong>th</strong>at interacts wi<strong>th</strong> an external potential V so <strong>th</strong>at <strong>th</strong>e wavefunction becomes<br />
˜ψ(τ) =<br />
∫ τ<br />
−∞<br />
dτ ′ e −H(τ−τ ′) V ψ , (5.8)<br />
63
where H is <strong>th</strong>e Wheeler–DeWitt operator. Note <strong>th</strong>at<br />
H ˜ψ =<br />
∫ τ<br />
−∞<br />
dτ ′ ∂<br />
(<br />
)<br />
∂τ ′ e −H(τ−τ ′) V ψ = V ψ ≠ 0 , (5.9)<br />
so <strong>th</strong>e condition Hψ = 0 is not preserved under time evolution.<br />
5.2. Strings in D Dimensions: String Theory as 2D Euclidean Quantum Gravity<br />
Nonlinear σ-Model Approach. The particle Lagrangian (5.1) can be generalized<br />
to a string Lagrangian, which we recognize as a 2D nonlinear σ-model, and <strong>th</strong>e quantum<br />
<strong>th</strong>eory involves a pa<strong>th</strong> integral over surfaces. To describe strings propagating in general<br />
manifolds we should in principle consider arbitrary 2d quantum field <strong>th</strong>eories:<br />
S σ = 1 ∫<br />
d 2 z √ (<br />
)<br />
g T (X) + R (2) D(X) + g ab ∂ a X µ ∂ b X ν G µν (X) + · · · , (5.10)<br />
4π<br />
where X µ=1,...,D parametrize a D-dimensional spacetime target space and <strong>th</strong>e ellipsis indicates<br />
a sum over a possibly infinite set of irrelevant operators.<br />
<br />
Pertinent operators?<br />
We are expanding here around <strong>th</strong>e Gaussian fixed point, since we <strong>th</strong>ink of each coordinate<br />
X µ as a Gaussian field. Including arbitrary interactions is a very formal procedure<br />
which must be made well-defined. An infinite sum of irrelevant operators might not be<br />
irrelevant at all, but might be <strong>th</strong>e effect of expanding around <strong>th</strong>e wrong fixed point.<br />
In standard treatments of string <strong>th</strong>eory [70], it is shown <strong>th</strong>at a consistent string <strong>th</strong>eory<br />
can be formulated from models of <strong>th</strong>e above type when <strong>th</strong>ey are conformally invariant (more<br />
precisely, BRST invariant).<br />
vanish, <strong>th</strong>at is, when <strong>th</strong>e spacetime equations of motion,<br />
The model is conformally invariant when <strong>th</strong>e β–functions<br />
βµν G =R µν + 2∇ µ ∇ ν D − ∇ µ T ∇ ν T + · · · = 0<br />
β D = 26 − d − R + 4(∇D) 2 − 4∇ 2 D + (∇T ) 2 − 2T 2 + · · · = 0<br />
3<br />
β T = − 2∇ 2 T + 4∇D ∇T − 4T + · · · = 0 ,<br />
(5.11)<br />
are satisfied. The dots indicate higher order (in <strong>th</strong>e string tension α ′ ) corrections, including<br />
tachyon interactions. These β–function equations <strong>th</strong>emselves follow from an action 19 [72]<br />
S = 1 ∫<br />
2πκ 2 d d x √ (<br />
G e −2D R + 4(∇D) 2 + 26 − d<br />
)<br />
− (∇T ) 2 + 2T 2 + · · · , (5.12)<br />
3<br />
19 The nonderivative dependence on T follows from very general considerations [71].<br />
64
where κ is <strong>th</strong>e string coupling.<br />
Consider <strong>th</strong>e case when <strong>th</strong>e matter conformal field <strong>th</strong>eory S CFT is a product of Gaussian<br />
models,<br />
∫<br />
S CFT =<br />
d 2 z √ ĝ 1<br />
8π ( ˆ∇X µ ) 2 , (5.13)<br />
toge<strong>th</strong>er wi<strong>th</strong> one CTFF field φ (<strong>th</strong>e Chodos–Thorn/Feigen–Fuks field described in<br />
sec. 1.4).<br />
Identifying φ wi<strong>th</strong> a spacetime coordinate in (5.10), we read off from comparison of<br />
(3.2) wi<strong>th</strong> (5.10):<br />
〈T 〉 = 0 , 〈D〉 = Q 2 φ , 〈G µν〉 = δ µν . (5.14)<br />
Substituting (5.14) into (5.11) and working to lowest order in 〈T 〉 shows <strong>th</strong>at β = 0 is<br />
satisfied provided <strong>th</strong>e KPZ formulae described in chapt. 2 are satisfied, so in particular<br />
Q = 2 γ + γ = √ (26 − d)/3 (where d = c + 1 in <strong>th</strong>e critical string interpretation).<br />
Now let us replace <strong>th</strong>e CTFF field by a Liouville field, i.e. instead of a free field we<br />
now have <strong>th</strong>e Liouville interaction term. Comparing actions (3.2) wi<strong>th</strong> (5.10), we find <strong>th</strong>e<br />
same dilaton and metric expectation values as in (5.14), but a new tachyon expectation<br />
value:<br />
〈T 〉 = µ<br />
2γ 2 eγφ , 〈D〉 = Q 2 φ , 〈G µν〉 = δ µν . (5.15)<br />
Conformal background?<br />
The background (5.15) no longer solves <strong>th</strong>e lowest order β-function equations (5.11).<br />
This has been blamed ei<strong>th</strong>er on <strong>th</strong>e possibility of field redefinitions [17], or on <strong>th</strong>e fact <strong>th</strong>at<br />
<strong>th</strong>e above equations are only <strong>th</strong>e lowest order terms in <strong>th</strong>e β-function. We never<strong>th</strong>eless<br />
continue wi<strong>th</strong> <strong>th</strong>is review, since <strong>th</strong>e Liouville <strong>th</strong>eory is conformal.<br />
More subtleties<br />
There are many o<strong>th</strong>er subtleties and caveats associated wi<strong>th</strong> <strong>th</strong>ese assertions. For<br />
example, due to <strong>th</strong>e difficulties of treating <strong>th</strong>eories wi<strong>th</strong> matter central charge c > 1 for<br />
µ > 0, we can really understand only <strong>th</strong>e case of a single gaussian model in (5.13).<br />
The construction of a consistent string <strong>th</strong>eory can be carried out for any conformal<br />
field <strong>th</strong>eory wi<strong>th</strong> total central charge c = 26. In <strong>th</strong>e case of a tensor product of Gaussian<br />
models, we identify each Gaussian model field wi<strong>th</strong> a macroscopic spacetime dimension.<br />
An arbitrary CFT is an abstract version of target spaces made from products of Gaussian<br />
models. The minimal models wi<strong>th</strong> c < 1, for example, can be <strong>th</strong>ought of as generalized<br />
65
Euclidean signature spacetimes. They can be augmented to c = 26 and converted to<br />
consistent target spaces for string propagation by coupling to a Liouville <strong>th</strong>eory since<br />
<strong>th</strong>e Liouville mode has a tunable central charge. 20 For example, by introducing a free<br />
CTFF field we can tune to lower dimensional critical string <strong>th</strong>eories [74]. We have already<br />
discussed some aspects of tensor products of Liouville and matter sectors in sec. 2.1, and<br />
pointed out <strong>th</strong>e relation between critical strings in d = D + 1 dimensions and “non-critical<br />
strings” in D dimensions (when <strong>th</strong>e latter interpretation exists, see footnote after (2.22)).<br />
5.3. 2D String Theory: Euclidean Signature<br />
It is useful to recall at <strong>th</strong>is point <strong>th</strong>e dual interpretations of <strong>th</strong>e <strong>th</strong>eories we consider:<br />
i) matter coupled to 2D quantum gravity.<br />
ii) critical strings moving <strong>th</strong>rough specific background geometries.<br />
In particular, as described in <strong>th</strong>e previous section, gravity coupled to a c = 1 Gaussian<br />
model can be interpreted as a d = 2 critical string <strong>th</strong>eory. The critical string interpretation<br />
of <strong>th</strong>e c = 1 matrix model is subtle and still changing 21 . Our specific action (4.11) describes<br />
strings moving in two Euclidean spacetime dimensions (X, φ), and in <strong>th</strong>e next section we<br />
shall consider its Minkowskian continuations.<br />
In general, <strong>th</strong>e KPZ formula (2.26, 4.6) <strong>th</strong>at determines <strong>th</strong>e gravitational dressing<br />
for an operator coupled to 2d gravity has a dual interpretation as <strong>th</strong>e Euclidean on-shell<br />
condition for string propagation in <strong>th</strong>e critical string target space picture. Recall <strong>th</strong>at for<br />
c = 1, we have Q = 2 √ 2. Thus <strong>th</strong>e operators in (4.12),<br />
e iqX/√2 e √ 2(1∓ 1 2 |q|)φ = e ip X X e 1 2 Qφ+iEφ ,<br />
create states <strong>th</strong>at satisfy <strong>th</strong>e Euclidean on-shell condition<br />
E 2 + p 2 X = 0<br />
for a massless particle (where p φ<br />
= E and p X<br />
= q/ √ 2 are <strong>th</strong>e φ and X momenta). We<br />
recognize <strong>th</strong>at <strong>th</strong>e KPZ formula (written in <strong>th</strong>e form (4.6)) is <strong>th</strong>e dispersion relation for<br />
massless propagation.<br />
20 But it always has <strong>th</strong>e same number of field <strong>th</strong>eoretic degrees of freedom. This remarkable<br />
aspect of Liouville <strong>th</strong>eory has been explored in detail in [7,73].<br />
21 June 1992<br />
66
It should come as no surprise to find massless propagation in d = 2 critical string<br />
<strong>th</strong>eory. In <strong>th</strong>e light cone gauge approach to string <strong>th</strong>eory, <strong>th</strong>ere are physical excitations<br />
associated wi<strong>th</strong> <strong>th</strong>e motion of <strong>th</strong>e string center of mass and wi<strong>th</strong> <strong>th</strong>e transverse oscillations<br />
of <strong>th</strong>e string. In two spacetime dimensions, <strong>th</strong>ere are no transverse oscillations so we expect<br />
to find a single field <strong>th</strong>eoretic degree of freedom. The center of mass degree of freedom,<br />
which is identified wi<strong>th</strong> <strong>th</strong>e tachyon field T (X µ ) of <strong>th</strong>e 26-dimensional critical string, has<br />
mass-squared m 2 = (2 − d)/12 in d dimensions. Thus, in two dimensional string <strong>th</strong>eory<br />
we expect to find one massless field <strong>th</strong>eoretic excitation.<br />
One way to confirm <strong>th</strong>e lightcone statements is to consider <strong>th</strong>e β-function derived<br />
spacetime action (5.12) in a general linear dilaton background in d dimensions, i.e. (5.14)<br />
or (5.15) wi<strong>th</strong> Q 2 = (26 − d)/3 (again d = c + 1 in <strong>th</strong>e critical string interpretation).<br />
Changing variables in (5.12) to T = e D b(x, φ), we find <strong>th</strong>at <strong>th</strong>e tachyon field has action<br />
S T = 1 ∫<br />
dx dφ e −2D( )<br />
(∇T ) 2 − 2T 2 + · · ·<br />
2πκ 2<br />
= 1<br />
2πκ 2 ∫<br />
(<br />
dx dφ (∇b) 2 + ( (∇D) 2 − ∇ 2 D − 2 ) )<br />
b 2 + · · ·<br />
= 1 ∫ (<br />
2πκ 2 dx dφ (∇b) 2 − 2 − d<br />
)<br />
12 b2 + interactions .<br />
In particular for d = 2, <strong>th</strong>e field b is massless.<br />
(5.16)<br />
Remark: We can view <strong>th</strong>e KPZ formula as <strong>th</strong>e on-shell condition for <strong>th</strong>e Euclidean<br />
target space propagator as well for c ≠ 1. Indeed from (4.6) we have<br />
− 1 (<br />
α − Q ) 2<br />
+ ∆X + 1 − c<br />
2 2<br />
24 = 0 , (5.17)<br />
which we read as <strong>th</strong>e Euclidean on-shell condition:<br />
1<br />
2 E2 + 1 2 p2 + 1 2 m2 = 0 . (5.18)<br />
In <strong>th</strong>e c = 1 model, we have seen just above <strong>th</strong>at <strong>th</strong>e analogy<br />
∆ X ←→ 1 1 − c<br />
2 p2 ←→ 1<br />
24<br />
2 m2 (5.19)<br />
is exact, wi<strong>th</strong> m 2 = 0.<br />
Following <strong>th</strong>e particle example we can — in <strong>th</strong>e minisuperspace approximation —<br />
immediately discuss <strong>th</strong>e propagator<br />
∫ ∞<br />
1<br />
G(l 1 , p; l 2 , −p) = dE<br />
E 2 + p 2 + m ψ E(l 2 1 ) ψ E (l 2 )<br />
0<br />
= θ(l 2 − l 1 ) I ωp (2 √ µ l 1 ) K ωp (2 √ µ l 2 ) + [1 ↔ 2] ,<br />
(5.20)<br />
where ω p ≡ + √ p 2 + m 2 . This is <strong>th</strong>e 2D gravity analog of (5.6) in <strong>th</strong>e minisuperspace<br />
approximation. From <strong>th</strong>e point of view of 2D quantum gravity, <strong>th</strong>is is <strong>th</strong>e universe–universe<br />
propagator of <strong>th</strong>ird quantization [75–77].<br />
67
5.4. 2D String Theory: Minkowskian Signature<br />
(4.11).<br />
Now we consider <strong>th</strong>e possible Minkowskian continuations of our Euclidean action<br />
A) X is Euclidean time.<br />
studied.<br />
The c = 1 model has <strong>th</strong>e clearest target space interpretation of <strong>th</strong>e models we have<br />
In particular if we rotate X → it, we can consider t as a Minkowskian time<br />
coordinate. Taking account of <strong>th</strong>e tachyon condensate, we have seen how to get <strong>th</strong>e target<br />
space wavefunctions (e.g. (4.8)). Then <strong>th</strong>e on-shell wavefunctions are, at tree level,<br />
(<br />
e iEt K iE (l) = e iEt l iE<br />
)<br />
Γ(1 + iE) − l−iE<br />
+ O(l 2 ) . (5.21)<br />
Γ(1 − iE)<br />
Physically <strong>th</strong>ese wavefunctions describe <strong>th</strong>e reflection scattering of an incoming tachyon<br />
by <strong>th</strong>e Liouville wall. Since <strong>th</strong>e Bessel function is a sum of incoming and outgoing waves<br />
we may, wi<strong>th</strong>out fur<strong>th</strong>er ado, read off <strong>th</strong>e genus zero 1 → 1 scattering amplitude in <strong>th</strong>e<br />
<strong>th</strong>eory:<br />
S(E) = − Γ(iE)<br />
Γ(−iE) .<br />
In sec. 13.5 below, we will calculate <strong>th</strong>e full nonperturbative S-matrix for <strong>th</strong>is <strong>th</strong>eory.<br />
The scattering cohomology classes are<br />
where ω > 0. Note <strong>th</strong>at<br />
coupling constant<br />
{}}{<br />
V ω ± = c¯c e (±iωt+iωφ)/√ 2<br />
} {{ }<br />
e √ 2φ<br />
wavefunction<br />
, (5.22)<br />
i) In quantum mechanics wavefunctions depend on time as ψ ∼ e −iEt where E ≥ 0 is a<br />
positive energy. When calculating scattering matrices in a pa<strong>th</strong> integral formalism [78]<br />
we insert ψ ∗ out and ψ in respectively for outgoers and incomers. Therefore <strong>th</strong>e vertex<br />
operators create scattering states according to:<br />
V − ω<br />
: incoming rightmover<br />
V + ω : outgoing leftmover . (5.23)<br />
Since we are effectively discussing scattering <strong>th</strong>eory in a half-space, incomers are<br />
rightmovers and outgoers are leftmovers. This is <strong>th</strong>e spacetime version of <strong>th</strong>e Seiberg<br />
bound (3.40).<br />
68
ii) We must work wi<strong>th</strong> macroscopic states to have (plane-wave) normalizable wavefunctions<br />
in Minkowski space, required to set up a sensible scattering <strong>th</strong>eory.<br />
B) φ is Euclidean time.<br />
In <strong>th</strong>is case we must rotate φ → it to obtain a Minkowskian interpretation. Unfortunately,<br />
<strong>th</strong>e rotation is problematic for µ > 0 [8]. The reason is evident from <strong>th</strong>e zero-mode<br />
part of <strong>th</strong>e Liouville pa<strong>th</strong> integral (3.60). If µ > 0, <strong>th</strong>en in <strong>th</strong>e complex φ 0 plane (i.e. zero<br />
modes of φ) <strong>th</strong>ere is a series of “ridges” along <strong>th</strong>e lines Im(φ 0 ) = (2n + 1)π/γ, n ∈ Z,<br />
which invalidate any contour rotation: <strong>th</strong>e right answer cannot be obtained by rotating<br />
φ → it and expanding in a series of δ-functions (except, perhaps, by dumb luck).<br />
These objections disappear if we consider <strong>th</strong>e “free Liouville <strong>th</strong>eory” wi<strong>th</strong> µ = 0.<br />
There is no obstruction to rotating φ → it, where t is a timelike coordinate. The natural<br />
BRST classes are<br />
T ± k = c¯c eik(X±t)/√2 e i√2t , (5.24)<br />
which now have <strong>th</strong>e interpretation<br />
: incoming leftmover k < 0<br />
: outgoing leftmover k > 0<br />
T − k : incoming rightmover k > 0 . : outgoing rightmover k < 0<br />
(5.25)<br />
T + k<br />
T + k<br />
T − k<br />
Since <strong>th</strong>ere is no wall at µ = 0, we can have bo<strong>th</strong> leftmovers and rightmovers. Moreover,<br />
<strong>th</strong>e string coupling becomes time-dependent, κ(t) = κ 0 e i√2t , and <strong>th</strong>e dilaton field is purely<br />
imaginary. 22 Clearly <strong>th</strong>e physics of <strong>th</strong>is model is ra<strong>th</strong>er different from case A) and any<br />
relation between <strong>th</strong>e models is only ma<strong>th</strong>ematical. We will return to <strong>th</strong>is world briefly in<br />
sec. 14.2 .<br />
5.5. Heterodox remarks regarding <strong>th</strong>e “special states”<br />
There are <strong>th</strong>ree reasons why <strong>th</strong>e infinite class of special states is exciting and interesting:<br />
1) They correspond to a large unbroken symmetry group of <strong>th</strong>e string gauge group.<br />
22 In conventional closed string field <strong>th</strong>eory [44], one imposes reality conditions on <strong>th</strong>e string<br />
field forcing <strong>th</strong>e dilaton to be real.<br />
69
2) The only difference in degrees of freedom between strings and fields in 2D is in <strong>th</strong>e<br />
special states. The spacetime meaning of <strong>th</strong>e special states is not understood and<br />
should be stringy and interesting.<br />
3) They enter non-trivially into <strong>th</strong>e 2d black hole metric.<br />
Let us elaborate on <strong>th</strong>ese <strong>th</strong>ree points:<br />
1) In <strong>th</strong>e 26-dimensional bosonic string wi<strong>th</strong> Minkowski space background <strong>th</strong>ere is an<br />
analog of <strong>th</strong>e special states. They are all at zero momentum and <strong>th</strong>eir physical interpretation<br />
is clear. The linearized gauge symmetry of string field <strong>th</strong>eory is<br />
Ψ → Ψ + QΛ + κ[Ψ, Λ] + · · · , (5.26)<br />
where <strong>th</strong>e last term is <strong>th</strong>e string product described in [44], and <strong>th</strong>e infinitesimal<br />
symmetry generator Λ has ghost number G = 1. Ψ represents deviations of <strong>th</strong>e<br />
fields from background values, so a symmetry of <strong>th</strong>e background should take Ψ =<br />
0 → Ψ = 0 and <strong>th</strong>erefore satisfy QΛ = 0, i.e., <strong>th</strong>e symmetry should act linearly on<br />
small deviations from <strong>th</strong>e background, as follows from (5.26). Moreover, modifying<br />
Λ → Λ + Qɛ doesn’t change <strong>th</strong>e linearized action on <strong>th</strong>e on-shell fields. Therefore <strong>th</strong>e<br />
nontrivial BRST classes of ghost number G = 1 correspond to on-shell symmetries of<br />
<strong>th</strong>e string background [66]. In <strong>th</strong>e case of <strong>th</strong>e “special states” of Minkowski space,<br />
<strong>th</strong>ey correspond to <strong>th</strong>e unbroken translation symmetries of <strong>th</strong>e vacuum defined by<br />
Minkowski space. 23 Reasoning by analogy, it would seem <strong>th</strong>at <strong>th</strong>e infinite number of<br />
special states in <strong>th</strong>e 2D string correspond to a much larger symmetry group. It has<br />
been suggested in [69] <strong>th</strong>at <strong>th</strong>is is also related to <strong>th</strong>e fact <strong>th</strong>at in <strong>th</strong>e 2d string <strong>th</strong>ere<br />
are far fewer states in <strong>th</strong>e <strong>th</strong>eory.<br />
2) The vertex operators representing small changes in <strong>th</strong>e tachyon background are just<br />
<strong>th</strong>ose given in (4.12). The question <strong>th</strong>us arises as to <strong>th</strong>e spacetime meaning of <strong>th</strong>e<br />
special state operators. It has been suggested in [49] <strong>th</strong>at <strong>th</strong>ese represent global modes<br />
of spacetime fields which have no propagating degrees of freedom. The basic idea<br />
can be seen by considering 1 + 1 dimensional gauge <strong>th</strong>eories of electromagnetism and<br />
gravity. In 1+1 dimensional (classical) electromagnetism and gravitation, for example,<br />
<strong>th</strong>e fields A µ (x) and G µν (x) have no propagating modes, yet <strong>th</strong>e background electric<br />
holonomy ∮ dσA σ and <strong>th</strong>e circumference of <strong>th</strong>e world ∮ dX √ G XX are gauge-invariant<br />
observables when X is compactified.<br />
23 Toge<strong>th</strong>er wi<strong>th</strong> dual symmetries for <strong>th</strong>e B-field.<br />
70
3) There are indications <strong>th</strong>at understanding special state correlators would aid in <strong>th</strong>e<br />
search for a model wi<strong>th</strong> bo<strong>th</strong> <strong>th</strong>e black hole mass and <strong>th</strong>e cosmological constant<br />
turned on.<br />
For <strong>th</strong>ese reasons <strong>th</strong>e “special states” have been <strong>th</strong>e subject of intense investigation for <strong>th</strong>e<br />
past year and a half. Sadly, some of <strong>th</strong>ese investigations have been ra<strong>th</strong>er misguided.<br />
When we compute BRST cohomology, we must pay proper attention to <strong>th</strong>e boundary<br />
conditions of <strong>th</strong>e fields representing BRST cohomology. In electromagnetism in four<br />
dimensions, for example, BRST cohomology will be represented by plane-wave states of<br />
<strong>th</strong>e gauge field A µ ∼ ξ µ e ik·x , k 2 = k · ξ = 0, representing transverse photons. Of course,<br />
k is real because we want only to consider plane-wave normalizable states. In addition<br />
<strong>th</strong>ere are o<strong>th</strong>er BRST invariant field configurations which are not plane-wave normalizable.<br />
For example, in 1 + 1 electrodynamics on IR 2 we can work in A 1 = 0 gauge, but<br />
<strong>th</strong>en A 0 = Ex for E constant is not normalizable. This corresponds to <strong>th</strong>e Coulomb force.<br />
We should <strong>th</strong>erefore distinguish <strong>th</strong>e scattering cohomology representing states for which<br />
one can scatter and compute an S-matrix, from <strong>th</strong>e background cohomology which represents<br />
gauge-invariant global information which cannot be changed by small wavelike field<br />
perturbations.<br />
This discussion applies to <strong>th</strong>e 2D string. As we have seen, when rotating <strong>th</strong>e coordinate<br />
X to Minkowskian time, <strong>th</strong>e primary matter fields have negative conformal weight.<br />
Thus, since ω must be real to provide plane wave normalizable incoming and outgoing<br />
wavefunctions, <strong>th</strong>e only BRST cohomology classes in <strong>th</strong>e Minkowskian <strong>th</strong>eory wi<strong>th</strong> ω > 0<br />
are <strong>th</strong>ose in (5.22). This reasoning breaks down for <strong>th</strong>e case of zero t-momentum. On <strong>th</strong>e<br />
o<strong>th</strong>er hand before looking for <strong>th</strong>e effects of special state operators like<br />
c¯c P 0,r (∂ ∗ t) P 0,r (∂ ∗ t) 2(1±r)φ<br />
e√<br />
, (5.27)<br />
(where <strong>th</strong>e Seiberg bound implies we must take 1 − r), we must require <strong>th</strong>at <strong>th</strong>e wavefunctions<br />
in question do not change <strong>th</strong>e asymptotic behavior of <strong>th</strong>e lagrangian of <strong>th</strong>e <strong>th</strong>eory.<br />
In fact <strong>th</strong>is is only <strong>th</strong>e case for <strong>th</strong>e operator ∂t ∂t. The o<strong>th</strong>er states have non-normalizable<br />
wavefunctions and <strong>th</strong>us belong to <strong>th</strong>e background cohomology groups. We cannot form<br />
well-defined wavepackets for <strong>th</strong>em and <strong>th</strong>ey will not be changed by scattering processes<br />
since such processes involve wavepacket normalizable quanta from <strong>th</strong>e scattering cohomology.<br />
The special states are very interesting for string <strong>th</strong>eory, but <strong>th</strong>ey have no place in<br />
<strong>th</strong>e wall S-matrix of <strong>th</strong>e 2D string. To paraphrase a warning to previous generations [79]:<br />
71
Those who look for special states in <strong>th</strong>e singularities of <strong>th</strong>e c = 1 S-matrix are like <strong>th</strong>e<br />
man who settled in Casablanca for <strong>th</strong>e waters. They were misinformed.<br />
The situation is ra<strong>th</strong>er more confused for <strong>th</strong>e bulk–scattering matrix described in<br />
chapt. 14.<br />
5.6. Bosonic String Amplitudes and <strong>th</strong>e “c > 1 problem”<br />
In <strong>th</strong>is section we consider some of <strong>th</strong>e “tachyonic” divergences <strong>th</strong>at occur in bosonic<br />
string <strong>th</strong>eories.<br />
First Description<br />
Let us return to <strong>th</strong>e operator formalism description of string amplitudes. In general,<br />
<strong>th</strong>e amplitudes A h,n are meaningless because of <strong>th</strong>e singularities of <strong>th</strong>e string density Ω<br />
on <strong>th</strong>e boundaries of moduli space. A traditional way of avoiding <strong>th</strong>is problem has been<br />
<strong>th</strong>e introduction of supersymmetry. An alternative way around <strong>th</strong>e problem is provided<br />
by low dimensional string <strong>th</strong>eory[4], since in low dimensions <strong>th</strong>e tachyon (which causes <strong>th</strong>e<br />
divergences) becomes massless or massive as we have seen in (5.16). We can see how <strong>th</strong>is<br />
comes about by considering <strong>th</strong>e one-loop partition function in <strong>th</strong>e example of a general<br />
linear dilaton background (i.e. non-zero Q in (5.15)) coupled to some matter conformal<br />
field <strong>th</strong>eory C,<br />
Ω 1,0 ∼ dτ η 2 ∧ d¯τ ¯η 2 Z Liouville⊗C (q, ¯q) . (5.28)<br />
(Note <strong>th</strong>e leading η 2¯η 2 is from <strong>th</strong>e ghosts.) The behavior of <strong>th</strong>e partition function as<br />
q → 0, which accounts for <strong>th</strong>e tachyon divergences of <strong>th</strong>e <strong>th</strong>eory, is obtained by writing<br />
<strong>th</strong>e partition function as a sum over eigenstates of L 0 , L 0 :<br />
Z Liouville⊗C = ∑ i<br />
∫ ∞<br />
0<br />
dE f i (E) (q¯q) 1 2 E2 + 1 8 Q2 +∆ i −26/24 , (5.29)<br />
where f i (E) represents <strong>th</strong>e density of Liouville states in (3.31). Including also <strong>th</strong>e leading<br />
(q¯q) 2/24 from <strong>th</strong>e ghosts in (5.28), we arrive at <strong>th</strong>e condition [7,73] for no tachyonic<br />
divergences:<br />
min ∆i ∈C<br />
{<br />
}<br />
1<br />
2 E2 + 1 8 Q2 + ∆ i − 1 ≥ 0 =⇒ c eff (C) ≡ c − 24 min∆ i ≤ 1 . (5.30)<br />
From <strong>th</strong>is point of view, we see <strong>th</strong>at <strong>th</strong>e problem is not necessarily <strong>th</strong>at c > 1 per se, but<br />
is ra<strong>th</strong>er an issue involving <strong>th</strong>e value of c toge<strong>th</strong>er wi<strong>th</strong> <strong>th</strong>e spectrum of <strong>th</strong>e <strong>th</strong>eory.<br />
72
The condition (5.30) is of course only a necessary condition. We should also worry<br />
about <strong>th</strong>e existence of divergences when operators approach each o<strong>th</strong>er. In <strong>th</strong>is case <strong>th</strong>e<br />
softening of <strong>th</strong>e Liouville operator product expansion discussed above explains <strong>th</strong>e lack of<br />
divergences on <strong>th</strong>e boundaries of punctured moduli space. In particular, if we look at <strong>th</strong>e<br />
operator product of two dressed matter primaries Φ 1 e αφ and Φ 2 e βφ , <strong>th</strong>en from (3.59) we<br />
have<br />
Φ 1 e αφ (z, ¯z) Φ 2 e βφ (w, ¯w)<br />
∼ ∑ ∫ ∞<br />
dE c 1,2,(X,E) |z − w| 2( 1 2 E2 + 1 8 Q2 +∆ X −2) Φ ∆X V E (w, ¯w)<br />
∆ X<br />
0<br />
(5.31)<br />
(where c 1,2,(X,E) is <strong>th</strong>e coefficient of <strong>th</strong>e field Φ ∆X<br />
and its gravitational dressing V E (w, ¯w)<br />
in <strong>th</strong>e operator product expansion of <strong>th</strong>e two above operators). The worst singularity at<br />
z = w comes from <strong>th</strong>e contribution near E = 0,<br />
1<br />
|z − w| 2 |z − w| 1 12 (1−c eff (C)) ,<br />
and is integrable when <strong>th</strong>e condition (5.30) is satisfied. (The case c eff (C) = 1 is a borderline<br />
case. In <strong>th</strong>e c = 1 model, it turns out <strong>th</strong>at c 1,2,E → 0 as E → 0.)<br />
Based on <strong>th</strong>ese two examples, we may guess <strong>th</strong>at all bosonic string amplitudes in fact<br />
do exist when (5.30) is satisfied.<br />
The matrix model approach to 2D string <strong>th</strong>eory has<br />
<strong>th</strong>e great virtue of confirming <strong>th</strong>is, and moreover gives an infinite dimensional space of<br />
background perturbations.<br />
Second Description<br />
We can also describe <strong>th</strong>ese divergences from <strong>th</strong>e point of view of <strong>th</strong>e spacetime <strong>th</strong>eory<br />
by interpreting <strong>th</strong>e norm of <strong>th</strong>e plumbing fixture coordinate q as |q| = e −s , where s is a<br />
proper time coordinate such as introduced following (5.3) for <strong>th</strong>e field <strong>th</strong>eory propagator.<br />
From <strong>th</strong>is point of view, we see <strong>th</strong>at <strong>th</strong>e divergences are due to on-shell tachyons and<br />
massless particles. When (5.30) is satisfied as a strict inequality, we see <strong>th</strong>at <strong>th</strong>e amplitudes<br />
are finite because only zero-momentum massive particles flow.<br />
particles present a special case at c = 1, but <strong>th</strong>ey are derivatively coupled.<br />
73<br />
As usual, <strong>th</strong>e massless
Fig. 10: The case of <strong>th</strong>e exploding worldsheet. Since every order in perturbation<br />
<strong>th</strong>eory adds a hole to <strong>th</strong>e surface, <strong>th</strong>is is an overly optimistic rendering. Summing<br />
up such a perturbation expansion, <strong>th</strong>e worldsheet on scales larger <strong>th</strong>an <strong>th</strong>e cutoff<br />
is all holes [7].<br />
Third Description [7]<br />
We may also consider <strong>th</strong>e above phenomenon from <strong>th</strong>e worldsheet point of view. We<br />
consider <strong>th</strong>e Liouville <strong>th</strong>eory coupled to some conformal field <strong>th</strong>eory C such <strong>th</strong>at <strong>th</strong>e total<br />
central charge is 26. The conformal field <strong>th</strong>eory C is assumed to have a spectrum bounded<br />
from below: <strong>th</strong>at is, we are considering strings in Euclidean space. In general we expect<br />
Euclidean propagators in Liouville <strong>th</strong>eory to have <strong>th</strong>e form<br />
∫<br />
f(E)<br />
dE<br />
E 2 + p 2 + m ψ 2 E (l 1) ψ E (l 2 ) , (5.32)<br />
where as explained in sec. 5.3 we identify<br />
p 2 + m 2 = ∆ X + 1 − c<br />
24<br />
. (5.33)<br />
Suppose <strong>th</strong>e unit operator flows <strong>th</strong>rough <strong>th</strong>e loop and c > 1. Then <strong>th</strong>ere is a zero in <strong>th</strong>e<br />
propagator for E real. That is, <strong>th</strong>ere exists an on-shell, normalizable (macroscopic) state<br />
74
in Euclidean space. As in 1D, we should suspect <strong>th</strong>at <strong>th</strong>ere are tachyons in <strong>th</strong>e <strong>th</strong>eory.<br />
Recalling <strong>th</strong>e semiclassical Liouville pictures discussed in chapt. 3, <strong>th</strong>e troubles caused<br />
by <strong>th</strong>ese states have a graphical worldsheet illustration. Insertion of an operator dressed<br />
by a macroscopic Liouville state is not a local disturbance to <strong>th</strong>e surface: it creates a<br />
macroscopic hole and tears <strong>th</strong>e surface apart. In any lattice description of a c > 1 model,<br />
unless we fine-tune <strong>th</strong>ere will be nonzero couplings to <strong>th</strong>e operator <strong>th</strong>at creates <strong>th</strong>e onshell<br />
macroscopic state whose existence we have established. In particular, using <strong>th</strong>e<br />
KPZ dressing formulae of sec. 2.2, we see <strong>th</strong>at <strong>th</strong>e cosmological constant operator itself<br />
becomes a macroscopic state. Bringing down any such operators from <strong>th</strong>e exponential in a<br />
perturbative expansion of <strong>th</strong>e pa<strong>th</strong> integral, we see <strong>th</strong>at <strong>th</strong>e typical resulting “worldsheet”<br />
would look as depicted in fig. 10. Evidently a worldsheet description of <strong>th</strong>e physics is no<br />
longer most appropriate. Once more, <strong>th</strong>e condition <strong>th</strong>at would prevent <strong>th</strong>is explosion is<br />
(5.30).<br />
6. Discretized surfaces, matrix models, and <strong>th</strong>e continuum limit<br />
Now <strong>th</strong>at we have some idea of <strong>th</strong>e physics we are looking for, we will study <strong>th</strong>e<br />
“experimental” results of <strong>th</strong>e matrix model. The next four chapters are devoted to defining<br />
<strong>th</strong>e continuum limit for <strong>th</strong>e models of c < 1 matter coupled to gravity associated wi<strong>th</strong> <strong>th</strong>e<br />
one matrix model. We mention matrix chains briefly. We will emphasize bo<strong>th</strong> <strong>th</strong>e role of<br />
macroscopic loops and also <strong>th</strong>e fermionic formulation of <strong>th</strong>e matrix model, which lies at<br />
<strong>th</strong>e heart of <strong>th</strong>e exactly solvable nature of <strong>th</strong>ese models.<br />
6.1. Discretized surfaces<br />
We begin by considering a “D = 0 dimensional string <strong>th</strong>eory”, i.e. a pure <strong>th</strong>eory of<br />
surfaces wi<strong>th</strong> no coupling to additional “matter” degrees of freedom on <strong>th</strong>e string worldsheet.<br />
This is equivalent to <strong>th</strong>e propagation of strings in a non-existent embedding space.<br />
For partition function we take<br />
Z = ∑ ∫<br />
Dg e −βA + γχ , (6.1)<br />
h<br />
where <strong>th</strong>e sum over topologies is represented by <strong>th</strong>e summation over h, <strong>th</strong>e number of<br />
handles of <strong>th</strong>e surface, and <strong>th</strong>e action consists of couplings to <strong>th</strong>e area A = ∫ √ g, and to<br />
∫<br />
<strong>th</strong>e Euler character χ = 1 √<br />
4π g R = 2 − 2h.<br />
75
Fig. 11: A piece of a random triangulation of a surface. Each of <strong>th</strong>e triangular<br />
faces is dual to a <strong>th</strong>ree point vertex of a quantum mechanical matrix model.<br />
The integral ∫ Dg over <strong>th</strong>e metric on <strong>th</strong>e surface in (6.1) is difficult to calculate in<br />
general. The most progress in <strong>th</strong>e continuum has been made via <strong>th</strong>e Liouville approach<br />
which we briefly reviewed in chapt. 2. If we discretize <strong>th</strong>e surface, on <strong>th</strong>e o<strong>th</strong>er hand, it<br />
turns out <strong>th</strong>at (6.1) is much easier to calculate, even before removing <strong>th</strong>e finite cutoff. We<br />
consider in particular a “random triangulation” of <strong>th</strong>e surface [80], in which <strong>th</strong>e surface is<br />
constructed from triangles, as in fig. 11. The triangles are designated to be equilateral, 24<br />
so <strong>th</strong>at <strong>th</strong>ere is negative (positive) curvature at vertices i where <strong>th</strong>e number N i of incident<br />
triangles is more (less) <strong>th</strong>an six, and zero curvature when N i = 6. The summation over<br />
all such random triangulations is <strong>th</strong>us <strong>th</strong>e discrete analog to <strong>th</strong>e integral ∫ Dg over all<br />
possible geometries,<br />
∑<br />
genus h<br />
∫<br />
Dg<br />
→<br />
∑<br />
random<br />
triangulations<br />
. (6.2)<br />
The discrete counterpart to <strong>th</strong>e infinitesimal volume element √ g is σ i = N i /3, so <strong>th</strong>at<br />
<strong>th</strong>e total area |S| = ∑ i σ i just counts <strong>th</strong>e total number of triangles, each designated to<br />
have unit area. (The factor of 1/3 in <strong>th</strong>e definition of σ i is because each triangle has <strong>th</strong>ree<br />
24 We point out <strong>th</strong>at <strong>th</strong>is constitutes a basic difference from <strong>th</strong>e Regge calculus, in which <strong>th</strong>e<br />
link leng<strong>th</strong>s are geometric degrees of freedom.<br />
Here <strong>th</strong>e geometry is encoded entirely into <strong>th</strong>e<br />
coordination numbers of <strong>th</strong>e vertices. This restriction of degrees of freedom roughly corresponds<br />
to fixing a coordinate gauge, hence we integrate only over <strong>th</strong>e gauge-invariant moduli of <strong>th</strong>e<br />
surfaces.<br />
76
vertices and is counted <strong>th</strong>ree times.) The discrete counterpart to <strong>th</strong>e Ricci scalar R at<br />
vertex i is R i = 2π(6 − N i )/N i , so <strong>th</strong>at<br />
∫ √g R →<br />
∑<br />
i<br />
4π(1 − N i /6) = 4π(V − 1 F ) = 4π(V − E + F ) = 4πχ .<br />
2<br />
Here we have used <strong>th</strong>e simplicial definition which gives <strong>th</strong>e Euler character χ in terms of<br />
<strong>th</strong>e total number of vertices, edges, and faces V , E, and F of <strong>th</strong>e triangulation (and we<br />
have used <strong>th</strong>e relation 3F = 2E obeyed by triangulations of surfaces, since each face has<br />
<strong>th</strong>ree edges each of which is shared by two faces).<br />
In <strong>th</strong>e above, triangles do not play an essential role and may be replaced by any set<br />
of polygons. General random polygonulations of surfaces wi<strong>th</strong> appropriate fine tuning of<br />
couplings may, as we shall see, have more general critical behavior, but can in particular<br />
always reproduce <strong>th</strong>e pure gravity behavior of triangulations in <strong>th</strong>e continuum limit.<br />
6.2. Matrix models<br />
We now demonstrate how <strong>th</strong>e integral over geometry in (6.1) may be performed in<br />
its discretized form as a sum over random triangulations. The trick is to use a certain<br />
matrix integral as a generating functional for random triangulations. The essential idea<br />
goes back to work [81] on <strong>th</strong>e large N limit of QCD, followed by work on <strong>th</strong>e saddle point<br />
approximation [82].<br />
We first recall <strong>th</strong>e (Feynman) diagrammatic expansion of <strong>th</strong>e (0-dimensional) field<br />
<strong>th</strong>eory integral.<br />
∫ ∞<br />
−∞<br />
dϕ<br />
√<br />
2π<br />
e −ϕ2 /2 + λϕ 4 /4! , (6.3)<br />
where ϕ is an ordinary real number. 25 In a formal perturbation series in λ, we would need<br />
to evaluate integrals such as<br />
λ n<br />
n!<br />
Up to overall normalization we can write<br />
∫<br />
ϕ<br />
e −ϕ2 /2 ϕ 2k =<br />
∂2k<br />
∫<br />
∂J 2k ∫ϕ<br />
ϕ<br />
e −ϕ2 /2 ( ϕ 4 ) n<br />
. (6.4)<br />
4!<br />
e −ϕ2 /2 + Jϕ ∣ ∣ ∣∣<br />
J=0<br />
= ∂2k<br />
∂J 2k eJ 2 /2 ∣ ∣<br />
∣∣J=0 . (6.5)<br />
25 The integral is understood to be defined by analytic continuation to negative λ.<br />
77
Since<br />
∂ /2<br />
eJ2<br />
∂J<br />
=<br />
Je J2 /2 , applications of ∂/∂J in <strong>th</strong>e above need to be paired so <strong>th</strong>at<br />
any factors of J are removed before finally setting J = 0. Therefore if we represent each<br />
“vertex” λϕ 4 diagrammatically as a point wi<strong>th</strong> four emerging lines (see fig. 12b), <strong>th</strong>en (6.4)<br />
simply counts <strong>th</strong>e number of ways to group such objects in pairs. Diagrammatically we<br />
represent <strong>th</strong>e possible pairings by connecting lines between paired vertices. The connecting<br />
line is known as <strong>th</strong>e propagator 〈ϕ ϕ〉 (see fig. 12a) and <strong>th</strong>e diagrammatic rule we have<br />
described for connecting vertices in pairs is known in field <strong>th</strong>eory as <strong>th</strong>e Wick expansion.<br />
(a)<br />
(b)<br />
Fig. 12: (a) <strong>th</strong>e scalar propagator. (b) <strong>th</strong>e scalar four-point vertex.<br />
When <strong>th</strong>e number of vertices n becomes large, <strong>th</strong>e allowed diagrams begin to form<br />
a mesh reminiscent of a 2-dimensional surface. Such diagrams do not yet have enough<br />
structure to specify a Riemann surface. The additional structure is given by widening <strong>th</strong>e<br />
propagators to ribbons (to give so-called “fat” graphs). From <strong>th</strong>e standpoint of (6.3), <strong>th</strong>e<br />
required extra structure is given by replacing <strong>th</strong>e scalar ϕ by an N × N hermitian matrix<br />
M i j. The analog of (6.5) is given by adding indices and traces:<br />
∫<br />
M<br />
e −trM 2 /2 M<br />
i 1<br />
j1 · · · M i n<br />
jn =<br />
=<br />
∂ ∂<br />
· · · e −trM 2 /2 + trJM ∣ ∣∣J=0<br />
∂J j1 i 1<br />
∂J jn i n<br />
∂ ∂<br />
· · · e trJ 2 /2 ∣ ∣∣J=0 ,<br />
∂J j1 i 1<br />
∂J jn i n<br />
(6.6)<br />
where <strong>th</strong>e source J i j is as well now a matrix. The measure in (6.6) is <strong>th</strong>e invariant dM =<br />
∏<br />
i dM i ∏<br />
i i
indices is represented in fig. 13 by <strong>th</strong>e double lines 26 and it is understood <strong>th</strong>at <strong>th</strong>e sense of<br />
<strong>th</strong>e arrows is to be preserved when linking toge<strong>th</strong>er vertices. The resulting diagrams are<br />
similar to <strong>th</strong>ose of <strong>th</strong>e scalar <strong>th</strong>eory, except <strong>th</strong>at each external line has an associated index<br />
i, and each internal closed line corresponds to a summation over an index j = 1, . . . , N.<br />
The “<strong>th</strong>ickened” structure is now sufficient to associate a Riemann surface to each diagram,<br />
because <strong>th</strong>e closed internal loops uniquely specify locations and orientations of faces.<br />
−→−− −−←−<br />
→− ↑↓ →− −←<br />
↑<br />
↓ −←<br />
(a)<br />
(b)<br />
Fig. 13: (a) <strong>th</strong>e hermitian matrix propagator. (b) <strong>th</strong>e hermitian matrix four-point vertex.<br />
To make contact wi<strong>th</strong> <strong>th</strong>e random triangulations discussed earlier, we consider <strong>th</strong>e<br />
diagrammatic expansion of <strong>th</strong>e matrix integral<br />
e Z ∫<br />
=<br />
− 1 2 trM 2 +<br />
dM e<br />
g √<br />
N<br />
trM 3 (6.8)<br />
(wi<strong>th</strong> M an N × N hermitian matrix, and <strong>th</strong>e integral again understood to be defined by<br />
analytic continuation in <strong>th</strong>e coupling g.) The term of order g n in a power series expansion<br />
counts <strong>th</strong>e number of diagrams constructed wi<strong>th</strong> n 3-point vertices. The dual to such a<br />
diagram (in which each face, edge, and vertex is associated respectively to a dual vertex,<br />
edge, and face) is identically a random triangulation inscribed on some orientable Riemann<br />
surface (fig. 11). We see <strong>th</strong>at <strong>th</strong>e matrix integral (6.8) automatically generates all such<br />
random triangulations. 27<br />
Since each triangle has unit area, <strong>th</strong>e area of <strong>th</strong>e surface is just n. We can <strong>th</strong>us make<br />
formal identification wi<strong>th</strong> (6.1) by setting g = e −β . Actually <strong>th</strong>e matrix integral generates<br />
bo<strong>th</strong> connected and disconnected surfaces, so we have written e Z on <strong>th</strong>e left hand side of<br />
26 This is <strong>th</strong>e same notation employed in <strong>th</strong>e large N expansion of QCD [81].<br />
27 Had we used real symmetric matrices ra<strong>th</strong>er <strong>th</strong>an <strong>th</strong>e hermitian matrices M, <strong>th</strong>e two indices<br />
would be indistinguishable and <strong>th</strong>ere would be no arrows in <strong>th</strong>e propagators and vertices of fig. 13.<br />
Such orientationless vertices and propagators generate an ensemble of bo<strong>th</strong> orientable and nonorientable<br />
surfaces, and have been studied, e.g., in [83].<br />
79
(6.8). As familiar from field <strong>th</strong>eory, <strong>th</strong>e exponential of <strong>th</strong>e connected diagrams generates<br />
all diagrams, so Z as defined above represents contributions only from connected surfaces.<br />
We see <strong>th</strong>at <strong>th</strong>e free energy from <strong>th</strong>e matrix model point of view is actually <strong>th</strong>e partition<br />
function Z from <strong>th</strong>e 2d gravity point of view.<br />
There is additional information contained in N, <strong>th</strong>e size of <strong>th</strong>e matrix. If we change<br />
variables M → M √ N in (6.8), <strong>th</strong>e matrix action becomes N tr(− 1 2 trM 2 +gtrM 3 ), wi<strong>th</strong> an<br />
overall factor of N. 28 This normalization makes it easy to count <strong>th</strong>e power of N associated<br />
to any diagram. Each vertex contributes a factor of N, each propagator (edge) contributes<br />
a factor of N −1 (because <strong>th</strong>e propagator is <strong>th</strong>e inverse of <strong>th</strong>e quadratic term), and each<br />
closed loop (face) contributes a factor of N due to <strong>th</strong>e associated index summation. Thus<br />
each diagram has an overall factor<br />
N V −E+F = N χ = N 2−2h , (6.9)<br />
where χ is <strong>th</strong>e Euler character of <strong>th</strong>e surface associated to <strong>th</strong>e diagram. We observe <strong>th</strong>at<br />
<strong>th</strong>e value N = e γ makes contact wi<strong>th</strong> <strong>th</strong>e coupling γ in (6.1). In conclusion, if we take<br />
g = e −β and N = e γ , we can formally identify <strong>th</strong>e continuum limit of <strong>th</strong>e partition function<br />
Z in (6.8) wi<strong>th</strong> <strong>th</strong>e Z defined in (6.1). The metric for <strong>th</strong>e discretized formulation is not<br />
smoo<strong>th</strong>, but one can imagine how an effective metric on larger scales could arise after<br />
averaging over local irregularities. In <strong>th</strong>e next section, we shall see explicitly how <strong>th</strong>is<br />
works.<br />
(Actually (6.8) automatically calculates (6.1) wi<strong>th</strong> <strong>th</strong>e measure factor in (6.2) corrected<br />
to ∑ 1<br />
S |G(S)|<br />
, where |G(S)| is <strong>th</strong>e order of <strong>th</strong>e (discrete) group of symmetries of <strong>th</strong>e<br />
triangulation S. This is familiar from field <strong>th</strong>eory where diagrams wi<strong>th</strong> symmetry result<br />
in an incomplete cancellation of 1/n!’s such as in (6.4) and (6.7). The symmetry group<br />
G(S) is <strong>th</strong>e discrete analog of <strong>th</strong>e isometry group of a continuum manifold.)<br />
The graphical expansion of (6.8) enumerates graphs as shown in fig. 11, where <strong>th</strong>e<br />
triangular faces <strong>th</strong>at constitute <strong>th</strong>e random triangulation are dual to <strong>th</strong>e 3-point vertices.<br />
Had we instead used 4-point vertices as in fig. 13b, <strong>th</strong>en <strong>th</strong>e dual surface would have square<br />
faces (a “random squarulation” of <strong>th</strong>e surface), and higher point vertices (g k /N k/2−1 )trM k<br />
in <strong>th</strong>e matrix model would result in more general “random polygonulations” of surfaces.<br />
28 Al<strong>th</strong>ough we could as well rescale M → M/g to pull out an overall factor of N/g 2 , note <strong>th</strong>at<br />
N remains distinguished from <strong>th</strong>e coupling g in <strong>th</strong>e model since it enters as well into <strong>th</strong>e traces<br />
via <strong>th</strong>e N × N size of <strong>th</strong>e matrix.<br />
80
(The powers of N associated wi<strong>th</strong> <strong>th</strong>e couplings are chosen so <strong>th</strong>at <strong>th</strong>e rescaling M →<br />
M √ N results in an overall factor of N multiplying <strong>th</strong>e action. The argument leading to<br />
(6.9) <strong>th</strong>us remains valid, and <strong>th</strong>e power of N continues to measure <strong>th</strong>e Euler character of<br />
a surface constructed from arbitrary polygons.) The different possibilities for generating<br />
vertices constitute additional degrees of freedom <strong>th</strong>at can be realized as <strong>th</strong>e coupling of 2d<br />
gravity to different varieties of matter in <strong>th</strong>e continuum limit.<br />
6.3. The continuum limit<br />
From (6.9), it follows <strong>th</strong>at we may expand Z in powers of N,<br />
Z(g) = N 2 Z 0 (g) + Z 1 (g) + N −2 Z 2 (g) + . . . = ∑ N 2−2h Z h (g) , (6.10)<br />
where Z h gives <strong>th</strong>e contribution from surfaces of genus h. In <strong>th</strong>e conventional large N<br />
limit, we take N → ∞ and only Z 0 , <strong>th</strong>e planar surface (genus zero) contribution, survives.<br />
Z 0 itself may be expanded in a perturbation series in <strong>th</strong>e coupling g, and for large order n<br />
behaves as (see [84] for a review)<br />
Z 0 (g) ∼ ∑ n<br />
n Γ str−3 (g/g c ) n ∼ (g c − g) 2−Γ str<br />
. (6.11)<br />
These series <strong>th</strong>us have <strong>th</strong>e property <strong>th</strong>at <strong>th</strong>ey diverge as g approaches some critical coupling<br />
g c . We can extract <strong>th</strong>e continuum limit of <strong>th</strong>ese surfaces by tuning g → g c . This is because<br />
<strong>th</strong>e expectation value of <strong>th</strong>e area of a surface is given by<br />
〈A〉 = 〈n〉 = ∂ ∂g ln Z 0(g) ∼ 1<br />
g − g c<br />
(recall <strong>th</strong>at <strong>th</strong>e area is proportional to <strong>th</strong>e number of vertices n, which appears as <strong>th</strong>e<br />
power of <strong>th</strong>e coupling in <strong>th</strong>e factor g n associated to each graph). As g → g c , we see <strong>th</strong>at<br />
A → ∞ so <strong>th</strong>at we may rescale <strong>th</strong>e area of <strong>th</strong>e individual triangles to zero, <strong>th</strong>us giving a<br />
continuum surface wi<strong>th</strong> finite area. Intuitively, by tuning <strong>th</strong>e coupling to <strong>th</strong>e point where<br />
<strong>th</strong>e perturbation series diverges, <strong>th</strong>e integral becomes dominated by diagrams wi<strong>th</strong> infinite<br />
numbers of vertices, and <strong>th</strong>is is precisely what we need to define continuum surfaces.<br />
There is no direct proof as yet <strong>th</strong>at <strong>th</strong>is procedure for defining continuum surfaces is<br />
“correct,” i.e. <strong>th</strong>at it coincides wi<strong>th</strong> <strong>th</strong>e continuum definition (6.1). We are able, however,<br />
to compare properties of <strong>th</strong>e partition function and correlation functions calculated by<br />
matrix model me<strong>th</strong>ods wi<strong>th</strong> <strong>th</strong>ose (few) properties <strong>th</strong>at can be calculated directly in <strong>th</strong>e<br />
81
continuum, as reviewed in preceding chapters. This gives implicit confirmation <strong>th</strong>at <strong>th</strong>e<br />
matrix model approach is sensible and gives reason to believe o<strong>th</strong>er results derivable by<br />
matrix model techniques (e.g. for higher genus) <strong>th</strong>at are not obtainable at all by continuum<br />
me<strong>th</strong>ods. In sec. 8.2 , we shall give a more precise formulation of what we mean by <strong>th</strong>e<br />
continuum limit.<br />
One of <strong>th</strong>e properties of <strong>th</strong>ese models derivable via <strong>th</strong>e continuum Liouville approach<br />
is a “critical exponent” Γ str , defined in terms of <strong>th</strong>e area dependence of <strong>th</strong>e partition<br />
function for surfaces of fixed large area A as<br />
Z(A) ∼ A (Γ str−2)χ/2−1 . (6.12)<br />
Recall <strong>th</strong>at <strong>th</strong>e unitary discrete series of conformal field <strong>th</strong>eories is labelled by an integer<br />
m ≥ 2 and has central charge D = 1 − 6/m(m + 1) (for a review, see e.g. [20]), where <strong>th</strong>e<br />
central charge is normalized such <strong>th</strong>at D = 1 corresponds to a single free boson. If we<br />
couple conformal field <strong>th</strong>eories wi<strong>th</strong> <strong>th</strong>ese fractional values of D to 2d gravity, we see from<br />
(2.22) <strong>th</strong>e continuum Liouville <strong>th</strong>eory prediction for <strong>th</strong>e exponent Γ str<br />
Γ str = 1 12<br />
(<br />
D − 1 −<br />
√<br />
(D − 1)(D − 25)<br />
)<br />
= −<br />
1<br />
m . (6.13)<br />
The case m = 2, for example, corresponds to D = 0 and hence Γ str = − 1 2<br />
for pure gravity.<br />
The next case m = 3 corresponds to D = 1/2, i.e. to a 1/2–boson or fermion. This is <strong>th</strong>e<br />
conformal field <strong>th</strong>eory of <strong>th</strong>e critical Ising model, and we learn from (6.13) <strong>th</strong>at <strong>th</strong>e Ising<br />
model coupled to 2d gravity has Γ str = − 1 3 .<br />
In chapt. 7 we shall present <strong>th</strong>e solution to <strong>th</strong>e matrix model formulation of <strong>th</strong>e problem,<br />
and <strong>th</strong>e value of <strong>th</strong>e exponent Γ str provides a coarse means of determining which<br />
specific continuum model results from taking <strong>th</strong>e continuum limit of a particular matrix<br />
model. Indeed <strong>th</strong>e coincidence of Γ str and o<strong>th</strong>er scaling exponents (defined in chapt. 2)<br />
calculated from <strong>th</strong>e two points of view were originally <strong>th</strong>e only evidence <strong>th</strong>at <strong>th</strong>e continuum<br />
limit of matrix models was a suitable definition for <strong>th</strong>e continuum problem of interest.<br />
Subsequently, <strong>th</strong>e simplicity of matrix model results for correlation functions has spurred<br />
a rapid evolution of continuum Liouville technology so <strong>th</strong>at as well many correlation functions<br />
can be computed in bo<strong>th</strong> approaches and are found to coincide. 29<br />
29 In particular, following <strong>th</strong>e confirmation <strong>th</strong>at <strong>th</strong>e matrix model approach reproduced <strong>th</strong>e<br />
scaling results of [33], some 3-point couplings for order parameters at genus zero were calculated<br />
82
6.4. A first look at <strong>th</strong>e double scaling limit<br />
Thus far we have discussed <strong>th</strong>e naive N → ∞ limit which retains only planar surfaces.<br />
It turns out <strong>th</strong>at <strong>th</strong>e successive coefficient functions Z h (g) in (6.10) as well diverge at<br />
<strong>th</strong>e same critical value of <strong>th</strong>e coupling g = g c (<strong>th</strong>is should not be surprising since <strong>th</strong>e<br />
divergence of <strong>th</strong>e perturbation series is a local phenomenon and should not depend on<br />
global properties such as <strong>th</strong>e effective genus of a diagram). As we saw in (2.21), for <strong>th</strong>e<br />
higher genus contributions (6.11) is generalized to<br />
Z h (g) ∼ ∑ n<br />
n (Γ str−2)χ/2−1 (g/g c ) n ∼ (g c − g) (2−Γ str)χ/2 . (6.14)<br />
We see <strong>th</strong>at <strong>th</strong>e contributions from higher genus (χ < 0) are enhanced as g → g c . This<br />
suggests <strong>th</strong>at if we take <strong>th</strong>e limits N → ∞ and g → g c not independently, but toge<strong>th</strong>er<br />
in a correlated manner, we may compensate <strong>th</strong>e large N high genus suppression wi<strong>th</strong> a<br />
g → g c enhancement. This would result in a coherent contribution from all genus surfaces<br />
[4–6].<br />
To see how <strong>th</strong>is works explicitly, we write <strong>th</strong>e leading singular piece of <strong>th</strong>e Z h (g) as<br />
Z h (g) ∼ f h (g − g c ) (2−Γstr)χ/2 .<br />
Then in terms of<br />
κ −1 ≡ N(g − g c ) (2−Γ str)/2 , (6.15)<br />
<strong>th</strong>e expansion (6.10) can be rewritten 30<br />
Z = κ −2 f 0 + f 1 + κ 2 f 2 + . . . = ∑ h<br />
κ 2h−2 f h . (6.16)<br />
The desired result is <strong>th</strong>us obtained by taking <strong>th</strong>e limits N → ∞, g → g c while holding<br />
fixed <strong>th</strong>e “renormalized” string coupling κ of (6.15). This is known as <strong>th</strong>e “double scaling<br />
limit.” (6.16) is an asymptotic expansion for κ → 0. In secs. 7.3, 7.4 below, we show how<br />
to find a function Z(κ) wi<strong>th</strong> identically <strong>th</strong>at asymptotic expansion.<br />
in [85] from <strong>th</strong>e standpoint of ADE face models on fluctuating lattices. The connection to KdV (to<br />
be reviewed in sec. 7.4 here) was made in [34], and <strong>th</strong>en general correlations of order parameters<br />
(not yet known in <strong>th</strong>e continuum) were calculated in [50]. Using techniques described in sec. 3.9,<br />
continuum calculations of <strong>th</strong>e correlation functions (when <strong>th</strong>ey can be done) have been found to<br />
agree wi<strong>th</strong> <strong>th</strong>e matrix model (for a review, see [14]). For D = 1, <strong>th</strong>e matrix model approach of<br />
[86–89] was used in [90,91] (also [92,93]) to calculate a variety of correlation functions. These<br />
were also calculated in <strong>th</strong>e collective field approach [76–97] where up to 6-point amplitudes were<br />
derived, and found to be in agreement wi<strong>th</strong> <strong>th</strong>e Liouville results of [48].<br />
30 Strictly speaking <strong>th</strong>e first two terms here have additional non-universal pieces <strong>th</strong>at need to<br />
be subtracted off.<br />
83
7. Matrix Model Technology I: Me<strong>th</strong>od of Or<strong>th</strong>ogonal Polynomials<br />
The large N limit of <strong>th</strong>e matrix models considered here was originally solved by saddle<br />
point me<strong>th</strong>ods in [82]. In <strong>th</strong>is chapter we shall instead present <strong>th</strong>e or<strong>th</strong>ogonal polynomial<br />
solution to <strong>th</strong>e problem ([84] and references <strong>th</strong>erein) since it extends readily to subleading<br />
order in N (higher genus corrections).<br />
7.1. Or<strong>th</strong>ogonal polynomials<br />
In order to justify <strong>th</strong>e claims made at <strong>th</strong>e end of sec. 6.4, we introduce some formalism<br />
to solve <strong>th</strong>e matrix models. We begin by rewriting <strong>th</strong>e partition function (6.8) in <strong>th</strong>e form<br />
e Z ∫<br />
= dM e −trV (M) ∫<br />
=<br />
N<br />
∏<br />
i=1<br />
dλ i ∆ 2 (λ) e − ∑ i V (λ i)<br />
, (7.1)<br />
where we now allow a general polynomial potential V (M). In (7.1), <strong>th</strong>e λ i ’s are <strong>th</strong>e N<br />
eigenvalues of <strong>th</strong>e hermitian matrix M, and<br />
∆(λ) = ∏ j − λ i ) (7.2)<br />
i
The now-standard me<strong>th</strong>od for solving (7.1) makes use of an infinite set of polynomials<br />
P n (λ), or<strong>th</strong>ogonal wi<strong>th</strong> respect to <strong>th</strong>e measure<br />
∫ ∞<br />
−∞<br />
dλ e −V (λ) P n (λ) P m (λ) = h n δ nm . (7.3)<br />
The P n ’s are known as or<strong>th</strong>ogonal polynomials and are functions of a single real variable λ.<br />
Their normalization is given by having leading term P n (λ) = λ n + . . ., hence <strong>th</strong>e constant<br />
h n on <strong>th</strong>e r.h.s. of (7.3). Due to <strong>th</strong>e relation<br />
∆(λ) = det λ j−1<br />
i<br />
= det P j−1 (λ i ) (7.4)<br />
(recall <strong>th</strong>at arbitrary polynomials may be built up by adding linear combinations of<br />
preceding columns, a procedure <strong>th</strong>at leaves <strong>th</strong>e determinant unchanged), <strong>th</strong>e polynomials<br />
P n can be employed to solve (7.1). We substitute <strong>th</strong>e determinant det P j−1 (λ i ) =<br />
∑ (−1)<br />
π ∏ k P i k −1(λ k ) for each of <strong>th</strong>e ∆(λ)’s in (7.1) (where <strong>th</strong>e sum is over permutations<br />
i k and (−1) π is <strong>th</strong>e parity of <strong>th</strong>e permutation). The integrals over individual λ i ’s factorize,<br />
and due to or<strong>th</strong>ogonality <strong>th</strong>e only contributions are from terms wi<strong>th</strong> all P i (λ j )’s paired.<br />
There are N! such terms so (7.1) reduces to<br />
e Z N−1<br />
∏<br />
= N! h i = N! h N 0<br />
i=0<br />
N−1<br />
∏<br />
k=1<br />
f N−k<br />
k<br />
, (7.5)<br />
where we have defined f k ≡ h k /h k−1 .<br />
In <strong>th</strong>e naive large N limit (<strong>th</strong>e planar limit), <strong>th</strong>e rescaled index k/N becomes a<br />
continuous variable ξ <strong>th</strong>at runs from 0 to 1, and f k /N becomes a continuous function<br />
f(ξ). In <strong>th</strong>is limit, <strong>th</strong>e partition function (up to an irrelevant additive constant) reduces<br />
to a simple one-dimensional integral:<br />
1<br />
N 2 Z = 1 N<br />
∑<br />
(1 − k/N) ln f k ∼<br />
k<br />
∫ 1<br />
0<br />
dξ(1 − ξ) ln f(ξ) . (7.6)<br />
To derive <strong>th</strong>e functional form for f(ξ), we assume for simplicity <strong>th</strong>at <strong>th</strong>e potential<br />
V (λ) in (7.3) is even. Since <strong>th</strong>e P i ’s form a complete set of basis vectors in <strong>th</strong>e space<br />
of polynomials, it is clear <strong>th</strong>at λP n (λ) must be expressible as a linear combination of<br />
lower P i ’s, λP n (λ) = ∑ n+1<br />
i=0 a i P i (λ) (wi<strong>th</strong> a i = h −1 ∫<br />
i e −V λP n P i ). In fact, <strong>th</strong>e or<strong>th</strong>ogonal<br />
polynomials satisfy <strong>th</strong>e simple recursion relation,<br />
λP n = P n+1 + r n P n−1 , (7.7)<br />
85
wi<strong>th</strong> r n a scalar coefficient independent of λ. This is because any term proportional to P n<br />
in <strong>th</strong>e above vanishes due to <strong>th</strong>e assumption <strong>th</strong>at <strong>th</strong>e potential is even, ∫ e −V λ P n P n = 0.<br />
Terms proportional to P i for i < n − 1 also vanish since ∫ e −V P n λ P i = 0 (recall λP i is a<br />
polynomial of order at most i + 1 so is or<strong>th</strong>ogonal to P n for i + 1 < n).<br />
By considering <strong>th</strong>e quantity P n λP n−1 wi<strong>th</strong> λ paired alternately wi<strong>th</strong> <strong>th</strong>e preceding or<br />
succeeding polynomial, we derive<br />
∫<br />
e −V P n λ P n−1 = r n h n−1 = h n .<br />
This shows <strong>th</strong>at <strong>th</strong>e ratio f n = h n /h n−1 for <strong>th</strong>is simple case 32 is identically <strong>th</strong>e coefficient<br />
defined by (7.7), f n = r n . Similarly if we pair <strong>th</strong>e λ in P ′ n λ P n before and afterwards,<br />
integration by parts gives<br />
∫<br />
∫<br />
nh n = e −V P n ′ λ P n =<br />
e −V P ′ n r n P n−1 = r n<br />
∫<br />
This is <strong>th</strong>e key relation <strong>th</strong>at will allow us to determine r n .<br />
7.2. The genus zero partition function<br />
e −V V ′ P n P n−1 . (7.8)<br />
Our intent now is to find an expression for f n = r n and substitute into (7.6) to<br />
calculate a partition function. For definiteness, we take as example <strong>th</strong>e potential<br />
wi<strong>th</strong> derivative<br />
V (λ) = 1<br />
)<br />
(λ 2 + λ4<br />
2g N + b λ6<br />
,<br />
N 2<br />
gV ′ (λ) = λ + 2 λ3<br />
N<br />
+ 3b<br />
λ5<br />
N 2 . (7.9)<br />
The right hand side of (7.8) involves terms of <strong>th</strong>e form ∫ e −V λ 2p−1 P n P n−1 . According to<br />
(7.7), <strong>th</strong>ese may be visualized as “walks” of 2p − 1 steps (p − 1 steps up and p steps down)<br />
starting at n and ending at n − 1, where each step down from m to m − 1 receives a factor<br />
of r m and each step up receives a factor of unity. The total number of such walks is given<br />
by ( )<br />
2p−1<br />
p , and each results in a final factor of hn−1 (from <strong>th</strong>e integral ∫ e −V P n−1 P n−1 )<br />
which combines wi<strong>th</strong> <strong>th</strong>e r n to cancel <strong>th</strong>e h n on <strong>th</strong>e left hand side of (7.8). For <strong>th</strong>e potential<br />
(7.9), (7.8) <strong>th</strong>us gives<br />
gn = r n + 2 N r n(r n+1 + r n + r n−1 ) + 3b (10 rrr terms) . (7.10)<br />
N<br />
2<br />
32 In o<strong>th</strong>er models, e.g. multimatrix models, f n = h n /h n−1 has a more complicated dependence<br />
on recursion coefficients.<br />
86
(The 10 rrr terms start wi<strong>th</strong> r n (r 2 n + r 2 n+1 + r 2 n−1 + . . .) and may be found e.g. in [98].)<br />
As mentioned before (7.6), in <strong>th</strong>e large N limit <strong>th</strong>e index n becomes a continuous<br />
variable ξ, and we have r n /N → r(ξ) and r n±1 /N → r(ξ ± ε), where ε ≡ 1/N. To leading<br />
order in 1/N, (7.10) reduces to<br />
gξ = r + 6r 2 + 30br 3 = W (r)<br />
= g c + 1 2 W ′′ | r=rc<br />
(<br />
r(ξ) − rc<br />
) 2<br />
+ . . . .<br />
(7.11)<br />
In <strong>th</strong>e second line, we have expanded W (r) for r near a critical point r c at which W ′ | r=rc =<br />
0 (which always exists wi<strong>th</strong>out any fine tuning of <strong>th</strong>e parameter b), and g c ≡ W (r c ). We<br />
see from (7.11) <strong>th</strong>at<br />
r − r c ∼ (g c − gξ) 1/2 .<br />
For a general potential V (λ) = 1 ∑<br />
2g p a p λ 2p in (7.9), we would have<br />
W (r) = ∑ p<br />
a p<br />
(2p − 1)!<br />
(p − 1)! 2 rp . (7.12)<br />
To make contact wi<strong>th</strong> <strong>th</strong>e 2d gravity ideas of chapt. 6, let us suppose more generally<br />
<strong>th</strong>at <strong>th</strong>e leading singular behavior of f(ξ) ( = r(ξ) ) for large N is<br />
f(ξ) − f c ∼ (g c − gξ) −Γ str<br />
(7.13)<br />
for g near some g c (and ξ near 1). (We shall see <strong>th</strong>at Γ str in <strong>th</strong>e above coincides wi<strong>th</strong> <strong>th</strong>e<br />
critical exponent Γ str defined in (6.12).) The behavior of (7.6) for g near g c is <strong>th</strong>en<br />
∫<br />
1<br />
1<br />
N 2 Z ∼<br />
0<br />
dξ (1 − ξ)(g c − gξ) −Γ str<br />
∼ (1 − ξ)(g c − gξ) −Γ str+1<br />
∣ 1 +<br />
0<br />
∼ (g c − g) −Γ str+2 ∼ ∑ n<br />
∫ 1<br />
0<br />
n Γ str−3 (g/g c ) n .<br />
dξ (g c − gξ) −Γ str+1<br />
(7.14)<br />
Comparison wi<strong>th</strong> (6.12) shows <strong>th</strong>at <strong>th</strong>e large area (large n) behavior identifies <strong>th</strong>e exponent<br />
Γ str in (7.13) wi<strong>th</strong> <strong>th</strong>e critical exponent defined earlier.<br />
derivative of Z wi<strong>th</strong> respect to x = g c − g has leading singular behavior<br />
We also note <strong>th</strong>at <strong>th</strong>e second<br />
Z ′′ ∼ (g c − g) −Γ str<br />
∼ f(1) . (7.15)<br />
From (7.13) and (7.14) we see <strong>th</strong>at <strong>th</strong>e behavior in (7.11) implies a critical exponent<br />
Γ str = −1/2. From (6.13), we see <strong>th</strong>at <strong>th</strong>is corresponds to <strong>th</strong>e case D = 0, i.e. to pure<br />
87
gravity. It is natural <strong>th</strong>at pure gravity should be present for a generic potential. Wi<strong>th</strong><br />
fine tuning of <strong>th</strong>e parameter b in (7.9), we can achieve a higher order critical point, wi<strong>th</strong><br />
W ′ | r=rc = W ′′ | r=rc = 0, and hence <strong>th</strong>e r.h.s. of (7.11) would instead begin wi<strong>th</strong> an (r−r c ) 3<br />
term. By <strong>th</strong>e same argument starting from (7.13), <strong>th</strong>is would result in a critical exponent<br />
Γ str = −1/3.<br />
Wi<strong>th</strong> a general potential V (M) in (7.1), we have enough parameters to<br />
achieve an m <strong>th</strong> order critical point [99] at which <strong>th</strong>e first m − 1 derivatives of W (r) vanish<br />
at r = r c . The behavior is <strong>th</strong>en r − r c ∼ (g c − gξ) 1/m wi<strong>th</strong> associated critical exponent<br />
Γ str = −1/m. As anticipated at <strong>th</strong>e end of sec. 6.2, we see <strong>th</strong>at more general polynomial<br />
matrix interactions provide <strong>th</strong>e necessary degrees of freedom to result in matter coupled<br />
to 2d gravity in <strong>th</strong>e continuum limit.<br />
7.3. The all genus partition function<br />
We now search for ano<strong>th</strong>er solution to (7.10) and its generalizations <strong>th</strong>at describes <strong>th</strong>e<br />
contribution of all genus surfaces to <strong>th</strong>e partition function (7.6). We shall retain higher<br />
order terms in 1/N in (7.10) so <strong>th</strong>at e.g. (7.11) instead reads<br />
gξ = W (r) + 2r(ξ) ( r(ξ + ε) + r(ξ − ε) − 2r(ξ) )<br />
= g c + 1 2 W ′′ | r=rc<br />
(<br />
r(ξ) − rc<br />
) 2<br />
+ 2r(ξ)<br />
(<br />
r(ξ + ε) + r(ξ − ε) − 2r(ξ)<br />
)<br />
+ . . . .<br />
(7.16)<br />
As suggested at <strong>th</strong>e end of sec. 6.4, we shall simultaneously let N → ∞ and g → g c in<br />
a particular way.<br />
Since g − g c has dimension [leng<strong>th</strong>] 2 , it is convenient to introduce a<br />
parameter a wi<strong>th</strong> dimension leng<strong>th</strong> and let g − g c = κ −4/5 a 2 , wi<strong>th</strong> a → 0. Our ansatz<br />
for a coherent large N limit will be to take ε ≡ 1/N = a 5/2 so <strong>th</strong>at <strong>th</strong>e quantity κ −1 =<br />
(g − g c ) 5/4 N remains finite as g → g c and N → ∞.<br />
Moreover since <strong>th</strong>e integral (7.6) is dominated by ξ near 1 in <strong>th</strong>is limit, it is convenient<br />
to change variables from ξ to z, defined by g c − gξ = a 2 z. Our scaling ansatz in <strong>th</strong>is region<br />
is r(ξ) = r c + au(z). If we substitute <strong>th</strong>ese definitions into (7.11), <strong>th</strong>e leading terms are of<br />
order a 2 and result in <strong>th</strong>e relation u 2 ∼ z. To include <strong>th</strong>e higher derivative terms, we note<br />
<strong>th</strong>at<br />
r(ξ + ε) + r(ξ − ε) − 2r(ξ) ∼ ε 2 ∂2 r<br />
∂ξ 2 = a ∂2<br />
∂z 2 au(z) ∼ a2 u ′′ ,<br />
where we have used ε(∂/∂ξ) = −ga 1/2 (∂/∂z) (which follows from <strong>th</strong>e above change of<br />
variables from ξ to z). Substituting into (7.16), <strong>th</strong>e vanishing of <strong>th</strong>e coefficient of a 2<br />
implies <strong>th</strong>e differential equation<br />
z = u 2 − 1 3 u′′ (7.17)<br />
88
(after a suitable rescaling of u and z). In (7.15), we saw <strong>th</strong>at <strong>th</strong>e second derivative of <strong>th</strong>e<br />
partition function (<strong>th</strong>e “specific heat”) has leading singular behavior given by f(ξ) wi<strong>th</strong><br />
ξ = 1, and <strong>th</strong>us by u(z) for z = (g − g c )/a 2 = κ −4/5 . The solution to (7.17) characterizes<br />
<strong>th</strong>e behavior of <strong>th</strong>e partition function of pure gravity to all orders in <strong>th</strong>e genus expansion.<br />
(Notice <strong>th</strong>at <strong>th</strong>e leading term is u ∼ z 1/2 so after two integrations <strong>th</strong>e leading term in Z<br />
is z 5/2 = κ −2 , consistent wi<strong>th</strong> (6.16).)<br />
Eq. (7.17) is known in <strong>th</strong>e ma<strong>th</strong>ematical literature as <strong>th</strong>e Painlevé I equation. The<br />
perturbative solution in powers of z −5/2 = κ 2 takes <strong>th</strong>e form u = z 1/2 (1− ∑ k=1 u kz −5k/2 ),<br />
where <strong>th</strong>e u k are all positive. 33 This verifies for <strong>th</strong>is model <strong>th</strong>e claims made in eqs. (6.14–<br />
6.16). For large k, <strong>th</strong>e u k go asymptotically as (2k)!, so <strong>th</strong>e solution for u(z) is not Borel<br />
summable (for a review of <strong>th</strong>ese issues in <strong>th</strong>e context of 2d gravity, see e.g. [100]). Our<br />
arguments in chapt. 6 show only <strong>th</strong>at <strong>th</strong>e matrix model results should agree wi<strong>th</strong> 2d gravity<br />
order by order in perturbation <strong>th</strong>eory. How to insure <strong>th</strong>at we are studying nonperturbative<br />
gravity as opposed to nonperturbative matrix models is still an open question. Some of <strong>th</strong>e<br />
constraints <strong>th</strong>at <strong>th</strong>e solution to (7.17) should satisfy are reviewed in [101]. In particular it<br />
is known <strong>th</strong>at real solutions to (7.17) cannot satisfy <strong>th</strong>e Schwinger–Dyson (loop) equations<br />
for <strong>th</strong>e <strong>th</strong>eory.<br />
In <strong>th</strong>e case of <strong>th</strong>e next higher multicritical point, wi<strong>th</strong> b in (7.11) adjusted so <strong>th</strong>at<br />
W ′ = W ′′ = 0 at r = r c , we have W (r) ∼ g c + 1 6 W ′′′ | r=rc (r − r c ) 3 + . . . and critical<br />
exponent Γ str = −1/3. In general, we take g − g c = κ 2/(Γ str−2) a 2 , and ε = 1/N = a 2−Γ str<br />
so <strong>th</strong>at <strong>th</strong>e combination<br />
(g − g c ) 1−Γ str/2 N = κ −1 (7.18)<br />
is fixed in <strong>th</strong>e limit a → 0. The value ξ = 1 now corresponds to z = κ 2/(Γ str−2) , so <strong>th</strong>e<br />
string coupling κ 2 = z Γ str−2 . The general scaling scaling ansatz is r(ξ) = r c + a −2Γ str<br />
u(z),<br />
and <strong>th</strong>e change of variables from ξ to z gives ε(∂/∂ξ) = −ga −Γ str<br />
(∂/∂z).<br />
For <strong>th</strong>e case Γ str = −1/3, <strong>th</strong>is means in particular <strong>th</strong>at r(ξ) = r c + a 2/3 u(z), κ 2 =<br />
z −7/3 , and ε(∂/∂ξ) = −ga 1/3 ∂ . Substituting into <strong>th</strong>e large N limit of (7.10) gives (again<br />
∂z<br />
after suitable rescaling of u and z)<br />
z = u 3 − uu ′′ − 1 2 (u′ ) 2 + α u ′′′′ , (7.19)<br />
33 The first term, i.e. <strong>th</strong>e contribution from <strong>th</strong>e sphere, is dominated by a regular part which<br />
has opposite sign. This is removed by taking an additional derivative of u, giving a series all of<br />
whose terms have <strong>th</strong>e same sign — negative in <strong>th</strong>e conventions of (7.17). The o<strong>th</strong>er solution, wi<strong>th</strong><br />
leading term −z 1/2 , has an expansion wi<strong>th</strong> alternating sign which is presumably Borel summable,<br />
but not physically relevant.<br />
89
wi<strong>th</strong> α = 1 10 . The solution to (7.19) takes <strong>th</strong>e form u = z1/3 (1+ ∑ k u k z −7k/3 ). It turns out<br />
<strong>th</strong>at <strong>th</strong>e coefficients u k in <strong>th</strong>e perturbative expansion of <strong>th</strong>e solution to (7.19) are positive<br />
definite only for α < 1<br />
12 , so <strong>th</strong>e 3<strong>th</strong> order multicritical point does not describe a unitary<br />
<strong>th</strong>eory of matter coupled to gravity. Al<strong>th</strong>ough from (6.13) we see <strong>th</strong>at <strong>th</strong>e critical exponent<br />
Γ str = −1/3 coincides wi<strong>th</strong> <strong>th</strong>at predicted for <strong>th</strong>e (unitary) Ising model coupled to gravity,<br />
it turns out [102,98,103] <strong>th</strong>at (7.19) wi<strong>th</strong> α = 1<br />
10<br />
instead describes <strong>th</strong>e conformal field<br />
<strong>th</strong>eory of <strong>th</strong>e Yang–Lee edge singularity (a critical point obtained by coupling <strong>th</strong>e Ising<br />
model to a particular value of imaginary magnetic field) coupled to gravity. The specific<br />
heat of <strong>th</strong>e conventional critical Ising model coupled to gravity turns out (see sec. 7.5 ) to<br />
be as well determined by <strong>th</strong>e differential equation (7.19), but instead wi<strong>th</strong> α = 2<br />
27 .<br />
For <strong>th</strong>e general m <strong>th</strong> order critical point of <strong>th</strong>e potential W (r),<br />
W (r) = g c − α(r c − r) m , (7.20)<br />
we have seen <strong>th</strong>at <strong>th</strong>e associated model of matter coupled to gravity has critical exponent<br />
Γ str = −1/m. Wi<strong>th</strong> scaling ansatz r(ξ) = r c + a 2/m u(z), we find leading behavior u ∼<br />
z 1/m (and Z ∼ z 2+1/m = κ −2 as expected). The differential equation <strong>th</strong>at results from<br />
substituting <strong>th</strong>e double scaling behaviors given before (7.19) into <strong>th</strong>e generalized version<br />
of (7.10) turns out to be <strong>th</strong>e m <strong>th</strong> member of <strong>th</strong>e KdV hierarchy of differential equations (of<br />
which Painlevé I results for m = 2). In <strong>th</strong>e next section, we shall provide some marginal<br />
insight into why <strong>th</strong>is structure emerges.<br />
The one-matrix models reproduce <strong>th</strong>e (2, 2m−1) minimal models (in <strong>th</strong>e nomenclature<br />
mentioned after (2.22)) coupled to quantum gravity. The remaining (p, q) models coupled<br />
to gravity can be realized in terms of multi-matrix models (to be defined in sec. 7.6).<br />
7.4. The Douglas Equations and <strong>th</strong>e KdV hierarchy<br />
We now wish to describe superficially why <strong>th</strong>e KdV hierarchy of differential equations<br />
plays a role in 2d gravity. To <strong>th</strong>is end it is convenient to switch from <strong>th</strong>e basis of or<strong>th</strong>ogonal<br />
polynomials P n employed in sec. 7.1 to a basis of or<strong>th</strong>onormal polynomials Π n (λ) =<br />
P n (λ)/ √ h n <strong>th</strong>at satisfy<br />
∫ ∞<br />
−∞<br />
In terms of <strong>th</strong>e Π n , eq. (7.7) becomes<br />
λΠ n =<br />
√<br />
hn+1<br />
h n<br />
= Q nm Π m .<br />
Π n + r n<br />
√<br />
hn−1<br />
h n<br />
dλ e −V Π n Π m = δ nm . (7.21)<br />
Π n−1 = √ r n+1 Π n+1 + √ r n Π n−1<br />
90
In matrix notation, we write <strong>th</strong>is as λΠ = QΠ, where <strong>th</strong>e matrix Q has components<br />
Q nm = √ r m δ m,n+1 + √ r n δ m+1,n . (7.22)<br />
Due to <strong>th</strong>e or<strong>th</strong>onormality property (7.21), we see <strong>th</strong>at ∫ e −V λΠ n Π m = Q nm = Q mn , and<br />
Q is a symmetric matrix. In <strong>th</strong>e continuum limit, Q will <strong>th</strong>erefore become a hermitian<br />
operator.<br />
To see how <strong>th</strong>is works explicitly [34,104], we substitute <strong>th</strong>e scaling ansatz r(ξ) =<br />
r c + a 2/m u(z) for <strong>th</strong>e m <strong>th</strong> multicritical model into (7.22),<br />
Q → (r c + a 2/m u(z)) 1/2 e ε ∂ ∂ξ<br />
+ e<br />
−ε ∂ ∂ξ<br />
(rc + a 2/m u(z)) 1/2 .<br />
Wi<strong>th</strong> <strong>th</strong>e substitution ε ∂ ∂ξ → −ga1/m ∂ ∂z<br />
, we find <strong>th</strong>e leading terms<br />
Q = 2r 1/2<br />
c<br />
+ a2/m<br />
√<br />
rc<br />
(u + r c κ 2 ∂ 2 z ) , (7.23)<br />
of which <strong>th</strong>e first is a non-universal constant and <strong>th</strong>e second is a hermitian 2 nd order<br />
differential operator.<br />
The o<strong>th</strong>er matrix <strong>th</strong>at naturally arises is defined by differentiation,<br />
∂<br />
∂λ Π n = A nm Π m , (7.24)<br />
and automatically satisfies [A, Q] = 1. The matrix A does not have any particular symmetry<br />
or antisymmetry properties so it is convenient to correct it to a matrix P <strong>th</strong>at satisfies<br />
<strong>th</strong>e same commutator as A. From our definitions, it follows <strong>th</strong>at<br />
∫ ∂ (<br />
0 = Πn Π m e ) −V ⇒ A + A T = V ′ (Q) ,<br />
∂λ<br />
where we have differentiated term by term and used ∫ e −V λ l Π n Π m = (Q l ) nm . The matrix<br />
P ≡ A − 1 2 V ′ (Q) = 1 2 (A − AT ) is <strong>th</strong>erefore anti-symmetric and satisfies<br />
[<br />
P, Q<br />
]<br />
= 1 . (7.25)<br />
To determine <strong>th</strong>e order of <strong>th</strong>e differential operator Q in <strong>th</strong>e continuum limit, let<br />
us assume for example <strong>th</strong>at <strong>th</strong>e potential V is of order 2l, i.e. V = ∑ l<br />
k=0 a k λ 2k . For<br />
m > n, <strong>th</strong>e integral A mn = ∫ e −V Π n<br />
∂<br />
∂λ Π m = ∫ e −V V ′ Π n Π m may be nonvanishing for<br />
m − n ≤ 2l − 1. That means <strong>th</strong>at P mn ≠ 0 for |m − n| ≤ 2l − 1, and <strong>th</strong>us has enough<br />
91
parameters to result in a (2l−1) st order differential operator in <strong>th</strong>e continuum. The single<br />
condition W ′ = 0 results in P tuned to a 3 rd order operator, and <strong>th</strong>e l − 1 conditions<br />
W ′ = . . . = W (l−1) = 0 allow P to be realized as a (2l−1) st order differential operator. In<br />
(7.23), we see <strong>th</strong>at <strong>th</strong>e universal part of Q after suitable rescaling takes <strong>th</strong>e form Q = d 2 −u.<br />
For <strong>th</strong>e simple critical point W ′ = 0, <strong>th</strong>e continuum limit of P is <strong>th</strong>e antihermitian operator<br />
P = d 3 − 3 4<br />
{u, d}, and <strong>th</strong>e commutator<br />
1 = [P, Q] = 4R ′ 2 =<br />
( 3<br />
4 u2 − 1 4 u′′ ) ′<br />
(7.26)<br />
is easily integrated wi<strong>th</strong> respect to z to give an equation equivalent to (7.17), <strong>th</strong>e string<br />
equation for pure gravity. In (7.26), <strong>th</strong>e notation R 2 is conventional for <strong>th</strong>e first member<br />
of <strong>th</strong>e ordinary KdV hierarchy. The emergence of <strong>th</strong>e KdV hierarchy in <strong>th</strong>is context is<br />
due to <strong>th</strong>e natural occurrence of <strong>th</strong>e fundamental commutator relation (7.25), which also<br />
occurs in <strong>th</strong>e Lax representation of <strong>th</strong>e KdV equations. (The topological gravity approach<br />
has as well been shown at leng<strong>th</strong> to be equivalent to KdV, for a review see [105,106].)<br />
In general <strong>th</strong>e differential equations<br />
[P, Q] = 1 (7.27)<br />
<strong>th</strong>at follow from (7.25) may be determined directly in <strong>th</strong>e continuum. Given an operator Q,<br />
<strong>th</strong>e differential operator P <strong>th</strong>at can satisfy <strong>th</strong>is commutator is constructed as a “fractional<br />
power” of <strong>th</strong>e operator Q. This me<strong>th</strong>od of formulating <strong>th</strong>e continuum <strong>th</strong>eory has a beautiful<br />
generalization to a larger class of <strong>th</strong>eories, which is defined in <strong>th</strong>e following two sections.<br />
7.5. Ising Model<br />
The first extension of <strong>th</strong>e me<strong>th</strong>od of or<strong>th</strong>ogonal polynomials occurs in <strong>th</strong>e solution of<br />
<strong>th</strong>e Ising model. The partition function of <strong>th</strong>e Ising model on a random surface can be<br />
formulated using <strong>th</strong>e two-matrix model:<br />
e Z ∫<br />
=<br />
dU dV e −tr( U 2 + V 2 − 2c UV + g N (eH U 4 + e −H V 4 ) ) , (7.28)<br />
where U and V are hermitian N × N matrices and H is a constant. In <strong>th</strong>e diagrammatic<br />
expansion of <strong>th</strong>e right hand side, we now have two different quartic vertices of <strong>th</strong>e type depicted<br />
in fig. 13b, corresponding to insertions of U 4 and V 4 . The propagator is determined<br />
by <strong>th</strong>e inverse of <strong>th</strong>e quadratic term,<br />
( ) −1<br />
1 −c<br />
= 1 ( )<br />
1 c<br />
−c 1 1 − c 2 c 1<br />
92<br />
.
We see <strong>th</strong>at double lines connecting vertices of <strong>th</strong>e same type (ei<strong>th</strong>er generated by U 4 or<br />
V 4 ) receive a factor of 1/(1 − c 2 ), while <strong>th</strong>ose connecting U 4 vertices to V 4 vertices receive<br />
a factor of c/(1 − c 2 ).<br />
lattice.<br />
This is identically <strong>th</strong>e structure necessary to realize <strong>th</strong>e Ising model on a random<br />
Recall <strong>th</strong>at <strong>th</strong>e Ising model is defined to have a spin σ = ±1 at each site of<br />
a lattice, wi<strong>th</strong> an interaction σ i σ j between nearest neighbor sites 〈ij〉. This interaction<br />
takes one value for equal spins and ano<strong>th</strong>er value for unequal spins.<br />
Up to an overall<br />
additive constant to <strong>th</strong>e free energy, <strong>th</strong>e diagrammatic expansion of (7.28) results in <strong>th</strong>e<br />
2d partition function<br />
Z =<br />
∑<br />
lattices<br />
∑<br />
spin<br />
configurations<br />
e β ∑ 〈ij〉 σ i σ j + H ∑ i σ i<br />
where H is <strong>th</strong>e magnetic field. The weights for equal and unequal neighboring spins are<br />
e ±β , so fixing <strong>th</strong>e ratio e 2β = 1/c relates <strong>th</strong>e parameter c in (7.28) to <strong>th</strong>e temperature<br />
β. It turns out <strong>th</strong>at <strong>th</strong>e Ising model is much easier to solve summed over random lattices<br />
<strong>th</strong>an on a regular lattice, and in particular is solvable even in <strong>th</strong>e presence of a magnetic<br />
field. This is because <strong>th</strong>ere is much more symmetry after coupling to gravity, since <strong>th</strong>e<br />
complicating details of any particular lattice (e.g. square) are effectively integrated out.<br />
We briefly outline <strong>th</strong>e me<strong>th</strong>od for solving (7.28) (see [107,98,103] for more details).<br />
By me<strong>th</strong>ods similar to <strong>th</strong>ose used to derive (7.1), we can write (7.28) in terms of <strong>th</strong>e<br />
eigenvalues x i and y i of U and V ,<br />
e Z ∫<br />
=<br />
∆(x) ∆(y) ∏ i<br />
dx i dy i e −W (x i, y i ) .<br />
where W (x i , y i ) ≡ x 2 i + y2 i − 2c x iy i + g N (eH x 4 i + e−H yi 4 ). The polynomials we define for<br />
<strong>th</strong>is problem are or<strong>th</strong>ogonal wi<strong>th</strong> respect to <strong>th</strong>e bilocal measure<br />
∫<br />
dx dy e −W (x,y) P n (x) Q m (y) = h n δ nm<br />
(where P n ≠ Q n for H ≠ 0). The result for <strong>th</strong>e partition function is identical to (7.5),<br />
e Z ∝ ∏ i<br />
h i ∝ ∏ i<br />
f N−i<br />
i<br />
,<br />
93
and <strong>th</strong>e recursion relations for <strong>th</strong>is case generalize (7.7),<br />
x P n (x) = P n+1 + r n P n−1 + s n P n−3 ,<br />
y Q m (y) = Q m+1 + q m Q m−1 + t m Q m−3 .<br />
We still have f n ≡ h n /h n−1 , and f n can be determined in terms of <strong>th</strong>e above recursion<br />
coefficients (al<strong>th</strong>ough <strong>th</strong>e formulae are more complicated <strong>th</strong>an in <strong>th</strong>e one-matrix case).<br />
After we substitute <strong>th</strong>e scaling ansätze described in sec. 7.3, <strong>th</strong>e formula for <strong>th</strong>e scaling<br />
part of f is derived via straightforward algebra. The result is <strong>th</strong>at <strong>th</strong>e specific heat u ∝ Z ′′<br />
is given by (7.19) wi<strong>th</strong> α = 2<br />
27 .<br />
7.6. Multi-Matrix Models<br />
We now expand slightly <strong>th</strong>e class of models from single matrix to multi-matrix models.<br />
The free energy of a particular (q − 1)-matrix model, generalizing (7.1), may be written<br />
[108]<br />
Z = ln<br />
∫<br />
= ln<br />
∫ q−1<br />
∏<br />
i=1<br />
dM i e −tr(∑ q−1<br />
i=1 V i(M i ) − ∑ q−2<br />
i=1 c i M i M i+1<br />
)<br />
∏<br />
dλ (α)<br />
i<br />
i=1,q−1<br />
α=1,N<br />
∆(λ 1 ) e − ∑ i,α V (<br />
(α)<br />
i λ<br />
i<br />
)<br />
+<br />
∑i,α c i λ (α)<br />
i<br />
λ (α)<br />
i+1<br />
∆(λq−1 ) ,<br />
(7.29)<br />
where <strong>th</strong>e M i (for i = 1, . . . , q − 1) are N × N hermitian matrices, <strong>th</strong>e λ (α)<br />
i<br />
(α = 1, . . . , N)<br />
are <strong>th</strong>eir eigenvalues, and ∆(λ i ) = ∏ α
7.7. Continuum Solution of <strong>th</strong>e Matrix Chains<br />
Following [108], we can introduce operators Q i and P i <strong>th</strong>at represent <strong>th</strong>e insertions<br />
of λ i and d/dλ i respectively in <strong>th</strong>e integral (7.29). These operators necessarily satisfy<br />
[ ]<br />
Pi , Q i = 1. In <strong>th</strong>e N → ∞ limit, we have seen (following [34]) <strong>th</strong>at P and Q become<br />
differential operators of finite order, say p, q respectively (where we assume p > q), and<br />
<strong>th</strong>ese continue to satisfy (7.27). In <strong>th</strong>e continuum limit of <strong>th</strong>e matrix problem (i.e. <strong>th</strong>e<br />
“double” scaling limit, which here means couplings in (7.29) tuned to critical values), Q<br />
becomes a differential operator of <strong>th</strong>e form<br />
Q = d q + { v q−2 (z), d q−2} + · · · + 2v 0 (z) , (7.30)<br />
where d = d/dz. (By a change of basis of <strong>th</strong>e form Q → f −1 (z)Qf(z), <strong>th</strong>e coefficient of<br />
d q−1 may always be set to zero.) The continuum scaling limit of <strong>th</strong>e multi-matrix models<br />
is <strong>th</strong>us abstracted to <strong>th</strong>e ma<strong>th</strong>ematical problem of finding solutions to (7.27).<br />
The differential equations (7.27) may be constructed as follows. For p, q relatively<br />
prime, a p <strong>th</strong> order differential operator <strong>th</strong>at can satisfy (7.27) is constructed as a fractional<br />
power of <strong>th</strong>e operator Q of (7.30).<br />
Formally, a q <strong>th</strong> root may be represented wi<strong>th</strong>in an<br />
algebra of formal pseudo-differential operators (see, e.g. [109]) as<br />
Q 1/q = d +<br />
∞∑ {<br />
ei , d −i} , (7.31)<br />
i=1<br />
where d −1 is defined to satisfy d −1 f = ∑ ∞<br />
j=0 (−1)j f (j) d −j−1 . The differential equations<br />
describing <strong>th</strong>e (p, q) minimal model coupled to 2d gravity are given by<br />
[<br />
Q<br />
p/q<br />
+ , Q ] = 1 , (7.32)<br />
where P = Q p/q<br />
+ indicates <strong>th</strong>e part of Qp/q wi<strong>th</strong> only non-negative powers of d, and is a<br />
p <strong>th</strong> order differential operator.<br />
To illustrate <strong>th</strong>e procedure we reproduce now <strong>th</strong>e results for <strong>th</strong>e one-matrix models,<br />
which can be used to generate (p, q) of <strong>th</strong>e form (2l − 1, 2). From (7.23), <strong>th</strong>ese models are<br />
obtained by taking Q to be <strong>th</strong>e hermitian operator<br />
Q = K ≡ d 2 − u(z) . (7.33)<br />
95
The formal expansion of Q l−1/2 = K l−1/2 (an anti-hermitian operator) in powers of d is<br />
given by<br />
K l−1/2 = d 2l−1 − 2l − 1<br />
4<br />
(where only symmetrized odd powers of d appear in <strong>th</strong>is case).<br />
{<br />
u, d<br />
2l−3 } + . . . (7.34)<br />
We now decompose<br />
K l−1/2 = K l−1/2<br />
+ + K l−1/2<br />
− , where K l−1/2<br />
+ = d 2l−1 + . . . contains only non-negative powers<br />
of d, and <strong>th</strong>e remainder K l−1/2<br />
− has <strong>th</strong>e expansion<br />
K l−1/2<br />
− =<br />
∞∑ {<br />
e2i−1 , d −(2i−1)} = { R l , d −1} + O(d −3 ) + . . . . (7.35)<br />
i=1<br />
Here we have identified R l ≡ e 1 as <strong>th</strong>e first term in <strong>th</strong>e expansion of K l−1/2<br />
− . For K 1/2 ,<br />
for example, we find K 1/2<br />
+ = d and R 1 = −u/4.<br />
The prescription (7.32) wi<strong>th</strong> p = 2l−1 corresponds here to calculating <strong>th</strong>e commutator<br />
[<br />
K<br />
l−1/2<br />
+ , K ] . Since K commutes wi<strong>th</strong> K l−1/2 , we have<br />
[<br />
l−1/2 K + , K ] = [ K, K l−1/2 ]<br />
− . (7.36)<br />
But since K begins at d 2 , and since from <strong>th</strong>e l.h.s. above <strong>th</strong>e commutator can have only<br />
positive powers of d, only <strong>th</strong>e leading (d −1 ) term from <strong>th</strong>e r.h.s. can contribute, which<br />
results in<br />
[<br />
K<br />
l−1/2<br />
+ , K ] = leading piece of [ K, 2R l d −1] = 4R ′ l . (7.37)<br />
After integration, <strong>th</strong>e equation [ K l−1/2<br />
+ , K ] = 1 <strong>th</strong>us takes <strong>th</strong>e simple form<br />
c R l [u] = z , (7.38)<br />
where <strong>th</strong>e constant c may be fixed by suitable rescaling of z and u.<br />
Such a scaling is<br />
enabled by <strong>th</strong>e property <strong>th</strong>at all terms in R l have fixed grade, namely 2l, where <strong>th</strong>e grade<br />
of d is defined to be 1 (and u <strong>th</strong>erefore has grade 2). The grade of v q−α in (7.30) is α for an<br />
operator Q of overall grade q. As we shall see shortly, <strong>th</strong>is notion of grade is related to <strong>th</strong>e<br />
conventional scaling weights of operators. It can also be used to determine <strong>th</strong>e terms <strong>th</strong>at<br />
may appear in many equations, since only terms of overall equal grade may be related.<br />
The quantities R l in (7.35) are easily seen to satisfy a simple recursion relation. From<br />
K l+1/2 = KK l−1/2 = K l−1/2 K, we find<br />
K l+1/2<br />
+ = 1 (<br />
)<br />
K l−1/2<br />
+ K + KK l−1/2<br />
+ + { R l , d } .<br />
2<br />
96
Commuting bo<strong>th</strong> sides wi<strong>th</strong> K and using (7.37), simple algebra gives [110]<br />
R ′ l+1 = 1 4 R′′′ l − uR ′ l − 1 2 u′ R l . (7.39)<br />
While <strong>th</strong>is recursion formula only determines R ′ l , by demanding <strong>th</strong>at <strong>th</strong>e R l (l ≠ 0)<br />
vanish at u = 0, we obtain<br />
R 0 = 1 2 , R 1 = − 1 4 u , R 2 = 3<br />
16 u2 − 1 16 u′′ ,<br />
R 3 = − 5 32 u3 + 5 (<br />
uu ′′ + 1<br />
32<br />
2 u′2) − 1<br />
64 u(4) .<br />
(7.40)<br />
We summarize as well <strong>th</strong>e first few K l−1/2<br />
+ ,<br />
K 1/2<br />
+ = d , K 3/2<br />
+ = d 3 − 3 {u, d} ,<br />
4<br />
K 5/2<br />
+ = d 5 − 5 4 {u, d3 } + 5 16<br />
{<br />
(3u 2 + u ′′ ), d } .<br />
(7.41)<br />
After rescaling, we recognize R 3 in (7.40) as eq. (7.19) wi<strong>th</strong> α = 1<br />
10<br />
, i.e. <strong>th</strong>e equation<br />
for <strong>th</strong>e (2,5) minimal model coupled to gravity. In general, <strong>th</strong>e equations determined<br />
by (7.27) for general p, q characterize <strong>th</strong>e partition function of <strong>th</strong>e (p, q) minimal model<br />
(mentioned after (2.22)) coupled to gravity. To realize <strong>th</strong>ese equations in <strong>th</strong>e continuum<br />
limit turns out [111,112] to require only a two-matrix model of <strong>th</strong>e type (7.29).<br />
argument given after (7.25) for <strong>th</strong>e one-matrix case is easily generalized to <strong>th</strong>e recursion<br />
relations for <strong>th</strong>e two-matrix case and shows <strong>th</strong>at for high enough order potentials, <strong>th</strong>ere<br />
are enough couplings to tune <strong>th</strong>e matrices P and Q to become p <strong>th</strong> and q <strong>th</strong> order differential<br />
operators. It is also possible to realize a c = 1 <strong>th</strong>eory coupled to gravity in terms of a<br />
two-matrix model formulation of <strong>th</strong>e 6-vertex model on a random lattice (see e.g. [11]). In<br />
[113], it is argued <strong>th</strong>at one can as well realize a wide variety of D < 1 <strong>th</strong>eories by means<br />
of a one-matrix model coupled to an external potential.<br />
It is possible to define a larger space of models defined by taking linear combinations<br />
of <strong>th</strong>e above models. In <strong>th</strong>e case where Q = K = d 2 − u, for example, we can consider<br />
∑<br />
k<br />
The<br />
t (k)<br />
[<br />
K<br />
k−1/2<br />
+ , K ] = 1 , (7.42)<br />
which results after suitable rescaling in <strong>th</strong>e “string equation” [4,6,104] describing a general<br />
massive model interpolating between multicritical points,<br />
z = ∑ k=1<br />
(<br />
k +<br />
1<br />
2)<br />
t(k) R k [u] . (7.43)<br />
97
If we consider <strong>th</strong>e higher operators K k−1/2<br />
+ as perturbations on pure gravity, P =<br />
K 3/2<br />
+ + ∑ j t (j)K j−1/2<br />
+ , <strong>th</strong>en <strong>th</strong>eir scaling weights follow from a simple argument. Since<br />
u ∼ z 1/2 for pure gravity and u has grade 2, we see <strong>th</strong>at a coupling of grade α scales<br />
as [z] α/4 , giving ∆ = α/4 as <strong>th</strong>e gravitationally dressed scaling weight of its conjugate<br />
operator. Now <strong>th</strong>e grade of t (j) is 3 − (2j − 1) = 4 − 2j, so it couples to an operator wi<strong>th</strong><br />
weight 1 − j/2.<br />
In <strong>th</strong>e case of unitary minimal models, z couples to <strong>th</strong>e area, so is proportional to <strong>th</strong>e<br />
cosmological constant. In general, however, z couples to <strong>th</strong>e lowest dimensional operator<br />
in <strong>th</strong>e <strong>th</strong>eory. From <strong>th</strong>e point of view of a perturbed (2, 2m − 1) model, we have u ∼ z 1/m ,<br />
<strong>th</strong>e grade of t (j) is (2m − 1) − (2j − 1) = 2m − 2j, and K j−1/2<br />
+ scales as (m − j)/m wi<strong>th</strong><br />
respect to <strong>th</strong>e lowest dimensional operator, i.e. corresponding to α/α 0 ra<strong>th</strong>er <strong>th</strong>an α/γ in<br />
(2.25). If we wish to compare to Liouville scaling wi<strong>th</strong> respect to <strong>th</strong>e area, on <strong>th</strong>e o<strong>th</strong>er<br />
hand, we must multiply by a factor of α 0 /γ = m/2, which results in<br />
1<br />
2<br />
(m − j) (7.44)<br />
for <strong>th</strong>e scaling of K j−1/2<br />
+ viewed as a perturbation of a (2, 2m − 1) model (a result we shall<br />
use when we expand macroscopic loops in terms of local operators in <strong>th</strong>ese <strong>th</strong>eories).<br />
We can also consider <strong>th</strong>e operators <strong>th</strong>at correspond to <strong>th</strong>ese perturbations from <strong>th</strong>e<br />
standpoint of <strong>th</strong>e underlying one-matrix model. The parameters t (j) in (7.43) correspond<br />
to perturbations to j <strong>th</strong> order multicritical potentials of <strong>th</strong>e form (7.20), and are given in<br />
turn by matrix operator perturbations of <strong>th</strong>e form<br />
tr V (j) (M) = tr<br />
∫ 1<br />
0<br />
dt<br />
t W ( )<br />
(j) t(1 − t)M<br />
2<br />
,<br />
wi<strong>th</strong> W (j) (r) = g c − α(r c − r) j .<br />
(7.45)<br />
(Note <strong>th</strong>at <strong>th</strong>e integral in <strong>th</strong>e first line just inverts <strong>th</strong>e expression leading from V to W in<br />
(7.12).)<br />
Finally, we note <strong>th</strong>at for a general (p, q) model, <strong>th</strong>e grade of <strong>th</strong>e l.h.s. of (7.27) is<br />
p + q, so z will be set equal (i.e. following one integration) to a quantity wi<strong>th</strong> grade<br />
p + q − 1. A coupling wi<strong>th</strong> grade α <strong>th</strong>erefore scales as [z] α/(p+q−1) , giving d = α/(p + q − 1)<br />
as <strong>th</strong>e gravitationally dressed scaling weight of its conjugate operator.<br />
v q−2<br />
(The grade of<br />
= 〈PP〉 is always 2, where P is <strong>th</strong>e puncture operator whose two-point function<br />
calculates <strong>th</strong>e 2 nd derivative of <strong>th</strong>e partition function, hence giving <strong>th</strong>e string susceptibility<br />
Γ str = −2/(p+q−1).) If we perturb P → P +t Q |pr−qs|/q−1<br />
+ , <strong>th</strong>en t has grade p+q−|pr−qs|,<br />
and hence couples to an operator of scaling weight coincident wi<strong>th</strong> (2.28) (after multiplying<br />
<strong>th</strong>e latter by γ/α q−1,p−1 = 2q/(p + q − 1) to take into account <strong>th</strong>e coupling of z to <strong>th</strong>e<br />
lowest dimensional operator ra<strong>th</strong>er <strong>th</strong>an to area).<br />
98
Exercise. Scaling of Lax operators<br />
a) Show <strong>th</strong>at if Q is <strong>th</strong>e Lax operator of order q, defining <strong>th</strong>e (p, q) series, <strong>th</strong>en <strong>th</strong>e<br />
operators in (4.10) are<br />
Q n/q∣ ∣+ . (7.46)<br />
Parametrizing n = kq+α, 1 ≤ α ≤ q−1, we identify <strong>th</strong>ese operators wi<strong>th</strong> <strong>th</strong>e topological<br />
field <strong>th</strong>eory operators σ k (O α ) which appear in [106].<br />
b) Calculate <strong>th</strong>e spectrum of indices ν of (4.7) we expect to find for <strong>th</strong>e Wheeler–<br />
DeWitt wavefunctions of <strong>th</strong>e Lian–Zuckerman states.<br />
8. Matrix Model Technology II: Loops on <strong>th</strong>e Lattice<br />
8.1. Lattice Loop Operators<br />
Fig. 14: Insertion of tr Φ M into <strong>th</strong>e matrix generating functional results in a vertex<br />
emanating M “spokes”.<br />
Consider <strong>th</strong>e one hermitian matrix model, as discussed in chapt. 7 (where we now<br />
use Φ ra<strong>th</strong>er <strong>th</strong>an M to denote <strong>th</strong>e N × N hermitian matrix). In <strong>th</strong>e Feynman diagram<br />
expansion, <strong>th</strong>e insertion of <strong>th</strong>e operator<br />
1<br />
M tr ΦM (8.1)<br />
creates a vertex emanating M “spokes,” as shown in fig. 14. On <strong>th</strong>e dual triangulated<br />
surface, <strong>th</strong>is operator has inserted a hole wi<strong>th</strong> M boundary leng<strong>th</strong>s, and <strong>th</strong>us has leng<strong>th</strong><br />
aM, where a is <strong>th</strong>e lattice spacing. The factor of 1/M in (8.1) is needed to take account<br />
99
of <strong>th</strong>e symmetry factor for <strong>th</strong>e Feynman diagram. In <strong>th</strong>e following we will generally work<br />
wi<strong>th</strong> marked loops and discard <strong>th</strong>is factor.<br />
To obtain macroscopic loop amplitudes we must take <strong>th</strong>e continuum limit a → 0<br />
and to maintain a finite physical leng<strong>th</strong>, we must simultaneously take M → ∞, holding<br />
some combination of a and M fixed. From <strong>th</strong>is point of view, <strong>th</strong>e local operators of <strong>th</strong>e<br />
<strong>th</strong>eory discussed in sec. 7.7, namely linear combinations of operators of <strong>th</strong>e form (8.1) (e.g.<br />
(7.45)), correspond instead to “microscopic loops”, i.e. loops only a few lattice spacings in<br />
leng<strong>th</strong>.<br />
Before proceeding to make precise sense of <strong>th</strong>e continuum limit, we begin wi<strong>th</strong> some<br />
heuristic remarks. Recall <strong>th</strong>at in <strong>th</strong>e or<strong>th</strong>ogonal polynomial formalism discussed in sec. 7.4,<br />
<strong>th</strong>e action of Φ M was equivalent to (Φ cr + Q) M , where<br />
Q → 2r 1/2<br />
c + a 2/m r −1/2<br />
c (u + r c κ 2 ∂ 2 z ) . (8.2)<br />
Hence if we hold Ma 2/m = 2r c l fixed, we expect <strong>th</strong>e loop operator to become <strong>th</strong>e heat<br />
kernel operator,<br />
1<br />
M tr(ΦM ) → e lQ . (8.3)<br />
After rescaling, we can write Q = κ 2 d 2 /dz 2 − u(z, κ), where Q is <strong>th</strong>e Schrödinger operator<br />
associated to <strong>th</strong>e model, and κ is <strong>th</strong>e topological coupling. A rigorous discussion of <strong>th</strong>e<br />
above limiting procedure makes use of <strong>th</strong>e free fermion formalism, implicit in <strong>th</strong>e or<strong>th</strong>ogonal<br />
polynomial technique (and described in chapt. 9).<br />
An alternative formulation of <strong>th</strong>e lattice loop operator is<br />
where L is a “chemical potential” for <strong>th</strong>e leng<strong>th</strong>.<br />
W (L) = 1 N Tr(eLΦ ) , (8.4)<br />
The limiting form (8.2) shows <strong>th</strong>at<br />
<strong>th</strong>ese loop operators have <strong>th</strong>e same continuum limit, up to a non-universal multiplicative<br />
renormalization. This is a useful observation for making sense of examples where r c = 0.<br />
operator<br />
To analyze macroscopic loop amplitudes on <strong>th</strong>e lattice, we introduce <strong>th</strong>e resolvent<br />
Ŵ (ζ) =<br />
∫ ∞<br />
0<br />
dL e −ζL W (L) . (8.5)<br />
Defining ζ = e ρ , we may interpret ρ as a bare boundary cosmological constant, and (8.5)<br />
should be regarded as <strong>th</strong>e lattice analog of <strong>th</strong>e relation<br />
∞∑<br />
Z(µ B ) = e −µ BA Z(A) . (8.6)<br />
A=1<br />
between fixed area and fixed cosmological constant partition functions.<br />
100
8.2. Precise definition of <strong>th</strong>e continuum limit<br />
Making use of <strong>th</strong>e resolvent operator (8.5), we can now present a more technical<br />
description of <strong>th</strong>e continuum limit discussed in chapts. 6,7. To take <strong>th</strong>e continuum limit of<br />
<strong>th</strong>e lattice expressions, we first study <strong>th</strong>e N → ∞ asymptotics of <strong>th</strong>e correlation functions<br />
〈 ∏ B<br />
N| Ŵ (ζ i )|N 〉 ∫<br />
≡ Z −1<br />
i=1<br />
∼<br />
∑<br />
χ=2−2h−B<br />
B<br />
−Ntr V (Φ)<br />
dΦ e<br />
∏<br />
Ŵ (ζ i )<br />
i=1<br />
N χ F χ [V ; ζ i ] ,<br />
(8.7)<br />
where V (Φ) = ∑ j≥0 T j Φ j is a polynomial interaction for Φ. This is a generating functional<br />
for correlation functions of B operators. As explained in sec. 6.1, at fixed topology <strong>th</strong>e<br />
functionals F χ [V ; ζ i ] have a lattice expansion<br />
F χ [V ; ζ i ] =<br />
∑<br />
F i ,L i ≥0<br />
∑<br />
D χ [F i ,L i ]<br />
∏<br />
(Ti / √ T 2 ) F i<br />
∏<br />
ζ<br />
−L i<br />
i<br />
, (8.8)<br />
where D χ [F i , L i ] is <strong>th</strong>e set of distinct “triangulations” of a surface into F i i-sided polygons<br />
wi<strong>th</strong> boundaries of lattice leng<strong>th</strong> L i . Using me<strong>th</strong>ods described below (in particular <strong>th</strong>e loop<br />
equations (8.17)), one can show <strong>th</strong>at <strong>th</strong>e functions F χ have <strong>th</strong>e following ma<strong>th</strong>ematical<br />
properties, familiar from <strong>th</strong>e study of phase transitions in statistical mechanical models.<br />
T n<br />
T2<br />
multicritical phase<br />
S: pure gravity<br />
higher multicritical point<br />
T 1<br />
Fig. 15: Subspaces of successively higher codimension in coupling constant space<br />
corresponding to multicritical domains.<br />
101
1) The expansions in V and ζ −1<br />
i<br />
are convergent in a sufficiently small neighborhood of<br />
<strong>th</strong>e origin. To specify a neighborhood of <strong>th</strong>e origin for <strong>th</strong>e potential V , we first define<br />
a filtration on <strong>th</strong>e space of potentials by requiring <strong>th</strong>at V be in V (n) , <strong>th</strong>e space of<br />
polynomials of degree ≤ n wi<strong>th</strong> no constant or linear term. Considered as a function<br />
on <strong>th</strong>e space of polynomials of degree ≤ n, we have convergence in a neighborhood of<br />
<strong>th</strong>e origin.<br />
2) The expansions have a finite radius of convergence. Defining <strong>th</strong>e lattice area A =<br />
∑<br />
Fi , it can be established <strong>th</strong>at <strong>th</strong>e asymptotic behavior of <strong>th</strong>e number of distinct<br />
∑<br />
triangulations goes as |D χ | ∼ e µ ∗A+ ρ ∗L i<br />
i A θ L θ′<br />
i for A → ∞, L i → ∞. Thus, <strong>th</strong>e<br />
series will diverge as V approaches a real codimension one subvariety, <strong>th</strong>e singular<br />
subvariety S of V (n) , and as ζ → ζ c from above.<br />
3) A priori S and ζ c could depend on <strong>th</strong>e Euler character χ and <strong>th</strong>e choice of boundary<br />
component, but turn out independent of <strong>th</strong>em.<br />
4) The variety S has subspaces of successively higher codimension corresponding to multicritical<br />
domains, as depicted in fig. 15. For <strong>th</strong>e space V 2n , <strong>th</strong>e highest multicritical<br />
behavior corresponds to a point, given in [6]<br />
∫ 1<br />
V ∗ (n) (Φ) =<br />
0<br />
(This is just (7.45) wi<strong>th</strong> normalization g c = α = r c = 1.)<br />
dt<br />
(<br />
1 − ( 1 − t(1 − t) Φ 2) ) n<br />
. (8.9)<br />
t<br />
5) The critical exponents θ, θ ′ in (2) are “universal,” which means <strong>th</strong>at <strong>th</strong>ey only depend<br />
on <strong>th</strong>e multicritical domain in S. On <strong>th</strong>e o<strong>th</strong>er hand, µ ∗ , ρ ∗ are “non-universal” which<br />
means <strong>th</strong>ey can vary from point to point in S.<br />
We can use <strong>th</strong>ese ma<strong>th</strong>ematical properties of <strong>th</strong>e functions F χ to learn about <strong>th</strong>e physics of<br />
smoo<strong>th</strong> surfaces as follows. We would like to distinguish “universal” phenomena — associated<br />
wi<strong>th</strong> smoo<strong>th</strong> continuum surfaces and not wi<strong>th</strong> <strong>th</strong>e details of lattice decompositions<br />
— by using <strong>th</strong>e nonanalytic behavior of F χ as we approach singular values of V, ζ. The<br />
idea is based on <strong>th</strong>e remark <strong>th</strong>at <strong>th</strong>e contribution of a hole wi<strong>th</strong> a finite lattice size, or a<br />
surface wi<strong>th</strong> a finite number of polygons, to (8.7) will always be analytic in ζ, V . Thus,<br />
<strong>th</strong>e nonanalyticity in ζ, V must “arise from holes and surfaces whose perimeter and area<br />
is infinity in lattice units,” <strong>th</strong>at is, from smoo<strong>th</strong> continuum surfaces. 34<br />
By turning <strong>th</strong>e above reasoning on its head, we define <strong>th</strong>e continuum limit by isolating<br />
<strong>th</strong>e nonanalytic dependence on ζ i , V . More precisely, we must define scaling functions as<br />
34 The extent to which <strong>th</strong>ese surfaces really are smoo<strong>th</strong> is an interesting question. See [114].<br />
102
we approach singular values and define <strong>th</strong>e continuum quantities in terms of <strong>th</strong>ese scaling<br />
functions. Note <strong>th</strong>at if we use such a definition, continuum quantities are ambiguous by<br />
terms which are purely analytic in <strong>th</strong>e coupling constants.<br />
8.3. The Loop Equations<br />
The correlation functions of Ŵ (ζ) may be determined by <strong>th</strong>e “Schwinger–Dyson” or<br />
loop equations [99,115,116] for <strong>th</strong>e matrix model.<br />
The loop equations are derived by requiring <strong>th</strong>at <strong>th</strong>e matrix model pa<strong>th</strong> integral be<br />
independent of a change of variables. A convenient way to organize arbitrary analytic<br />
changes of variables is to consider <strong>th</strong>e simple transformation<br />
1<br />
φ → φ + ɛ<br />
ζ − φ . (8.10)<br />
Under (8.10), we have<br />
tr V (φ) → tr V (φ) + ɛ tr V ′ (φ)(ζ − φ) −1<br />
(<br />
dφ → dφ 1 + ɛN 2( Ŵ (ζ) ) ) 2<br />
(8.11)<br />
.<br />
Inserting (8.11) into ∫ dφ e −NtrV (Φ) and equating first order terms in ɛ gives<br />
〈(Ŵ ) 2 〉 1 〈<br />
(ζ) = tr V ′ (φ)(ζ − φ) −1〉 . (8.12)<br />
N<br />
We wish to expand <strong>th</strong>e above in 1/N. Ŵ is normalized so <strong>th</strong>at <strong>th</strong>e expansion in 1/N<br />
of 〈Ŵ 〉 begins at O(1), corresponding to <strong>th</strong>e disk geometry,<br />
〈Ŵ 〉 〈Ŵ<br />
(ζ) ∼ (ζ)<br />
〉h=0 + 1 〈Ŵ 〉<br />
(ζ)<br />
N 2 h=1 + · · · . (8.13)<br />
By considering <strong>th</strong>e relevant topologies contributing to 〈 (Ŵ (ζ)) 2〉 , we see <strong>th</strong>at <strong>th</strong>e leading<br />
term on <strong>th</strong>e left hand side of (8.12) has <strong>th</strong>e topology of two disks. In general, we may<br />
separate <strong>th</strong>e contribution of connected and disconnected geometries:<br />
〈(Ŵ ) 2 〉 〈Ŵ 〉<br />
(ζ) = ( (ζ) ) 2 + 1 〈(Ŵ ) 2 〉<br />
(ζ)<br />
N , (8.14)<br />
2 c<br />
where <strong>th</strong>e second term corresponds to connected geometries. Expanding V ′ (φ) as a polynomial<br />
in ζ −φ wi<strong>th</strong> coefficients which are polynomials in ζ, we see <strong>th</strong>at 〈 Ŵ (ζ) 〉 satisfies<br />
h=0<br />
a quadratic equation,<br />
〈Ŵ 〉 2<br />
(ζ) − V ′ (ζ) 〈 Ŵ (ζ) 〉 + Q(ζ, V ) = 0 , (8.15)<br />
h=0 h=0<br />
where<br />
Q(ζ, V ) = 〈 Q(ζ, φ) 〉<br />
Q(ζ, φ) = ∑ k≥1<br />
1<br />
k! V (k+1) (ζ) 1 N tr(φ − ζ)k−1 .<br />
(8.16)<br />
Q is a polynomial in ζ of degree deg (V ) − 2 whose coefficients are linear combinations of<br />
c j (V ) ≡ 〈 tr Φ j〉 , j ≤ deg (V ) − 2. The disk amplitude is obtained by solving (8.15).<br />
103
V′<br />
( ) + + + . . .<br />
∂<br />
∂L<br />
L<br />
= ∫ L<br />
0<br />
+ ∫ L<br />
0<br />
L<br />
L′ L′<br />
dL′ + . . . +<br />
L − L′<br />
L − L′<br />
L<br />
L′ L − L′ L′<br />
L − L′<br />
dL′ + + . . .<br />
Fig. 16: Pictorial representation of V ′ (∂/∂L) 〈 W (L) 〉 c .<br />
Geometrical Interpretation: SD Equations as Loop Equations:<br />
The loop equations have a beautiful geometrical interpretation fur<strong>th</strong>er justifying <strong>th</strong>e<br />
identification of W (L) wi<strong>th</strong> a loop operator. Write (8.12) in <strong>th</strong>e form<br />
V ′ (ζ) 〈 Ŵ (ζ) 〉 + Q(ζ, V ) = ( 〈 Ŵ (ζ) 〉 c )2 + 1<br />
N 2 〈<br />
( Ŵ (ζ)) 2〉 2<br />
c . (8.17)<br />
Taking an inverse Laplace transform, we obtain<br />
V ′( ∂<br />
) ∫ 〈W L<br />
(L)<br />
∂L<br />
〉c ( = dL ′ 〈<br />
W (L ′ ) 〉 〈<br />
c W (L−L ′ ) 〉 + 1 〈<br />
c W (L ′ ) W (L−L ′ ) 〉 )<br />
, (8.18)<br />
0<br />
N 2 c<br />
which has <strong>th</strong>e pictorial representation shown in fig. 16.<br />
Exercise.<br />
Derive (8.18).<br />
transform.<br />
Note <strong>th</strong>at a polynomial in ζ does not have an inverse Laplace<br />
The full set of loop equations may be elegantly summarized by introducing a “source”<br />
coupling to <strong>th</strong>e loop operator:<br />
∫<br />
Z[J] ≡<br />
∫<br />
dΦ e −Ntr V (Φ)+ ∞<br />
dL ′ J(L ′ ) W (L ′)<br />
0<br />
. (8.19)<br />
Making <strong>th</strong>e change of variables Φ → Φ + ɛ e LΦ and using <strong>th</strong>e above procedures, we obtain<br />
( ∂<br />
) ∫ 〈W L ( 〈W<br />
V ′ (L)<br />
∂L<br />
〉c [J] = dL ′ (L ′ ) 〉 [J] 〈 W (L − L ′ ) 〉 [J]<br />
c c<br />
0<br />
+ 1 〈<br />
W (L ′<br />
N 2 ) W (L − L ′ ) 〉 )<br />
+ 1 ∫ ∞<br />
c<br />
N 2 dL ′ L 〈 ′ W (L + L ′ ) 〉 (8.20)<br />
c [J] ,<br />
104<br />
0
for which one may draw a similar pictorial representation.<br />
Remark: It is possible, al<strong>th</strong>ough slightly subtle [115,117], to take <strong>th</strong>e continuum limit<br />
of <strong>th</strong>e loop equations we have derived here to write analogous equations for <strong>th</strong>e continuum<br />
amplitudes. These continuum loop equations have many important applications, including<br />
for example <strong>th</strong>e elimination [115] of unphysical solutions to <strong>th</strong>e string equations (7.43).<br />
9. Matrix Model Technology III: Free Fermions from <strong>th</strong>e Lattice<br />
The equivalence of matrix models to <strong>th</strong>eories of free fermions is <strong>th</strong>e underlying reason<br />
for <strong>th</strong>e solvability of matrix models. In <strong>th</strong>is chapter we describe <strong>th</strong>e free fermion formalism.<br />
9.1. Lattice Fermi Field Theory<br />
The free-fermion formalism provides <strong>th</strong>e basis for a rigorous description of <strong>th</strong>e doublescaling<br />
limit of macroscopic loop operators. The formalism is also a very efficient way<br />
for calculating loops, bo<strong>th</strong> on <strong>th</strong>e lattice and in <strong>th</strong>e continuum. The formalism was first<br />
applied to macroscopic loop amplitudes in [104].<br />
In sec. 7.1, or<strong>th</strong>ogonal polynomials were introduced and it was shown <strong>th</strong>at correlation<br />
functions are integrals of powers of λ multiplying a Vandermonde determinant. Interpreting<br />
<strong>th</strong>is determinant as a Slater determinant for a <strong>th</strong>eory of free fermions, we introduce <strong>th</strong>e<br />
second-quantized Fermi field<br />
Ψ(λ) =<br />
∞∑<br />
a n ψ n (λ) , (9.1)<br />
n=0<br />
where ψ n are <strong>th</strong>e or<strong>th</strong>onormal wavefunctions built from <strong>th</strong>e or<strong>th</strong>ogonal polynomials:<br />
ψ n (λ) ≡ 1 √<br />
hn<br />
P n (λ) e − 1 2 NV (λ) , (9.2)<br />
and {a n , a † m} = δ n,m .<br />
Correlation functions in <strong>th</strong>e matrix model wi<strong>th</strong> N × N matrices are obtained by<br />
calculating correlation functions in <strong>th</strong>e Fermi sea defined by<br />
a n |N〉 = 0 n ≥ N<br />
(9.3)<br />
a † n |N〉 = 0 n < N .<br />
To see <strong>th</strong>is, introduce <strong>th</strong>e second-quantized operator for multiplication by λ n ,<br />
∫<br />
Ψ †ˆλn Ψ = dλ Ψ † (λ) λ n Ψ(λ) , (9.4)<br />
105
and <strong>th</strong>e main observation is<br />
〈∏<br />
i<br />
〉<br />
∫<br />
tr Φ n i<br />
≡ Z −1<br />
matrix model<br />
= 〈 N| ∏ i<br />
dΦ ∏ tr Φ n i −Ntr V (Φ)<br />
e<br />
i<br />
(<br />
Ψ<br />
†ˆλn i<br />
Ψ ) |N 〉 .<br />
(9.5)<br />
where V = ∑ i≥2 g i Φ i and all but finitely many g i = 0. The proof of <strong>th</strong>is identity uses <strong>th</strong>e<br />
or<strong>th</strong>ogonal polynomial techniques, e.g. for <strong>th</strong>e one-point function:<br />
〈tr Φ n 〉 =<br />
=<br />
=<br />
∫ ∏ dλi ∆ 2 (λ i )( ∑ λ n i ) ∏ i e−NV (λ i)<br />
N! ∏ h i<br />
∫<br />
N ∏<br />
N! ∏ (<br />
dλi det Pj−1 (λ i ) ) ∏<br />
2<br />
λ<br />
n<br />
h 1<br />
i<br />
N−1<br />
N<br />
N! ∏ ∑<br />
(N − 1)!<br />
h i<br />
j=0<br />
∏<br />
hi<br />
h j<br />
N−1<br />
∑<br />
= 〈ψ j |λ n |ψ j 〉 = 〈N| Ψ †ˆλn Ψ |N〉 .<br />
j=0<br />
∫<br />
i<br />
e −NV (λ i)<br />
dλ ( P j (λ) ) 2<br />
λ n −NV (λ)<br />
e<br />
(9.6)<br />
Exercise.<br />
a) Prove (9.5) for two point functions using <strong>th</strong>e same steps as in (9.6).<br />
b) Find a general proof of (9.5).<br />
c) Show <strong>th</strong>at <strong>th</strong>e lattice loop operator (8.4) and resolvent may be realized in <strong>th</strong>e<br />
fermion formalism as<br />
W (L) = 1 N Ψ† e LˆλΨ<br />
Ŵ (ζ) =<br />
∞∑<br />
n=0<br />
ζ −n−1 Ψ †ˆλn Ψ = Ψ † 1<br />
ζ − ˆλ Ψ (9.7)<br />
9.2. Eigenvalue distributions<br />
We will now justify some of <strong>th</strong>e statements made in sec. 8.2 and indicate why <strong>th</strong>e<br />
lattice (and hence continuum) correlation functions are computable.<br />
By Wick’s <strong>th</strong>eorem, we can express all amplitudes in terms of <strong>th</strong>e fermion two-point<br />
function<br />
K N (λ 1 , λ 2 ) ≡ 〈N|Ψ † (λ 1 )Ψ(λ 2 )|N〉 . (9.8)<br />
106
Therefore, in order to define <strong>th</strong>e double scaling limit we must study <strong>th</strong>e N → ∞ asymptotic<br />
behavior of <strong>th</strong>e kernel K N . As explained in sec. 6.2, matrix model correlation functions<br />
have an asymptotic expansion in 1/N, and are obtained from <strong>th</strong>e asymptotic expansion:<br />
K N ∼ ∑ j≥0<br />
N 1−j K j (λ 1 , λ 2 ) (9.9)<br />
The functions K j have support on an interval 35 I which is independent of j. As a special<br />
case, note in particular <strong>th</strong>at <strong>th</strong>e diagonal of <strong>th</strong>is kernel is <strong>th</strong>e eigenvalue density:<br />
ρ(λ) = K N (λ, λ) . (9.10)<br />
By (9.10) we may identify <strong>th</strong>e interval I wi<strong>th</strong> support of <strong>th</strong>e eigenvalue density in perturbation<br />
<strong>th</strong>eory.<br />
Exercise. Eigenvalue Density<br />
Show <strong>th</strong>at ρ(λ) is <strong>th</strong>e probability for finding an eigenvalue wi<strong>th</strong> value λ in a random<br />
matrix ensemble described by V (λ). That is, show <strong>th</strong>at it is <strong>th</strong>e matrix expectation value<br />
of<br />
1<br />
N<br />
N∑<br />
δ(λ − λ i ) . (9.11)<br />
i=1<br />
The easiest way to prove our assertions about <strong>th</strong>e nature of <strong>th</strong>e eigenvalue densities<br />
proceeds by studying <strong>th</strong>e correlation functions of <strong>th</strong>e resolvent operators Ŵ (ζ). Note <strong>th</strong>at<br />
Ŵ (ζ) is only defined for ζ off <strong>th</strong>e real axis since φ has real eigenvalues. Moreover, <strong>th</strong>e<br />
discontinuity of Ŵ (ζ) across <strong>th</strong>e real axis is equal to <strong>th</strong>e eigenvalue density.<br />
Solving <strong>th</strong>e quadratic equation we see <strong>th</strong>at <strong>th</strong>e roots of <strong>th</strong>e polynomials define several<br />
branch points for 〈 Ŵ (ζ) 〉 , and since ρ(λ) is <strong>th</strong>e discontinuity of 〈 Ŵ (ζ) 〉 , <strong>th</strong>e<br />
h=0 h=0<br />
support of <strong>th</strong>e genus zero eigenvalue density must lie on an interval or finite union of<br />
intervals.<br />
35 In more complicated cases <strong>th</strong>e support can be on unions of intervals.<br />
107
2<br />
λ<br />
Fig. 17: The Wigner semicircle distribution.<br />
Exercise. Derivation of <strong>th</strong>e Wigner Distribution<br />
As an example of an eigenvalue distribution, we consider <strong>th</strong>e Gaussian matrix<br />
model. The leading term in <strong>th</strong>e large N asymptotics of <strong>th</strong>e eigenvalue distribution is<br />
<strong>th</strong>e famous Wigner distribution<br />
K 1 (λ, λ) = 1 √<br />
2 − λ<br />
4π<br />
2 θ(2 − λ 2 ) , (9.12)<br />
shown in fig. 17.<br />
Derive <strong>th</strong>e Wigner distribution from <strong>th</strong>e “Schwinger–Dyson” equations of <strong>th</strong>e matrix<br />
model using <strong>th</strong>e above procedure. First show <strong>th</strong>at for <strong>th</strong>e Gaussian potential we<br />
have<br />
〈Ŵ(ζ) 〉<br />
= 1 ( √<br />
h=0 ζ − ζ2 − 2 ) , (9.13)<br />
2<br />
and from <strong>th</strong>is obtain <strong>th</strong>e Wigner distribution (9.12).<br />
The finiteness of <strong>th</strong>e support of <strong>th</strong>e kernels has important implications for <strong>th</strong>e nonanalyticity<br />
in ζ. Consider for example <strong>th</strong>e one-point function<br />
〈Ŵ 〉 1<br />
〈<br />
1<br />
〉<br />
(ζ) ≡ tr = 1 ∫<br />
K(λ, λ)<br />
dλ<br />
N ζ − φ N ζ − λ<br />
∼ ∑ ∫<br />
N χ dλ K χ(λ, λ)<br />
.<br />
χ I ζ − λ<br />
(9.14)<br />
The nonanalytic dependence on ζ we are looking for comes from <strong>th</strong>e contributions in<br />
<strong>th</strong>e λ integrals from <strong>th</strong>e integrals near <strong>th</strong>e edge of <strong>th</strong>e support I of <strong>th</strong>e eigenvalue distribution.<br />
In <strong>th</strong>e last expression we may take ζ real and ζ > ζ c . We encounter nonanalytic<br />
behavior as ζ hits <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution.<br />
Example: Let us verify <strong>th</strong>e statements about analytic dependence on ζ in <strong>th</strong>e example<br />
of a Gaussian potential. Expanding ζ = ζ c + δζ = √ 2 + δζ, or equivalently, expanding λ<br />
around <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution, we obtain a nonanalytic function of (δζ) 1/2<br />
corresponding (formally) to <strong>th</strong>e one-loop amplitude 〈 W (l) 〉 = l −3/2 .<br />
108
9.3. Double–Scaled Fermi Theory<br />
More generally, to prove (9.9) and to investigate <strong>th</strong>e scaling limit of K near <strong>th</strong>e edge<br />
of I more <strong>th</strong>oroughly, note <strong>th</strong>at using <strong>th</strong>e recursion relation<br />
we may write<br />
〈N|Ψ † (λ 1 )Ψ(λ 2 )|N〉 =<br />
λψ n = √ r n+1 ψ n+1 + √ r n ψ n−1 , (9.15)<br />
N−1<br />
∑<br />
n=0<br />
ψ n (λ 1 )ψ n (λ 2 )<br />
= √ r N+1<br />
ψ N+1<br />
(λ 1 )ψ N<br />
(λ 2 ) − ψ N+1<br />
(λ 2 )ψ N<br />
(λ 1 )<br />
λ 1 − λ 2<br />
,<br />
(9.16)<br />
and <strong>th</strong>erefore we should study <strong>th</strong>e scaling limit of <strong>th</strong>e or<strong>th</strong>onormal wavefunctions <strong>th</strong>emselves.<br />
As discussed in sec. 8.2, <strong>th</strong>e recursion functions r n [V ] have singular behavior as V → S.<br />
Moreover, using <strong>th</strong>e recursion relations for or<strong>th</strong>ogonal polynomials, as elegantly summarized<br />
in <strong>th</strong>e statement [P, Q] = 1, <strong>th</strong>e large n asymptotics determines a consistent ansatz<br />
for <strong>th</strong>e following behavior. If V ∗<br />
(m) (λ) is <strong>th</strong>e m <strong>th</strong> multicritical potential, <strong>th</strong>en we approach<br />
criticality by taking <strong>th</strong>e limit:<br />
V = e a2µ V (m)<br />
∗ a → 0<br />
n/N = 1 − a 2 (z − µ)<br />
Na 2+1/m = κ −1<br />
r n [V ] → r c + a 2/m u(z) .<br />
(9.17)<br />
The recursion relation (9.15) implies <strong>th</strong>at if ψ has well-behaved limiting behavior near <strong>th</strong>e<br />
edge of <strong>th</strong>e eigenvalue distribution λ c ,<br />
ψ n (λ c + a 2/m˜λ) → a θ ψ(z, ˜λ) (9.18)<br />
(here a θ is a normalization factor), <strong>th</strong>en <strong>th</strong>e ψ’s are eigenfunctions of <strong>th</strong>e Lax operator: 36<br />
Qψ = ( κ 2 d 2 /dz 2 − u(z, κ) ) ψ = ˜λψ . (9.19)<br />
The limiting form of ψ will be an eigenfunction of Q. 37<br />
36 In <strong>th</strong>e <strong>th</strong>eory of <strong>th</strong>e KdV hierarchy, such functions are known as Baker-Akhiezer functions.<br />
37 The eigenfunctions of Q are obtained by demanding appropriate asymptotic behavior in z<br />
and λ, insuring convergence of integrals as z → ∞ and exponential decay of eigenvalue density<br />
off <strong>th</strong>e perturbative cut as λ → ∞.<br />
109
Example. The Gaussian potential<br />
We will work <strong>th</strong>rough in detail <strong>th</strong>e fermionic formulation of <strong>th</strong>e double scaling limit<br />
for <strong>th</strong>e simplest matrix potential of all, <strong>th</strong>e Gaussian potential V (Φ) = Φ 2 . The Gaussian<br />
potential corresponds to <strong>th</strong>e so-called topological point or <strong>th</strong>e (1, 2) point in <strong>th</strong>e Lax<br />
operator classification described in sec. 7.7. Perturbations about <strong>th</strong>is point define <strong>th</strong>e<br />
correlation functions of topological gravity, described from <strong>th</strong>e point of view of topological<br />
field <strong>th</strong>eory in [106].<br />
t<br />
1 2<br />
Fig. 18: As λ → λ c , two stationary phase points coalesce at i/ √ 2.<br />
We now obtain <strong>th</strong>e full asymptotics in 1/N of <strong>th</strong>e contributions to (9.5) of <strong>th</strong>e integrals<br />
from <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution. We do <strong>th</strong>is first explicitly for <strong>th</strong>e case of <strong>th</strong>e<br />
Gaussian potential. In <strong>th</strong>e case of a gaussian matrix potential e −Ntrφ2 , <strong>th</strong>e or<strong>th</strong>onormal<br />
wavefunctions are simply<br />
ψ n (λ) =<br />
N 1/4<br />
2 n/2 π 1/4√ n! H n( √ Nλ) e −Nλ2 /2 , (9.20)<br />
where H n is a Hermite polynomial, and has <strong>th</strong>e integral representation<br />
H n (x) = 2n<br />
√ π<br />
∫ ∞<br />
−∞<br />
dt (x + it) n e −t2 . (9.21)<br />
Using <strong>th</strong>e stationary phase approximation, one finds two stationary points for λ 2 ≠ 2. For<br />
λ 2 < 2 we find an oscillatory function while for λ 2 > 2, <strong>th</strong>e wavefunction is zero to all<br />
orders of <strong>th</strong>e 1/N expansion. We are most interested in <strong>th</strong>e behavior of <strong>th</strong>e wavefunctions<br />
for λ infinitesimally close to ± √ 2, <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution. At <strong>th</strong>is point, <strong>th</strong>e<br />
two stationary phase points coalesce as in fig. 18, and by simultaneously scaling N → ∞<br />
and λ → λ c = √ 2 one can obtain a well-defined limit whose asymptotics captures <strong>th</strong>e<br />
110
contribution of <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution to <strong>th</strong>e entire perturbation series in<br />
1/N. This simultaneous scaling is <strong>th</strong>e fermionic version of <strong>th</strong>e double scaling limit.<br />
In detail, let<br />
Na 3 = κ −1 n/N = 1 − a 2 z λ = √ 2(1 + a 2˜λ) , (9.22)<br />
and let a → 0 holding z, κ, ˜λ fixed. Then we have<br />
∫ ∞<br />
lim<br />
a→0 a1/2 ψ n (λ) = κ−1/6<br />
dt e itκ−2/3 (˜λ+z)+it 3 /3<br />
π2 3/4 −∞<br />
(9.23)<br />
1<br />
( z +<br />
=<br />
2 1/4 κ Ai ˜λ<br />
)<br />
.<br />
1/6 κ 2/3<br />
That is, <strong>th</strong>e double scaling limit of <strong>th</strong>e Hermite functions of <strong>th</strong>e Gaussian model are Airy<br />
functions. 38<br />
λ − λ<br />
c<br />
λ c<br />
Fig. 19: A magnified view of <strong>th</strong>e eigenvalue distribution near <strong>th</strong>e endpoint. Note<br />
<strong>th</strong>e nearby exponential falloff and squareroot grow<strong>th</strong> far from <strong>th</strong>e endpoint.<br />
The edge of <strong>th</strong>e eigenvalue distribution is at λ = 0, and is given by<br />
ρ(λ) = (Ai ′ ) 2 − λκ −2/3 (Ai) 2 . (9.24)<br />
From <strong>th</strong>e asymptotics of Airy functions, we obtain <strong>th</strong>e picture of <strong>th</strong>e eigenvalue distribution<br />
depicted in fig. 19. This completes our example of <strong>th</strong>e Gaussian potential. 39<br />
38 The appearance of <strong>th</strong>ese functions is directly related to <strong>th</strong>e Airy functions which play a key<br />
role in <strong>th</strong>e Kontsevich matrix model.<br />
39 We could have deduced <strong>th</strong>e connection to Airy functions more directly using <strong>th</strong>e WKB<br />
analysis of wavefunctions in a harmonic oscillator potential, but <strong>th</strong>at argument does not generalize<br />
to o<strong>th</strong>er or<strong>th</strong>ogonal polynomials.<br />
111
Exercise.<br />
Using <strong>th</strong>e asymptotics of <strong>th</strong>e Airy function, show <strong>th</strong>at <strong>th</strong>e double-scaled eigenvalue<br />
density behaves like:<br />
ρ(λ) ∼ π 4 λ1/4 e − 2<br />
3κ λ3/2 λ → +∞<br />
∼ π(−λ) 1/2<br />
λ → −∞<br />
(9.25)<br />
Returning to <strong>th</strong>e general case, we can use (9.18) to find <strong>th</strong>e behavior of <strong>th</strong>e fermion<br />
two-point function in <strong>th</strong>e region of interest, namely, <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution.<br />
It is just<br />
where<br />
K(λ c + a 2/m λ 1 , λ c + a 2/m λ 2 ) → a −2/m K cont (λ 1 , λ 2 ) , (9.26)<br />
K cont =<br />
∫ ∞<br />
µ<br />
dz ψ(z, λ 1 ) ψ(z, λ 2 ) . (9.27)<br />
(To prove <strong>th</strong>is, note <strong>th</strong>at <strong>th</strong>e continuum limit of <strong>th</strong>e Darboux–Christoffel formula is<br />
K cont = ψ(µ, λ 1) ′ ψ(µ, λ 2 ) − ψ(µ, λ 2 ) ′ ψ(µ, λ 1 )<br />
λ 1 − λ 2<br />
. (9.28)<br />
Taking a derivative wi<strong>th</strong> respect to µ, we find ∂ µ K cont = −ψ(µ, λ 1 ) ψ(µ, λ 2 ), and integrating<br />
gives (9.27).) From <strong>th</strong>ese remarks we derive <strong>th</strong>e main statement of double-scaled Fermi<br />
<strong>th</strong>eory:<br />
The nonanalytic dependence on coupling constants in (9.5) comes from <strong>th</strong>e contributions<br />
in <strong>th</strong>e integrals over λ from <strong>th</strong>e edge of <strong>th</strong>e eigenvalue distribution. These contributions<br />
in turn may be studied by using <strong>th</strong>e double-scaled fermion field <strong>th</strong>eory, i.e., <strong>th</strong>e<br />
<strong>th</strong>eory of free fermions wi<strong>th</strong> expansion<br />
∫<br />
ˆψ(λ) =<br />
dz a(z) ψ(z, λ) , (9.29)<br />
where ψ(z, λ) is <strong>th</strong>e or<strong>th</strong>onormalized eigenfunction of <strong>th</strong>e Lax operator Q which is exponentially<br />
decaying for λ → +∞ and oscillatory for λ → −∞. The free oscillators satisfy<br />
a(z)|µ〉 = 0 (z < µ) a † (z)|µ〉 = 0 (z > µ)<br />
{a(z), a † (z ′ )} = δ(z − z ′ ) .<br />
(9.30)<br />
112
In particular, continuum loop amplitudes are obtained from <strong>th</strong>e double-scaled operator<br />
creating macroscopic loops<br />
∫<br />
W (l) = dλ e lλ ˆψ† ˆψ(λ) . (9.31)<br />
Al<strong>th</strong>ough we integrate λ over <strong>th</strong>e entire real axis, in fact <strong>th</strong>e Laplace transform converges.<br />
A detailed study of <strong>th</strong>e asymptotics of <strong>th</strong>e Baker-Akhiezer functions shows <strong>th</strong>at ψ decreases<br />
exponentially fast (as exp ( −λ m+ 1 2 /κ ) ) off <strong>th</strong>e region of perturbative support of<br />
<strong>th</strong>e eigenvalue density, and oscillates wi<strong>th</strong> an algebraically decaying envelope wi<strong>th</strong>in <strong>th</strong>e<br />
region of support. On <strong>th</strong>at region, <strong>th</strong>e eigenvalue density grows algebraically and is Laplace<br />
transformable.<br />
10. Loops and States in Matrix Model Quantum Gravity<br />
10.1. Computation of Macroscopic Loops<br />
We now use <strong>th</strong>e fermion formalism to calculate macroscopic loop amplitudes in <strong>th</strong>e<br />
one-matrix model. Beginning wi<strong>th</strong> <strong>th</strong>e one-loop amplitude we insert (9.31) to get one of<br />
<strong>th</strong>e beautiful results of [104], 40<br />
〈 〉<br />
W (l) = κ<br />
−1 + κ<br />
+<br />
κ 3 + . . .<br />
=<br />
=<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
µ<br />
dλ e lλ 〈µ|ψ † ψ(λ)|µ〉 =<br />
dz 〈 z|e lQ |z 〉 .<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
dλ e lλ dz ψ(z, λ) 2<br />
µ<br />
(10.1)<br />
Similarly, <strong>th</strong>e connected amplitude for two macroscopic loops is easily shown to be [104]<br />
〈<br />
W (l1 ) W (l 2 ) 〉 = + + + . . .<br />
κ 2 κ 4<br />
=<br />
∫ ∞<br />
µ<br />
∫ µ<br />
dx dy 〈 x|e l1Q |y 〉 〈 y|e l2Q |x 〉 .<br />
−∞<br />
(10.2)<br />
40 Notice <strong>th</strong>at <strong>th</strong>is is valid only when <strong>th</strong>e Lax operator has a continuous spectrum. This criterion<br />
can be used to select boundary conditions on <strong>th</strong>e physically appropriate solutions to <strong>th</strong>e string<br />
equation: for example, it selects <strong>th</strong>e solutions first isolated in [118].<br />
113
The formulae (10.1) and (10.2), while elegant, do not make manifest <strong>th</strong>e physics of<br />
<strong>th</strong>e models we are discussing. To address <strong>th</strong>is problem, we examine <strong>th</strong>ese formulae at<br />
genus zero.<br />
Since κ counts loops, we can regard <strong>th</strong>e expectation value in (10.1) as a<br />
“quantum-mechanical” expectation value, wi<strong>th</strong> κ playing <strong>th</strong>e role of ¯h, and obtain <strong>th</strong>e<br />
genus zero approximation to <strong>th</strong>e loop formulae as follows. Using <strong>th</strong>e Campbell–Baker–<br />
Hausdorff formula to separate exponentials of ˆp 2 and u, and <strong>th</strong>en inserting a complete set<br />
of eigenfunctions<br />
we obtain<br />
〈p|x〉 ≡ 1 √<br />
2πκ<br />
e ipx/κ , (10.3)<br />
〈<br />
W (l)<br />
〉h=0 = ∫ ∞<br />
µ<br />
dz 〈 z|e −lˆp2 e −lu |z 〉 = 1<br />
2πκ<br />
=<br />
∫ ∞<br />
µ<br />
∫<br />
1 ∞<br />
2 √ πκl 1/2<br />
∫ ∞<br />
dz dp e −lp2 e −lu<br />
−∞<br />
µ<br />
dz e −lu .<br />
(10.4)<br />
Let us consider <strong>th</strong>is formula first for <strong>th</strong>e case of pure gravity. If we wish to calculate <strong>th</strong>e<br />
expectation value of a loop wi<strong>th</strong> <strong>th</strong>e cosmological constant inserted, we take a derivative<br />
wi<strong>th</strong> respect to µ to bring <strong>th</strong>e operator down from <strong>th</strong>e action, yielding<br />
〈 ∫ 〉<br />
W (l) e γφ 1<br />
=<br />
2 √ πκl 1/2 e−lu(µ) . (10.5)<br />
In pure gravity, <strong>th</strong>e string equation is <strong>th</strong>e Painlevé I equation (7.17):<br />
u 2 − κ2<br />
3 u′′ = z , (10.6)<br />
and <strong>th</strong>e genus zero equation becomes simply u(z) = z 1/2 . The matrix model result for <strong>th</strong>e<br />
wavefunction of <strong>th</strong>e cosmological constant is <strong>th</strong>us<br />
l<br />
V<br />
=<br />
〈<br />
W (l)<br />
precisely as expected from <strong>th</strong>e continuum <strong>th</strong>eory (e.g. sec. 4.3).<br />
(10.7)<br />
Exercise. Spectrum of 2D Gravity<br />
Using <strong>th</strong>e KPZ formula (4.6), show <strong>th</strong>at <strong>th</strong>e spectrum of numbers ν in <strong>th</strong>e WdW<br />
equation (4.7) for <strong>th</strong>e case of pure gravity is ν = ν j = j + 1 2 , j ≥ 0.<br />
114
More generally, we may calculate <strong>th</strong>e one macroscopic loop amplitude for general<br />
perturbed (2, 2m − 1) models coupled to gravity along <strong>th</strong>e following lines. The genus zero<br />
limit of <strong>th</strong>e string equation (7.43) can be written<br />
∑<br />
t j u j = 0 . (10.8)<br />
j≥0<br />
(Note µ = −t 0 . Recall <strong>th</strong>at <strong>th</strong>e t j describe <strong>th</strong>e coupling to <strong>th</strong>e various scaling operators<br />
and if <strong>th</strong>e largest nonvanishing t j has j = m, we are describing 2D gravity coupled to <strong>th</strong>e<br />
(2, 2m − 1) minimal model.) Using (10.8), can explicitly evaluate <strong>th</strong>e loop amplitude as<br />
〈<br />
W (l)<br />
〉h=0 = 1 ∫ ∞<br />
dx e −lu(x;ti) = 1 ∫ ∞<br />
dy ( ∑<br />
∞ j t<br />
κl 1/2 −t 0<br />
κl 1/2 j y j−1) e −ly<br />
u j=1<br />
= 1 ∞∑<br />
j t j u j−1/2 ψ<br />
κl<br />
j−1 (˜l) ,<br />
j=1<br />
(10.9)<br />
where<br />
ψ j (x) ≡ j! x −j−1/2 (1 + x + x 2 /2! + · · · x j /j!) e −x , (10.10)<br />
and ˜l ≡ ul. (We assume here <strong>th</strong>at all but finitely many t j are nonzero.) The relation of<br />
<strong>th</strong>ese amplitudes to <strong>th</strong>e Bessel functions of <strong>th</strong>e continuum <strong>th</strong>eory is less evident <strong>th</strong>an in<br />
(10.7) and will be explained in sec. 10.3 .<br />
Similarly, we can study <strong>th</strong>e genus zero approximation to <strong>th</strong>e propagator as<br />
〈<br />
W (l1 )W (l 2 ) 〉 h=0 = e−(l 1+l 2 )u<br />
√<br />
l1 l 2 κ 2<br />
∫ ∞ ∫ −t0<br />
dx<br />
−t 0 −∞<br />
= 2 √ l 1 l 2<br />
e −u(µ)(l 1+l 2 )<br />
l 1 + l 2<br />
.<br />
dy e −(x−y)2 (l 1 +l 2 )/(4κ 2 l 1 l 2 )<br />
(10.11)<br />
Exercise.<br />
Prove (10.11) by inserting (10.3) into (10.2). Similarly, try to prove<br />
〈 W (l1 ) W (l 2 ) W (l 3 ) 〉 = 2 ∂u<br />
∂t 0<br />
√<br />
l 1 l 2 l 3 e −u(l 1+l 2 +l 3 ) . (10.12)<br />
(We will prove <strong>th</strong>is more efficiently below.)<br />
115
10.2. Loops to Local Operators<br />
By shrinking <strong>th</strong>e loops we can obtain correlation functions of <strong>th</strong>e local operators.<br />
This intuition comes from <strong>th</strong>e critical string example discussed in chapt. 1 and from <strong>th</strong>e<br />
expression for <strong>th</strong>e matrix model loop operator (8.4) which is manifestly an expansion in<br />
local operators.<br />
In eq. (7.44), we saw <strong>th</strong>at in <strong>th</strong>e matrix model <strong>th</strong>ere are scaling operators σ j ∝ K j−1/2<br />
+<br />
scaling like e αjφ wi<strong>th</strong> α j /γ = 1 (m − j), j = 0, 1, . . . . According to (3.88), <strong>th</strong>e macroscopic<br />
2<br />
loop operator has an expansion as in (3.87),<br />
W (l) = ∑ j≥0<br />
l j+ 1 2 σj . (10.13)<br />
Exercise. Exponents<br />
Use <strong>th</strong>e result (3.88) to verify <strong>th</strong>e expansion (10.13) using <strong>th</strong>e fact <strong>th</strong>at for pure<br />
gravity we have Q/γ = 5/2 and α j /γ = 1 − j/2, j = 0, 1, . . ..<br />
As we have already discussed in <strong>th</strong>e context of semiclassical Liouville <strong>th</strong>eory, <strong>th</strong>e<br />
expansion (10.13) is not strictly true and must be treated wi<strong>th</strong> care. As we see from<br />
(10.9), <strong>th</strong>ere can be negative powers of l in <strong>th</strong>e small l expansion of loop correlators. In<br />
sec. 3.11, we showed <strong>th</strong>at <strong>th</strong>e l → 0 behavior of loop amplitudes must satisfy certain<br />
rules which imply <strong>th</strong>at one can unambiguously extract <strong>th</strong>e correlators of local operators.<br />
The rules of sec. 3.11 are confirmed by explicit matrix model computations. For example,<br />
notice <strong>th</strong>at (10.9) can be written as<br />
〈<br />
W (l)<br />
〉h=0 m = ∑ 1<br />
j! t j<br />
j=0 l j+ 1 2<br />
+ ∑ n≥0<br />
l n+ 1 2<br />
Γ(n + 3 2 )〈σ n〉 . (10.14)<br />
The divergent terms in l are indeed analytic in <strong>th</strong>e coupling constants. Similarly, notice<br />
<strong>th</strong>at (10.11, 10.12) are smoo<strong>th</strong> as any loopleng<strong>th</strong> goes to zero. In general, wi<strong>th</strong> <strong>th</strong>e rules<br />
(1) and (2) from <strong>th</strong>e end of sec. 3.11 in mind, we can extract correlation functions of local<br />
operators by shrinking macroscopic loops.<br />
Exercise.<br />
Using (10.12), calculate 〈σ n1 σ n2 σ n3 〉.<br />
116
Exercise. The general amplitude<br />
Using rules (1) and (2) and <strong>th</strong>e genus zero KdV flow equations, we will prove <strong>th</strong>at<br />
〈 n<br />
∏<br />
i=1<br />
〉 ∏<br />
( ) n−3 −u·∑<br />
W (l i ) = l 1/2 ∂<br />
l<br />
i<br />
e i . (10.15)<br />
∂t 0<br />
For example, to prove <strong>th</strong>e <strong>th</strong>ree macroscopic loop formula proceed as follows:<br />
〈 W (l1 ) W (l 2 ) W (l 3 ) 〉 =<br />
=<br />
∞∑<br />
n=0<br />
∞∑<br />
n=0<br />
〈<br />
l n+1/2<br />
1 σn W (l 2 ) W (l 3 ) 〉<br />
l n+1/2<br />
1<br />
√<br />
= −2 l 2 l 3 e −u(l 2+l 3 )<br />
∂ 〈 W (l2 ) W (l 3 ) 〉<br />
∂t n<br />
∞∑<br />
(−1) n+1 l n+1/2<br />
1 ∂u<br />
n! ∂t 0<br />
n=0<br />
= 2 ∂u<br />
∂t 0<br />
√<br />
l 1 l 2 l 3 e −u(l 1 +l 2 +l 3 ) .<br />
(10.16)<br />
In <strong>th</strong>e last line we may obtain <strong>th</strong>e l dependence immediately since <strong>th</strong>e amplitude must<br />
be totally symmetric in l 1 , l 2 , l 3 . Give a complete proof of (10.15) along <strong>th</strong>ese lines by<br />
induction.<br />
This formula was first discovered in [119] from a different point of view and <strong>th</strong>en<br />
rediscovered in [36]. The strange fact <strong>th</strong>at <strong>th</strong>e amplitude is essentially only a function<br />
of <strong>th</strong>e sum of <strong>th</strong>e loop leng<strong>th</strong>s has never been given a simple explanation.<br />
10.3. Wavefunctions and Propagators from <strong>th</strong>e Matrix Model<br />
Let us finally match <strong>th</strong>e expectations of sec. 4.3 above, specifically <strong>th</strong>e Bessel function<br />
behavior of wavefunctions, wi<strong>th</strong> <strong>th</strong>e results of <strong>th</strong>e matrix model computations of sec. 10.1.<br />
At first <strong>th</strong>e results appear to be very different but <strong>th</strong>is turns out to be a matter of working<br />
in two different bases.<br />
One quick way to see <strong>th</strong>is is to use <strong>th</strong>e Gegenbauer addition formula to expand <strong>th</strong>e<br />
genus zero propagator (10.11) in terms of Bessel functions 41<br />
√<br />
l1 l 2<br />
e −u(l 1+l 2 )<br />
l 1 + l 2<br />
= θ(l 2 − l 1 )<br />
∞∑<br />
j=0<br />
(−1) j (2j + 1) I j+<br />
1 (l 1u) K<br />
2 j+<br />
1 (l 2u) + [1 ↔ 2] . (10.17)<br />
2<br />
41 Al<strong>th</strong>ough we have pulled <strong>th</strong>is identity out of a hat, it is quite natural. The Yukawa potential<br />
in <strong>th</strong>ree spatial dimensions, which is <strong>th</strong>e Green’s function for <strong>th</strong>e Helmholtz operator ∇ 2 − µ,<br />
is just e −µ|r 1 −r 2 | /|r 1 − r 2 |.<br />
Gegenbauer addition <strong>th</strong>eorem amounts to <strong>th</strong>at expansion.<br />
This Green’s function may be expanded in partial waves and <strong>th</strong>e<br />
117
This suggests <strong>th</strong>at instead of <strong>th</strong>e local operator expansion (10.13), we use a different<br />
expansion<br />
W (l) = ∑ j≥0<br />
l j+ 1 2 σj = 2<br />
∞∑<br />
I 1 (lu)<br />
ˆσ j (−1) j j+<br />
2<br />
(2j + 1) . (10.18)<br />
u j+1/2<br />
j=0<br />
That is, instead of expanding <strong>th</strong>e loop in terms of <strong>th</strong>e functions l 1/2 , l 3/2 , l 5/2 , . . . , we use<br />
<strong>th</strong>e basis functions I 1/2 , I 3/2 , . . . . Using <strong>th</strong>e properties of I, we see <strong>th</strong>at <strong>th</strong>is change of basis<br />
is upper triangular and hence <strong>th</strong>e operators ˆσ j are related to σ j by an upper triangular<br />
transformation whose coefficients are analytic functions of u 2 .<br />
Since we isolate continuum contributions from nonanalytic dependence on couplings<br />
like µ, we must not mix operators wi<strong>th</strong> coefficients <strong>th</strong>at are nonanalytic in µ. Conversely,<br />
we are always free to make redefinitions involving coefficients which are analytic in µ. Since<br />
<strong>th</strong>e critical exponents α j /γ are rational, <strong>th</strong>ere can be operator mixing, and hence <strong>th</strong>ere is<br />
no unique definition of scaling operators. In order for <strong>th</strong>e change of basis (10.18) to satisfy<br />
<strong>th</strong>is rule, u 2 must be an analytic function of <strong>th</strong>e couplings t k . One way <strong>th</strong>is can happen is<br />
by considering perturbations around <strong>th</strong>e pure gravity point where u 2 = 2µ. 42<br />
The wavefunctions of <strong>th</strong>e operators ˆσ j are given by shrinking one of <strong>th</strong>e loops,<br />
〈ˆσ j W (l)〉 = u j+ 1 2 Kj+ 1 (ul) , (10.19)<br />
2<br />
in complete agreement wi<strong>th</strong> <strong>th</strong>e Euclidean on-shell wavefunctions of sec. 4.3! Thus we have<br />
actually done better <strong>th</strong>an we had any right to expect, <strong>th</strong>e minisuperspace approximation<br />
to <strong>th</strong>e wavefunctions turned out to be exact for <strong>th</strong>ese boundary conditions.<br />
In <strong>th</strong>e <strong>th</strong>eory of a particle, <strong>th</strong>e propagator was written in terms of on-shell and off-shell<br />
states as in (5.6) above. Similarly here we may write <strong>th</strong>e matrix model propagator in a<br />
way which nicely summarizes <strong>th</strong>e spectrum of <strong>th</strong>e <strong>th</strong>eory<br />
〈<br />
W (l1 )W (l 2 ) 〉 ≡<br />
∫ ∞<br />
0<br />
dE<br />
2π G(E) ψ E(l 1 ) ψ E (l 2 ) , (10.20)<br />
where<br />
G(E) =<br />
∞∑<br />
j=0<br />
(−1) j (2j + 1)<br />
E 2 + (j + 1 2 )2 =<br />
π<br />
cosh πE . (10.21)<br />
42 More generally, one should look at <strong>th</strong>e so-called “conformal backgrounds,” which, as argued<br />
in [36], are <strong>th</strong>e precise matrix model couplings corresponding to a tensor product wi<strong>th</strong> a conformal<br />
(2, 2m−1) model. An understanding of <strong>th</strong>ese backgrounds was needed to resolve certain paradoxes<br />
about one- and two-point functions in 2d gravity [36].<br />
118
It is extremely interesting to note <strong>th</strong>at — even for pure gravity — <strong>th</strong>e <strong>th</strong>ird quantized<br />
universe propagator is not <strong>th</strong>e naive minisuperspace Wheeler–DeWitt propagator (5.20).<br />
In particular <strong>th</strong>e ultraviolet behavior of <strong>th</strong>e propagator (in E) is completely different from<br />
<strong>th</strong>e naive propagator (5.20). For example, <strong>th</strong>e 1/E 2 behavior in <strong>th</strong>e ultraviolet becomes<br />
e −πE behavior. Our understanding of why <strong>th</strong>is is so is incomplete. (Part of <strong>th</strong>e story is<br />
explained in <strong>th</strong>e next section.)<br />
In general, we can decompose amplitudes as<br />
〈<br />
W (l1 ) · · · W (l n ) 〉 ∫ ∏<br />
=<br />
i<br />
dE i ψ Ei<br />
(l i ) A(E 1 , . . . , E n ) , (10.22)<br />
which (in <strong>th</strong>e case of <strong>th</strong>e four-point amplitude) we depict as<br />
l 1<br />
l 2<br />
∫ ∏<br />
=<br />
i<br />
l 4 l 3<br />
By shrinking l i to zero we get an expansion in terms of local operators. Alternatively, and<br />
equivalently, by doing <strong>th</strong>e integral over <strong>th</strong>e E i we pick up residues corresponding to <strong>th</strong>e<br />
Euclidean on-shell states. A similar picture emerges for all <strong>th</strong>e multicritical points. The<br />
following exercise carries <strong>th</strong>is out in detail for <strong>th</strong>e Ising model.<br />
d<br />
Exercise. The Ising Model<br />
The Ising model has a Z 2 symmetry flipping up spins for down, which, in <strong>th</strong>e<br />
matrix model formulation described in sec. 7.5 is exchange of U ↔ V . Letting W ± (l)<br />
denote <strong>th</strong>e Z 2 odd/even loop operators show <strong>th</strong>at<br />
〈<br />
W± (l 1 )W ± (l 2 ) 〉 ∑<br />
= ± (j + 1/3) I j+1/3 (2 √ µl 1 ) K j+1/3 (2 √ µl 2 )<br />
j,±<br />
∑<br />
∓ (j + 2/3) I j+2/3 (2 √ µl 1 ) K j+2/3 (2 √ µl 2 ) .<br />
j,∓<br />
(10.23)<br />
By summing <strong>th</strong>e infinite series, show <strong>th</strong>at<br />
G(E, ±) = 2π(e πE ± 1 + e −πE sinh πE<br />
)<br />
sinh 3πE . (10.24)<br />
Remark: We have shown <strong>th</strong>at <strong>th</strong>e wavefunctions 〈 σW (l) 〉 satisfy a linear WdW equation.<br />
On <strong>th</strong>e o<strong>th</strong>er hand, from <strong>th</strong>e Schwinger–Dyson equations (8.18) and <strong>th</strong>eir continuum<br />
analogs, we see <strong>th</strong>at 〈 W (l) 〉 itself satisfies a nonlinear equation. A precise understanding<br />
of <strong>th</strong>e relation of <strong>th</strong>ese has never been given (except in special cases [120]). This is an<br />
interesting problem for <strong>th</strong>e future.<br />
119
Fig. 20: Two loops on a continuum surface collide.<br />
10.4. Redundant operators, singular geometries and contact terms<br />
One important and not generally discussed issue is <strong>th</strong>e contribution of singular geometries<br />
to <strong>th</strong>e pa<strong>th</strong> integral. In <strong>th</strong>e case of macroscopic loop amplitudes, <strong>th</strong>ere are geometries<br />
in which loops collide to make figure-eights as in fig. 20, and as well more complicated geometries.<br />
Our understanding of <strong>th</strong>e contributions of <strong>th</strong>ese geometries is very incomplete,<br />
but <strong>th</strong>ere is plenty of evidence <strong>th</strong>at such terms are responsible for several peculiarities of<br />
<strong>th</strong>e matrix model answers (e.g. <strong>th</strong>e cosh propagator discovered above) and perhaps lie<br />
at <strong>th</strong>e heart of a geometrical understanding of <strong>th</strong>e Lian–Zuckerman states. See also <strong>th</strong>e<br />
discussion at <strong>th</strong>e end of sec. 11.4 below.<br />
11. Loops and States in <strong>th</strong>e c = 1 Matrix Model<br />
11.1. Definition of <strong>th</strong>e c = 1 Matrix Model<br />
There are several approaches to defining a matrix model for gravity coupled to c = 1<br />
matter. The most direct me<strong>th</strong>od is <strong>th</strong>e discretization of <strong>th</strong>e Polyakov pa<strong>th</strong> integral for a<br />
one-dimensional target space,<br />
Z qg (κ, g) = ∑ Λ<br />
κ 2h−2 g |Λ|<br />
V ∏<br />
i=1<br />
∫<br />
dX i e − ∑ 〈ij〉 L(X i − X j ) ,<br />
(11.1)<br />
where |Λ| is <strong>th</strong>e number of vertices on <strong>th</strong>e lattice Λ which is summed over Euler character<br />
2 − 2h, and <strong>th</strong>e nearest neighbor interaction L(X i − X j ) between <strong>th</strong>e bosonic fields X i,j<br />
at vertices i, j is summed over links 〈ij〉 between vertices.<br />
120
The asymptotic expansion in κ of <strong>th</strong>e partition function (11.1) can be equivalently<br />
generated 43 from an integral over N × N matrices,<br />
∫<br />
Z(N; g) = ln<br />
DΦ e −Ng−1 tr [∫ dX dY 1 2 Φ(X) G−1 (X − Y ) Φ(Y ) + ∫ dX V ( Φ(X) )] ,<br />
(11.2)<br />
where Φ(X) is an N × N hermitian matrix field, V is a polynomial interaction of some<br />
fixed order, and <strong>th</strong>e propagator G(X) = exp −L(X). g is a loop counting parameter, and<br />
<strong>th</strong>erefore counts <strong>th</strong>e number of vertices in <strong>th</strong>e dual graph (identified wi<strong>th</strong> <strong>th</strong>e area of <strong>th</strong>e<br />
lattice). As in sec. 6.1, <strong>th</strong>e coefficient of N χ in an expansion of Z in a powers of N 2 gives<br />
<strong>th</strong>e sum of all connected Feynman diagrams wi<strong>th</strong> Euler character χ. As functions of g,<br />
<strong>th</strong>ese coefficients are all singular at a critical coupling g c where <strong>th</strong>e perturbation series<br />
diverges.<br />
The continuum limit can be extracted from <strong>th</strong>e leading singular behavior as<br />
g → g c , a limit which emphasizes graphs wi<strong>th</strong> an infinite number of vertices.<br />
Taking L(X i − X j ) = (X i − X j ) 2 in (11.1) leads to <strong>th</strong>e continuum limit form<br />
∫<br />
d 2 ξ √ g g ab ( ∇ a X)( ∇ b X), <strong>th</strong>us providing a standard discretization of <strong>th</strong>e Polyakov string<br />
[2] embedded in one dimension. This quadratic choice corresponds to a gaussian propagator,<br />
G(X) ∼ exp(−X 2 ), in <strong>th</strong>e matrix model (11.2). In momentum space, <strong>th</strong>e leading small<br />
momentum behavior of <strong>th</strong>e gaussian form G −1 (P ) ∼ exp P 2 coincides wi<strong>th</strong> <strong>th</strong>at of <strong>th</strong>e Feynman<br />
form G −1 (P ) ∼ 1+P 2 , which corresponds in position space to G(X) = exp(−|X|). As<br />
argued in [9], <strong>th</strong>is substitution (corresponding to L(X i − X j ) = |X i − X j |, wi<strong>th</strong> continuum<br />
form |g ab ∂ a X ∂ b X| 1/2 ), should not affect <strong>th</strong>e critical properties (e.g. critical exponents).<br />
Due to <strong>th</strong>e ultraviolet convergence of <strong>th</strong>e model, only its short distance, i.e. non-universal<br />
behavior, is affected. 44 For <strong>th</strong>e same reason, continuum answers should only depend on <strong>th</strong>e<br />
universality class of <strong>th</strong>e potential V , <strong>th</strong>e necessary conditions for which will be discussed<br />
below. For now we simply require V (φ) to go to +∞ for φ → ±∞ in order <strong>th</strong>at (11.2) is<br />
well-defined.<br />
For <strong>th</strong>e latter choice of propagators, i.e. <strong>th</strong>e Feynman propagator, after rescaling Φ<br />
we can write (11.2) as<br />
( ∫<br />
F mm (N; g, V ) ≡ lim T −1 ln DΦ(X) e −N ∫ T<br />
0 dX tr( ˙Φ 2 + g −1 V ( √ gΦ)) ) . (11.3)<br />
T →∞<br />
43 Note <strong>th</strong>at <strong>th</strong>is matrix model construction works equally well for bosons X µ , µ = 1, . . . , D, to<br />
generate strings embedded in D dimensions. For D > 1, however, <strong>th</strong>e matrix model representation<br />
is no longer solvable.<br />
44 Indeed we will see <strong>th</strong>at energies of order ɛ ∼ 1/N dominate <strong>th</strong>e continuum limit.<br />
121
Using Feynman diagrams to obtain <strong>th</strong>e large N asymptotics of <strong>th</strong>e function F mm , we write<br />
(as in (11.1))<br />
F(N; g, V ) mm ∼ ∑ ∫<br />
N 2−2h g |Λ|<br />
Λ<br />
∏<br />
dX i e −|X i−X j | , (11.4)<br />
The quantum mechanical model (11.3) was solved to leading order in large N in [82].<br />
Interpreting <strong>th</strong>e solution as <strong>th</strong>e partition function (11.4) of 2d gravity on a genus zero<br />
worldsheet coupled to a single gaussian massless field, it was shown in [9] <strong>th</strong>at <strong>th</strong>e string<br />
susceptibility exponent, defined by <strong>th</strong>e leading singular behavior Z(g) = (g c − g) 2−Γ str<br />
,<br />
satisfies Γ str = 0, in agreement wi<strong>th</strong> <strong>th</strong>e continuum prediction of [33]. The emergence<br />
of such physically reasonable answers in <strong>th</strong>e continuum limit supports <strong>th</strong>e assumption of<br />
universality wi<strong>th</strong> respect to <strong>th</strong>e choice of propagator.<br />
We will now study <strong>th</strong>e continuum limit of <strong>th</strong>e integral (11.3) to confirm and extend<br />
<strong>th</strong>e above discussion.<br />
〈ij〉<br />
11.2. Matrix Quantum Mechanics<br />
From general principles, we see <strong>th</strong>at (11.3) is simply <strong>th</strong>e ground state energy for <strong>th</strong>e<br />
quantum mechanics of an N × N matrix, and was analyzed from <strong>th</strong>is point of view in [82].<br />
The reduction to free fermions can be established quickly using a pa<strong>th</strong> integral argument<br />
given in [86]. In <strong>th</strong>e matrix quantum mechanics, we can discretize <strong>th</strong>e time coordinate<br />
X → X i and <strong>th</strong>en pass to <strong>th</strong>e action<br />
S = N ∑ i<br />
tr Φ(X i ) Φ(X i+1 ) + ∑ tr V ( Φ(X i ) ) . (11.5)<br />
We now analyze <strong>th</strong>e model as a matrix chain model as in sec. 7.6, diagonalizing<br />
Φ(X i ) = Ω i Λ i Ω −1<br />
i<br />
, (11.6)<br />
where Λ i = Diag(λ 1 (X i ), . . . λ N (X i )). The Vandermonde determinants all cancel except<br />
for <strong>th</strong>e first and last. Taking <strong>th</strong>e time lattice spacing to zero, we are left wi<strong>th</strong> a pa<strong>th</strong><br />
integral for N quantum mechanical degrees of freedom λ i (t),<br />
(∫<br />
F mm (N; g, V ) ≡ lim T ∏ N −1 log<br />
T →∞<br />
i=1<br />
Dλ i (t) ∆(λ i (0)) ∆(λ i (T ))<br />
∫ ∑<br />
· e −N T<br />
( dx N ˙λ2<br />
0 i=1 i +g−1 V ( √ ))<br />
gλ i )<br />
.<br />
(11.7)<br />
122
Thus we are studying <strong>th</strong>e quantum mechanics of N free fermions moving in a potential<br />
g −1 V (λ √ g) wi<strong>th</strong> Planck’s constant equal to ¯h = 1/N,<br />
F mm (N; g, V ) ≡ 1¯h E ground = N<br />
N∑<br />
ɛ i (¯h = 1/N; g, V ) , (11.8)<br />
i=1<br />
where<br />
To define <strong>th</strong>e double scaling limit we must:<br />
(<br />
− 1 d 2<br />
2N 2 dλ + 1 )<br />
2 g V (λ√ g) ψ i = ɛ i ψ i . (11.9)<br />
1) Compute <strong>th</strong>e asymptotic expansion as N → ∞,<br />
2) Isolate <strong>th</strong>e leading singular behavior of F h as g → g c .<br />
F mm ∼ ∑ N 2−2h F h (g, V ) . (11.10)<br />
3) Determine <strong>th</strong>e scaling variable and scaling functions as g → g c and N → ∞.<br />
V (λ)<br />
λ<br />
Fig. 21: Generic potential wi<strong>th</strong> quadratic maximum at λ = 0.<br />
We begin by studying <strong>th</strong>e function F 0 (g, V ) corresponding to genus zero surfaces.<br />
The potentials of interest are polynomials <strong>th</strong>at have a quadratic maximum. We place <strong>th</strong>is<br />
maximum at λ = 0 and shift V so <strong>th</strong>at V (0) = 0, so <strong>th</strong>at V might look as in fig. 21. To<br />
fix ideas, one can take<br />
V (λ) = − 1 2 λ2 + 1 4 λ4 , (11.11)<br />
but it is important to note <strong>th</strong>at <strong>th</strong>e results hold for a large class of potentials, <strong>th</strong>us providing<br />
evidence <strong>th</strong>at we are calculating true continuum results and not lattice artifacts.<br />
Since ¯h = 1/N in our problem, <strong>th</strong>e function F 0 can be calculated as <strong>th</strong>e leading term in<br />
a semiclassical expansion using <strong>th</strong>e WKB approximation. Classically <strong>th</strong>e energy becomes<br />
123
continuous and particle states are specified by points in fermion phase space (λ, p). By<br />
<strong>th</strong>e Pauli exclusion principle, we can put at most one fermion in each volume element of<br />
area 2π¯h in phase space. At <strong>th</strong>e same time we are putting O(1/¯h) distinct particles into<br />
<strong>th</strong>e sea, so <strong>th</strong>e sea covers a region of area O(1). In <strong>th</strong>e classical limit, <strong>th</strong>e state described<br />
by <strong>th</strong>e Fermi sea of <strong>th</strong>e free fermions is <strong>th</strong>us a region in phase space. By <strong>th</strong>e Liouville<br />
<strong>th</strong>eorem, <strong>th</strong>e time evolution of <strong>th</strong>e system preserves <strong>th</strong>e area of <strong>th</strong>is region, so we may<br />
<strong>th</strong>ink of <strong>th</strong>e region as a fluid in phase space. We will return to <strong>th</strong>is picture in chapters 12<br />
and 13 below. The fluid has a total area determined by <strong>th</strong>e Fermi level, which is in turn<br />
fixed by <strong>th</strong>e total number of fermions,<br />
∫ dp dλ<br />
N =<br />
2π¯h θ(ɛ F − ɛ) , (11.12)<br />
implying <strong>th</strong>at<br />
where<br />
1 =<br />
∫ dp dλ<br />
2π θ(ɛ F − ɛ) , (11.13)<br />
ɛ(p, λ) = 1 2 p2 + 1 g V (g√ λ) .<br />
Thus, we require <strong>th</strong>at <strong>th</strong>e fluid have total area one. The total energy is<br />
∫ dp dλ<br />
E ground =<br />
2π¯h ɛ θ(ɛ F − ɛ) , (11.14)<br />
which implies<br />
F 0 (g; V ) =<br />
∫ dp dλ<br />
2π ɛ θ(ɛ F − ɛ) . (11.15)<br />
Eq. (11.13) determines <strong>th</strong>e Fermi level ɛ F as a function of g, and <strong>th</strong>en (11.15) determines<br />
F 0 (g; V ).<br />
p<br />
−<br />
2<br />
g<br />
2<br />
g<br />
λ<br />
Fig. 22: Level curves in phase space. Filled Fermi levels are shaded.<br />
124
To see how singularities of F 0 can arise, let us consider <strong>th</strong>e specific potential (11.11)<br />
for which<br />
ɛ(p, λ) = 1 2 p2 − 1 2 λ2 + 1 4 gλ4 . (11.16)<br />
Level curves for ɛ are plotted in fig. 22. At small values of g, <strong>th</strong>e lattice expansion (11.4)<br />
(at fixed topology) converges. To define <strong>th</strong>e continuum limit, we look for <strong>th</strong>e leading<br />
singularity in F 0 due to <strong>th</strong>e singularity in g closest to <strong>th</strong>e origin. In general, as we vary<br />
g <strong>th</strong>e region of unit area defined by (11.13), and hence <strong>th</strong>e weighted area (11.15), vary<br />
analytically. As we tune g from small values (as in fig. 22) to large values, however, g<br />
passes <strong>th</strong>rough a value of order 1 where <strong>th</strong>e ɛ = 0 line surrounds <strong>th</strong>e unit area. At <strong>th</strong>is<br />
juncture <strong>th</strong>e shape of <strong>th</strong>e Fermi sea equipotential changes discontinuously, resulting in<br />
nonanalytic behavior in F 0 . Thus we are interested in <strong>th</strong>e limit g → g c , where g c is defined<br />
by equating <strong>th</strong>e Fermi level ɛ F wi<strong>th</strong> <strong>th</strong>e top of <strong>th</strong>e quadratic maximum (= 0 here by<br />
convention).<br />
(λ)<br />
λ<br />
p +<br />
(λ)<br />
p<br />
−<br />
Fig. 23: Blowup near origin of fig. 22.<br />
From <strong>th</strong>e above discussion, it is (intuitively) clear <strong>th</strong>at <strong>th</strong>e nonanalytic part of <strong>th</strong>e<br />
integrals (11.13, 11.15) comes from <strong>th</strong>e crossing region λ, p ∼ 0, and blowing up <strong>th</strong>is region<br />
gives <strong>th</strong>e picture in fig. 23. The singular dependence of F 0 can <strong>th</strong>erefore be determined by<br />
∫ λ2<br />
dλ<br />
( 1<br />
F 0 = 2<br />
λ 1<br />
2π 6 (p3 + − p 3 −) + (− 1 2 λ2 + g 4 λ4 )(p + − p − ))<br />
= 2 ∫ 1<br />
dλ(<br />
π<br />
√ −2ɛF 3 (ɛ F + 1 2 λ2 ) − 1 2 λ2 + g 4 λ4) √<br />
λ2 + 2ɛ F + · · · (11.17)<br />
= 1 (<br />
ɛ 2 F + 1 )<br />
π 4 ɛ3 F g log(−ɛ F ) + · · · .<br />
125
Here λ 1,2 are <strong>th</strong>e two turning points, p ± define <strong>th</strong>e upper and lower branches of <strong>th</strong>e Fermi<br />
surface, <strong>th</strong>e first line is exact, and <strong>th</strong>e terms omitted in <strong>th</strong>e subsequent lines are analytic in<br />
g and ɛ F . Since <strong>th</strong>e critical value for g is of order one, we immediately obtain <strong>th</strong>e leading<br />
nonanalytic behavior as g → g c as 45 F 0 = ɛ 2 F log(−ɛ F ) + · · ·<br />
N 2 F 0 = µ 2 log µ + · · · ,<br />
(11.18)<br />
where<br />
µ ≡ −Nɛ F , (11.19)<br />
and <strong>th</strong>e ellipsis in <strong>th</strong>e second line of (11.18) indicates terms which are analytic or less<br />
singular in µ. (In particular, we can write log µ ra<strong>th</strong>er <strong>th</strong>an log(−ɛ F ) in (11.18) since <strong>th</strong>e<br />
difference is just an analytic piece µ 2 log N.)<br />
Exercise. Doing <strong>th</strong>e integrals<br />
For <strong>th</strong>e specific example (11.16), all <strong>th</strong>e integrals can be done explicitly in terms of<br />
elliptic functions. Perform <strong>th</strong>ese integrals and verify <strong>th</strong>e statements about nonanalytic<br />
dependence from <strong>th</strong>e exact results.<br />
Four important remarks:<br />
1) From <strong>th</strong>e above derivation, it is clear <strong>th</strong>at <strong>th</strong>e critical properties are independent of<br />
<strong>th</strong>e detailed form of V and depend only on <strong>th</strong>e existence of a quadratic maximum.<br />
2) The result (11.18) suggests <strong>th</strong>at it is µ and not any power of g c − g <strong>th</strong>at should be<br />
taken as <strong>th</strong>e scaling variable. This is indeed <strong>th</strong>e case as we will confirm in <strong>th</strong>e next<br />
section. Thus <strong>th</strong>e double scaling limit is defined by varying g so <strong>th</strong>at µ = −ɛ F (g)N is<br />
held fixed. The free energy in <strong>th</strong>is limit takes <strong>th</strong>e form<br />
F ( N, g(N), V ) → F(µ) . (11.20)<br />
This definition of <strong>th</strong>e c = 1 double scaling limit was given in [86–89].<br />
3) The relation between <strong>th</strong>e bare cosmological constant g and <strong>th</strong>e scaling variable −ɛ F N<br />
is subtle. We can obtain <strong>th</strong>is relation, at tree level, from <strong>th</strong>e relation (11.13):<br />
∫ λ2<br />
1 = 2<br />
λ 1<br />
dλ<br />
2π (p + − p − ) = 2 ∫ √<br />
dλ ɛ<br />
π<br />
√ F + 1 2 λ2 + · · · . (11.21)<br />
−2ɛF<br />
45 We have made a constant rescaling to eliminate irrelevant numerical factors.<br />
126
Evaluating <strong>th</strong>e singular part of <strong>th</strong>e integral gives<br />
g − g c ∼ (−ɛ F ) log(−ɛ F ) + · · · . (11.22)<br />
Historically, <strong>th</strong>e peculiar relation (11.22) between <strong>th</strong>e “bare” cosmological constant<br />
and <strong>th</strong>e scaling variable caused a great deal of confusion. (For a discussion, see [15].)<br />
Unlike (7.18), a very complicated function of g − g c multiplies N to form <strong>th</strong>e scaling<br />
variable κ <strong>th</strong>at is held fixed in <strong>th</strong>e double scaling limit.<br />
An interpretation of <strong>th</strong>is<br />
result directly from <strong>th</strong>e continuum spacetime point of view has been given in [75], and<br />
we shall reinterpret <strong>th</strong>is understanding in terms of macroscopic loop field <strong>th</strong>eory in<br />
sec. 11.6 .<br />
4) Multicritical c = 1 Theories. At c < 1 one discovers an enormous space of multicritical<br />
points. At c = 1 <strong>th</strong>is has not been as extensively investigated, al<strong>th</strong>ough some results<br />
may be found in [87]. While <strong>th</strong>e spacetime interpretation of <strong>th</strong>ese <strong>th</strong>eories remains<br />
unclear, <strong>th</strong>ere is evidence <strong>th</strong>at perturbations of <strong>th</strong>e conventional c = 1 <strong>th</strong>eories by<br />
special state operators flow to <strong>th</strong>ese points.<br />
The reason is <strong>th</strong>at <strong>th</strong>e matrix model<br />
defines flows by operators ψ † λ r ψ which yield <strong>th</strong>e above multicritical behavior. As<br />
discussed below, <strong>th</strong>ese operators seem to be related to <strong>th</strong>e special states.<br />
11.3. Double-Scaled Fermi Field Theory<br />
Let us now turn to fermionic quantum mechanics and investigate to all orders of<br />
perturbation <strong>th</strong>eory. Formulated in terms of a fermionic quantum field <strong>th</strong>eory, <strong>th</strong>e <strong>th</strong>eory<br />
has <strong>th</strong>e action<br />
S = N<br />
∫ ∞<br />
and lattice Fermi operators<br />
−∞<br />
dX dλ ˆΨ †( −i d<br />
dX + d2<br />
dλ 2 + 1 g V ( λ √ g )) ˆΨ , (11.23)<br />
ˆΨ(λ, X) =<br />
∞∑<br />
a ɛ (ε i ) ψ ɛ (ε i , λ; V ) e −iεiX . (11.24)<br />
i=1<br />
The wavefunctions ψ ɛ (ε i , λ; V ) are eigenfunctions of <strong>th</strong>e Schrödinger equation (11.9), and<br />
we may take <strong>th</strong>e potential V to be symmetric since it effectively becomes so in <strong>th</strong>e doublescaling<br />
limit. Thus we also have an index ɛ = ± which refers to <strong>th</strong>e parity of <strong>th</strong>e wavefunction,<br />
and a repeated ɛ index will indicate a sum over parity states. The a’s anticommute<br />
and satisfy {a ɛ (ε i ), a † ɛ<br />
(ε ′ j )} = δ ij δ ɛ,ɛ ′. The Fermi sea |N〉 is defined as usual.<br />
127
As we have seen from <strong>th</strong>e tree-level analysis, <strong>th</strong>e singular terms in F come from <strong>th</strong>e<br />
behavior of <strong>th</strong>e <strong>th</strong>eory for λ ∼ O(N −1/2 ) and ɛ ∼ O(1/N). Thus we scale equation (11.9)<br />
by setting λ = ˜λ/ √ 2N and ɛ i = −ε/N, so <strong>th</strong>at (11.9) becomes<br />
( d<br />
2<br />
d˜λ + ˜λ 2<br />
)<br />
2 4 + O(N −1/2 ) ψ i = ε ψ i . (11.25)<br />
All <strong>th</strong>e details of V are in <strong>th</strong>e O(N −1/2 ) piece and what remains is <strong>th</strong>e parabolic cylinder<br />
equation. (See appendix A.) Two consequences of <strong>th</strong>is are:<br />
1) The density of states at <strong>th</strong>e Fermi level can be calculated to all orders of perturbation<br />
<strong>th</strong>eory. The WKB matching of <strong>th</strong>e parabolic cylinder function (valid at <strong>th</strong>e tip of<br />
<strong>th</strong>e potential) to <strong>th</strong>e WKB functions (valid in <strong>th</strong>e o<strong>th</strong>er regions of λ) involves <strong>th</strong>e large ˜λ<br />
asymptotics of <strong>th</strong>e cylinder function. Matching <strong>th</strong>e phases, we find from <strong>th</strong>e asymptotic<br />
formula (A.6) for <strong>th</strong>e behavior of <strong>th</strong>e parabolic cylinder function <strong>th</strong>at <strong>th</strong>e quantization<br />
condition Φ(ε n+1 ) − Φ(ε n ) = π implies <strong>th</strong>at<br />
ρ(ε) ≡ ∂n<br />
∂ε = 1 π Φ′ (ε) = − 1<br />
2π Re ψ( 1 2<br />
+ iε) , (11.26)<br />
where ψ(x) = d<br />
dx<br />
log Γ(x) is <strong>th</strong>e digamma function. Identifying <strong>th</strong>e density of states wi<strong>th</strong><br />
a second derivative of <strong>th</strong>e free energy, we confirm <strong>th</strong>e double scaling procedure mentioned<br />
at <strong>th</strong>e end of <strong>th</strong>e last section. In particular, <strong>th</strong>e semiclassical expansion of ρ is<br />
ρ(ε) = 1 (<br />
− ln ε +<br />
2π<br />
∞∑<br />
(2 2m−1 − 1) |B (<br />
2m| ¯h<br />
) ) 2m<br />
, (11.27)<br />
m 2ε<br />
m=1<br />
where <strong>th</strong>e B 2m are Bernoulli numbers. This expansion shows <strong>th</strong>at indeed Nɛ F is <strong>th</strong>e<br />
correct scaling variable and justifies <strong>th</strong>e definition of <strong>th</strong>e double scaling limit in (11.20).<br />
2) We expect <strong>th</strong>at, just as in <strong>th</strong>e one matrix model studied in <strong>th</strong>e sec. 9.3, <strong>th</strong>e<br />
wavefunctions <strong>th</strong>emselves have N → ∞ limits in terms of δ-function normalized parabolic<br />
cylinder functions, independent of <strong>th</strong>e details of V :<br />
ψ ɛ (−ε/N, λ/ √ ( ) 2πN<br />
1/2 1/2<br />
2N; V ) →<br />
ψ ± (ε,<br />
log N<br />
˜λ) (11.28)<br />
where <strong>th</strong>e wavefunctions ψ ± (ε, ˜λ) are given in (A.1, A.2).<br />
The prefactor takes proper<br />
account of <strong>th</strong>e normalizations of <strong>th</strong>e wavefunctions and can be verified by putting a nonuniversal<br />
wall at distance O(1) from <strong>th</strong>e maximum. Thus, as in <strong>th</strong>e one-matrix model, we<br />
128
may take <strong>th</strong>e double scaling limit of <strong>th</strong>e fermion operator by first defining an operator ˆΨ N<br />
wi<strong>th</strong> a smoo<strong>th</strong> N → ∞ limit,<br />
1 ( λ<br />
ˆΨ N (λ, x) ≡ ˆΨ √ , Nx ) , (11.29)<br />
(2N)<br />
1/4 2N<br />
where we have substituted x = X/N. 46 We now rescale<br />
a ɛ (ε i ) →<br />
a ɛ (ν)<br />
( 1<br />
π log √ 2N ) 1/2 , (11.30)<br />
so <strong>th</strong>at in <strong>th</strong>e N → ∞ limit we have<br />
∫<br />
ˆΨ N (λ, x) → ˆψ(λ, x) =<br />
dν e iνx a ɛ (ν) ψ ɛ (ν, λ) , (11.31)<br />
where ψ ɛ (ν, x) are normalized as in appendix A (eqns. (A.1, A.2)). The vacuum of <strong>th</strong>e<br />
double scaled field <strong>th</strong>eory is defined by<br />
a ɛ (ν)|µ〉 = 0 ν < µ<br />
(11.32)<br />
a † ɛ (ν)|µ〉 = 0 ν > µ ,<br />
where <strong>th</strong>e Fermi level µ is as in (11.19).<br />
11.4. Macroscopic Loops at c = 1<br />
The discussion of macroscopic loop operators given in chapters 8 and 9 above continues<br />
to hold at c = 1 wi<strong>th</strong> some minor modifications [90]. We now wish to compute <strong>th</strong>e<br />
continuum limit of <strong>th</strong>e macroscopic loop operator<br />
∫<br />
∫<br />
W lattice (L, q) = dX e iqX tr e LΦ(X) → dλ dx e iqx ψ † ψ(λ, x) e λl = W cont (l, q) . (11.33)<br />
In particular, <strong>th</strong>e boundary condition on <strong>th</strong>e loop is of Dirichlet type, x(σ) = x.<br />
A subtlety <strong>th</strong>at arises is <strong>th</strong>at <strong>th</strong>e eigenvalue density is concentrated on bo<strong>th</strong> sides of<br />
<strong>th</strong>e quadratic maximum, or, in double-scaled coordinates, on (−∞, −2 √ µ ] ∪ [2 √ µ, ∞).<br />
This means <strong>th</strong>ere are two (perturbatively) disjoint “worlds” and we cannot simultaneously<br />
46 Note <strong>th</strong>at in <strong>th</strong>is chapter we have used X to denote <strong>th</strong>e lattice target space coordinate and<br />
in what follows we use x to denote <strong>th</strong>e continuum target space coordinate, in minor conflict wi<strong>th</strong><br />
<strong>th</strong>e notation of chapt. 5 in which X denoted <strong>th</strong>e continuum target space coordinate.<br />
129
Laplace transform <strong>th</strong>e eigenvalue density wi<strong>th</strong> respect to bo<strong>th</strong>. We can get around <strong>th</strong>is difficulty<br />
by using a technical trick. We compute amplitudes instead for <strong>th</strong>e Fourier transform<br />
wi<strong>th</strong> respect to λ,<br />
ˆψ † e izˆλ ˆψ ≡<br />
∫ ∞<br />
dλ ˆψ † (λ, x) e izλ ˆψ(λ, x)<br />
−∞<br />
M(z i , x i ) ≡ 〈 (11.34)<br />
ˆψ† e iz 1ˆλ ˆψ · · · ˆψ† e iz nˆλ ˆψ〉 .<br />
We will find <strong>th</strong>at <strong>th</strong>e answer naturally splits into two pieces. In <strong>th</strong>e first piece we<br />
may continue z → il in <strong>th</strong>e upper half plane to obtain a convergent answer. This analytic<br />
continuation makes no sense in <strong>th</strong>e second piece, but <strong>th</strong>ere we can analytically continue<br />
z → −il in <strong>th</strong>e lower half plane. We interpret <strong>th</strong>e two pieces as <strong>th</strong>e contributions of <strong>th</strong>e<br />
two “worlds” defined by <strong>th</strong>e two eigenvalue cuts. Focusing on ei<strong>th</strong>er contribution, we can<br />
define macroscopic loop amplitudes for real loop leng<strong>th</strong>s. This ra<strong>th</strong>er strange reasoning<br />
can be checked in various ways. At <strong>th</strong>e level of tachyon correlation functions, for example,<br />
<strong>th</strong>e techniques of chapt. 13 make it possible to calculate even if we put an infinite wall at<br />
λ = 0, rendering ψ † e −lλ ψ rigorously well-defined. The resulting amplitudes agree to all<br />
orders.<br />
Now we describe <strong>th</strong>e calculation of <strong>th</strong>e amplitudes M(z i , q i ). The Euclidean Green’s<br />
functions of <strong>th</strong>e eigenvalue density ρ = ψ † ψ(λ, x), where x is <strong>th</strong>e “time” dimension of <strong>th</strong>e<br />
c = 1 matrix model, are defined by<br />
(<br />
G Euclidean (x 1 , λ 1 , . . . , x n , λ n ) ≡ 〈µ|T ˆψ† ˆψ(x1 , λ 1 ) · · · ˆψ<br />
)<br />
† ˆψ(xn , λ n ) |µ〉 c<br />
∫ ∏<br />
G Euclidean (q 1 , λ 1 , . . . q n , λ n ) ≡ dx i e iq ix i<br />
G(x 1 , λ 1 , . . . x n , λ n ) .<br />
i<br />
(11.35)<br />
Since <strong>th</strong>e fermions are noninteracting, <strong>th</strong>ese Green’s functions may be written in terms of<br />
<strong>th</strong>e Euclidean fermion propagator,<br />
S E (x 1 , λ 1 ; x 2 , λ 2 ) = e −µ∆x ∫ ∞<br />
−∞<br />
dp<br />
2π e−ip∆x I(p, λ 1 , λ 2 ) , (11.36)<br />
where I is <strong>th</strong>e resolvent for <strong>th</strong>e upside-down oscillator Hamiltonian H = 1 2 p2 − 1 8 λ2 , or,<br />
more generally, for a Hamiltonian H = 1 2 p2 + V (λ) wi<strong>th</strong> <strong>th</strong>e potential tuned in <strong>th</strong>e scaling<br />
region to differ from exact quadratic behavior. In particular, for q > 0,<br />
I(q, λ 1 , λ 2 ) = ( I(−q, λ 1 , λ 2 ) ) ∗ 1<br />
= 〈λ1 |<br />
H − µ − iq |λ 2〉 . (11.37)<br />
130
Using Wick’s <strong>th</strong>eorem, we evaluate (11.35) as a sum of ring diagrams and <strong>th</strong>ereby<br />
obtain <strong>th</strong>e integral representation for <strong>th</strong>e eigenvalue correlators<br />
G Euclidean (q i , λ i ) = 1 n<br />
∫<br />
n<br />
∏<br />
i=1<br />
= 1 n δ(∑ q i )<br />
dq i<br />
2π e−iq ix i<br />
∫ ∞<br />
−∞<br />
∑<br />
σ∈Σ n<br />
∏<br />
SE (x σ(i) , λ σ(i) ; x σ(i+1) , λ σ(i+1) )<br />
dq ∑<br />
n ∏<br />
σ∈Σ n k=1<br />
I(Q σ k , λ σ(k) , λ σ(k+1) ) ,<br />
where Q σ k ≡ q + q σ(1) + · · · + q σ(k) , and <strong>th</strong>e sum is over <strong>th</strong>e permutation group Σ n .<br />
(11.38)<br />
We can now obtain <strong>th</strong>e formula for loop amplitudes as follows. The resolvent of <strong>th</strong>e<br />
upside down oscillator may be given <strong>th</strong>e integral representation:<br />
1<br />
〈λ 1 |<br />
H − ζ |λ 2〉 = −i<br />
∫ −ɛ∞<br />
0<br />
ds e −isζ 1<br />
√ exp i ( λ<br />
2<br />
1 + λ 2 2<br />
4πi sinh s 4 tanh s − 2λ )<br />
1λ 2<br />
, (11.39)<br />
sinh s<br />
where ɛ = sgn(Im ζ). Therefore, <strong>th</strong>e calculation of (11.34) reduces to <strong>th</strong>e evaluation of a<br />
gaussian integral wi<strong>th</strong> <strong>th</strong>e result<br />
· · ·<br />
∂<br />
∂µ M(z i, q i ) = 1 2 in+1 δ( ∑ q i ) ∑ ∫ ∞<br />
σ∈Σ n<br />
∫ ɛn−1 ∞<br />
0<br />
k=1<br />
−∞<br />
dξ<br />
e iµξ<br />
| sinh ξ/2|<br />
( n−1<br />
∑ ) i<br />
ds n−1 exp − s k Q σ k exp(<br />
2 co<strong>th</strong>(ξ/2) ∑ )<br />
zi<br />
2<br />
(<br />
· exp i<br />
∑<br />
1≤i
The perturbative expansion of <strong>th</strong>e Hartle–Hawking wavefunction is obtained by expanding<br />
<strong>th</strong>e last factor in (11.41), and continuing z → il in an integral representation for <strong>th</strong>e Bessel<br />
function.<br />
b) Two macroscopic loops:<br />
M 2 = + κ 2 + κ 4<br />
+ . . .<br />
This gives <strong>th</strong>e propagator from which we may hope to understand <strong>th</strong>e spectrum of <strong>th</strong>e<br />
<strong>th</strong>eory.<br />
The integral formula for ∂<br />
∂µ M becomes<br />
Im<br />
∫ ∞<br />
0<br />
dξ eiµξ+ 1 2 i(z2 1 +z2 2 ) co<strong>th</strong>(ξ/2)<br />
sinh(ξ/2)<br />
∫ ∞<br />
0<br />
ds e −|q|s( )<br />
e i cosh(s−ξ/2) z<br />
sinh ξ/2 1 z 2<br />
− e i cosh(s+ξ/2) z<br />
sinh ξ/2 1 z 2<br />
.<br />
(11.43)<br />
This formula holds for z i real. If we wish to have real loop leng<strong>th</strong>s we replace Im → − 1 2 i<br />
and continue z i → il i , as discussed above.<br />
The integral over s may be written as<br />
−iπ|q|/2 sinh(|q|ξ/2)<br />
2πe J |q| (2α) +<br />
sin π|q|<br />
where α = z 1 z 2 /2 sinh(ξ/2).<br />
∞∑<br />
r=1<br />
4i r r<br />
r 2 − q 2 J r(2α) sinh(rξ/2) (11.44)<br />
The remaining integral over ξ can be done in terms of<br />
Whittaker functions to give nonperturbative answers. This is done in detail in [90].<br />
At genus zero, we have<br />
and <strong>th</strong>erefore<br />
〈<br />
W (l1 , p)W (l 2 , −p) 〉 = πp<br />
sin πp I p(2 √ µl 1 )K p (2 √ µl 2 )<br />
∞∑ 2(−1) r r 2<br />
+<br />
r 2 − p I r(2 √ µl 2 1 ) K r (2 √ µl 2 ) ,<br />
r=1<br />
〈<br />
W (l1 , p)W (l 2 , −p) 〉 ≡<br />
∫ ∞<br />
0<br />
(11.45)<br />
dE<br />
2π G(E, p) ψ E (l 1 )ψ E (l 2 ) , (11.46)<br />
wi<strong>th</strong><br />
G(E, p) =<br />
1 πE<br />
E 2 + p 2 sinh πE<br />
= πp<br />
sin πp<br />
1<br />
E 2 + p + ∑ ∞ 2<br />
r=1<br />
2(−1) r r 2<br />
r 2 − p 2 1<br />
E 2 + r 2 . (11.47)<br />
132
c) Three macroscopic loops:<br />
κ<br />
The same techniques as above can be applied to <strong>th</strong>ree and four macroscopic loop amplitudes.<br />
The final formulae are ra<strong>th</strong>er complicated (see e.g. [91]), while at genus zero <strong>th</strong>ere<br />
is considerable simplification.<br />
The entire genus zero amplitude, toge<strong>th</strong>er wi<strong>th</strong> <strong>th</strong>e integer powers of l, is summarized<br />
nicely in terms of macroscopic state wavefunctions [91]:<br />
〈∏ 〉 ∫ ∞ ∏ E i<br />
W (l i , q i ) = dE i K iEi (2 √ µl i ) (E 1 +E 2 +E 3 ) co<strong>th</strong> ( π<br />
E i − iq i 2 (E 1+E 2 +E 3 ) ) .<br />
i<br />
−∞<br />
Exercise. macroscopic → microscopic<br />
i<br />
Evaluate <strong>th</strong>e integrals in (11.48) using residues by closing <strong>th</strong>e E i contours in <strong>th</strong>e<br />
upper or lower half plane. (Warning: <strong>th</strong>is involves a certain amount of algebra — <strong>th</strong>e<br />
result is in [91].)<br />
(11.48)<br />
The results (11.46),(11.47), and (11.48) contain a weal<strong>th</strong> of information on <strong>th</strong>e nature<br />
of contact terms and singular geometries in <strong>th</strong>e pa<strong>th</strong> integral of <strong>th</strong>e c = 1 <strong>th</strong>eory. Note<br />
especially from (11.47) <strong>th</strong>at <strong>th</strong>e propagator agrees wi<strong>th</strong> <strong>th</strong>e naive Wheeler–DeWitt propagator<br />
at low values of |E|, but is quite distinct, indeed exponentially decaying, at large<br />
values of |E|. Correspondingly, in position space <strong>th</strong>e propagator turns out to be smoo<strong>th</strong> at<br />
l 1 = l 2 . From a quantum gravity point of view, <strong>th</strong>e only source of violation of <strong>th</strong>e naive<br />
WdW propagator in <strong>th</strong>e Euclidean quantum gravity pa<strong>th</strong> integral is <strong>th</strong>e contribution of<br />
singular geometries such as fig. 20. From a target space point of view, <strong>th</strong>e smoo<strong>th</strong>ness<br />
of <strong>th</strong>e propagator suggests <strong>th</strong>e existence of o<strong>th</strong>er degrees of freedom. This guess is confirmed<br />
by <strong>th</strong>e pole structure of <strong>th</strong>e propagator manifested in <strong>th</strong>e second line of (11.47).<br />
These extra degrees of freedom are clearly related to <strong>th</strong>e special states. The two ideas: 1)<br />
contributions of singular geometries, and 2) existence of new degrees of freedom related<br />
to special states, are tied toge<strong>th</strong>er by interpreting <strong>th</strong>e new degrees of freedom in terms<br />
of Liouville boundary operators, or, equivalently, in terms of redundant operators in <strong>th</strong>e<br />
matrix model. Recall from sec. 10.4 <strong>th</strong>at such operators contribute figure eights — <strong>th</strong>e<br />
lattice version of <strong>th</strong>e singular geometry of fig. 20. In [91] <strong>th</strong>is interpretation was elaborated<br />
upon, and partially carried out for <strong>th</strong>e data provided by <strong>th</strong>e <strong>th</strong>ree-point function (11.48).<br />
133
11.5. Wavefunctions and Wheeler–DeWitt Equations<br />
From (11.45) we can easily extract <strong>th</strong>e wavefunction of <strong>th</strong>e vertex operator V q by<br />
extracting <strong>th</strong>e coefficient of l |q|<br />
1 as l 1 → 0 to get<br />
〈<br />
W (l, −q) Vq<br />
〉<br />
= µ |q|/2 K q (2 √ µl) . (11.49)<br />
This result is analogous to (10.19), and is in complete accord wi<strong>th</strong> <strong>th</strong>e continuum answer.<br />
The wavefunctions to all orders of perturbation <strong>th</strong>eory are not much more complicated.<br />
We extract <strong>th</strong>e term proportional to z |q|<br />
1<br />
in (11.44) and perform <strong>th</strong>e remaining ξ integral<br />
in terms of Whittaker functions to find<br />
( ∫ |q|<br />
)<br />
ψ q (l) = 2Γ(−|q|)Im e 3πi<br />
4 (1+|q|) dt Γ( 1 2 − iµ + t) l−1 W iµ−t+<br />
1<br />
2<br />
0<br />
|q|, 1 2 |q| (il 2 ) , (11.50)<br />
where W η,ξ is a Whittaker function. In particular, <strong>th</strong>e function χ η,ξ (l) = l −1 W η,ξ (il 2 )<br />
satisfies an equation derived from <strong>th</strong>e Whittaker equation:<br />
(−(l ∂ ∂l )2 − 4iηl 2 + 4ξ 2 − l 4 )<br />
χ η,ξ = 0 . (11.51)<br />
Therefore <strong>th</strong>e all-orders Wheeler–DeWitt wavefunctions satisfy some simple differential<br />
relations generalizing <strong>th</strong>e genus zero Wheeler–DeWitt equation. The answer is especially<br />
simple at q = 0 where we find <strong>th</strong>e modified Wheeler–DeWitt equation:<br />
(−(l ∂ ∂l )2 + 4µl 2 − κ 2 l 4 )<br />
ψ q=0 = 0 , (11.52)<br />
where we have explicitly introduced <strong>th</strong>e topological coupling κ. The consequence of (11.51)<br />
is not so simple when q ≠ 0. 47<br />
11.6. Macroscopic Loop Field Theory and c = 1 scaling<br />
In sec. 5.3, we discussed <strong>th</strong>e tachyon field T (φ, X). On <strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>e formulae<br />
of <strong>th</strong>e previous section suggest <strong>th</strong>e existence of a macroscopic loop field <strong>th</strong>eory in which<br />
W (l, x) is a field. Since l and φ are related, we may suspect <strong>th</strong>at <strong>th</strong>e two fields T and W<br />
are essentially <strong>th</strong>e same (recall <strong>th</strong>at X from sec. 5.3 translates directly to <strong>th</strong>e continuum<br />
x we use in <strong>th</strong>is chapter).<br />
47 We disagree wi<strong>th</strong> a recent discussion of <strong>th</strong>e all orders WdW equation in [121].<br />
134
Treated as a field, W has a vacuum expectation value given by <strong>th</strong>e Hartle–Hawking<br />
wavefunction, and correlations of fluctuations δW are measured by higher correlation functions.<br />
On <strong>th</strong>e o<strong>th</strong>er hand, we see from (11.49) <strong>th</strong>at first order fluctuations δW correspond<br />
exactly to <strong>th</strong>e tachyon wavefunction, i.e. satisfy <strong>th</strong>e Wheeler–DeWitt equation, up to a<br />
factor of <strong>th</strong>e coupling constant. This suggests <strong>th</strong>e relation<br />
W (l, x) ∼ e − 1 2 Qφ T (φ, x) . (11.53)<br />
In <strong>th</strong>e next section we shall see <strong>th</strong>at tachyon S-matrix elements can be extracted directly<br />
from W correlators, fur<strong>th</strong>er corroborating <strong>th</strong>is result.<br />
Replacing (11.53) by an equality, we may transform <strong>th</strong>e standard free tachyon action<br />
∫ ∞<br />
S 0 = dx dφ e −Qφ( )<br />
(∂ φ T ) 2 + µ e γφ T 2 + T (H x − Q2<br />
4 )T (11.54)<br />
to <strong>th</strong>e action<br />
−∞<br />
S 0 =<br />
∫ ∞<br />
0<br />
dl<br />
l<br />
∫ ∞<br />
−∞<br />
dx W ( −∂ 2 φ + µ e γφ + H x<br />
)<br />
W . (11.55)<br />
The interactions are complicated, but can be deduced from <strong>th</strong>e formulae of sec. 11.4.<br />
Using <strong>th</strong>e tachyon wavefunction, we are now prepared to adapt Polchinski’s discussion<br />
48 [75] to interpret <strong>th</strong>e scaling law variation (11.22) and <strong>th</strong>e definition (11.19) in terms<br />
of <strong>th</strong>e continuum <strong>th</strong>eory. As we see from (11.49) wi<strong>th</strong> q = 0, <strong>th</strong>e wavefunction for <strong>th</strong>e<br />
static tachyon background is K 0 (2 √ µ l). To mimic <strong>th</strong>e discreteness of matrix models, we<br />
consider a cutoff scaling variable µ B (analogous to ɛ F ) and measure distances l in “lattice<br />
units”. Substituting in (11.53), we find <strong>th</strong>at <strong>th</strong>e static tachyon configuration in <strong>th</strong>e cutoff<br />
<strong>th</strong>eory is given by<br />
T (φ) ∼ (2 √ µ B l) 2 K 0 (2 √ µ B l) . (11.56)<br />
In <strong>th</strong>e σ-model approach (eq. (5.10)), <strong>th</strong>e spacetime one-point function 〈T 〉 plays <strong>th</strong>e role<br />
of <strong>th</strong>e cosmological constant. Wi<strong>th</strong> an ultraviolet cutoff φ ≥ 0 on <strong>th</strong>e <strong>th</strong>eory, <strong>th</strong>e value<br />
of <strong>th</strong>e tachyon field at <strong>th</strong>e cutoff is <strong>th</strong>us naturally interpreted as <strong>th</strong>e bare cosmological<br />
constant<br />
g − g c ∼ T (0) ∼ (2 √ µ B ) 2 K 0 (2 √ µ B ) , (11.57)<br />
since working at <strong>th</strong>e cutoff is equivalent in <strong>th</strong>e matrix model to multiplying <strong>th</strong>e bare<br />
cosmological constant by <strong>th</strong>e (unit) area of <strong>th</strong>e basic triangle. As µ B → 0, we have<br />
g − g c ∼ 2µ B log µ B (11.58)<br />
which is functionally equivalent to (11.22), giving <strong>th</strong>e relation between <strong>th</strong>e bare worldsheet<br />
cosmological constant and <strong>th</strong>e scaling variable.<br />
48 To go from <strong>th</strong>e variables ∆ and µ used in [75] to our conventions, let ∆ → g − g c , µ → −ɛ F .<br />
135
x<br />
Fig. 24: The function x 2 K 0 (x), wi<strong>th</strong> a peak at x ∼ O(1).<br />
The function x 2 K 0 (x) of (11.56) moreover behaves as in fig. 24, wi<strong>th</strong> a peak at x ∼<br />
O(1). (This is <strong>th</strong>e analog of <strong>th</strong>e soliton configuration of [75], al<strong>th</strong>ough in our argument<br />
we use <strong>th</strong>e Wheeler–DeWitt equation for macroscopic loops ra<strong>th</strong>er <strong>th</strong>an <strong>th</strong>e properties of<br />
an interacting tachyon <strong>th</strong>eory.) As µ B → 0, <strong>th</strong>e peak occurs at larger and larger lattice<br />
leng<strong>th</strong>s ¯l 2 = e √ 2 ¯φ ∼ 1/µ B . The scaling behavior at higher genus follows from perturbation<br />
<strong>th</strong>eory wi<strong>th</strong> <strong>th</strong>e effective action (5.12) in <strong>th</strong>e dilaton background 〈D〉 = (Q/2)φ = √ 2 φ.<br />
The effective string coupling is <strong>th</strong>us κ eff = κ e √ 2 ¯φ, where we have substituted <strong>th</strong>e value<br />
at <strong>th</strong>e peak of <strong>th</strong>e tachyon configuration which dominates <strong>th</strong>e string scattering. Since <strong>th</strong>e<br />
bare string coupling (genus counting parameter) in <strong>th</strong>e matrix models is κ = 1/N, we see<br />
<strong>th</strong>at holding fixed <strong>th</strong>e effective string coupling κ eff = κ e √ 2 ¯φ ∼ 1/(Nµ B ) in <strong>th</strong>e continuum<br />
limit defines <strong>th</strong>e c = 1 double scaling limit as in <strong>th</strong>e matrix models [86–89], where <strong>th</strong>e<br />
string coupling is as well related to <strong>th</strong>e bare cosmological constant via (11.58) ra<strong>th</strong>er <strong>th</strong>an<br />
via (7.18) as in <strong>th</strong>e c < 1 models coupled to gravity.<br />
Tachyon condensates<br />
In <strong>th</strong>is discussion we are identifying 〈T 〉 ∼ µl 2 K 0 (2 √ µ l), while in <strong>th</strong>e σ-model discussion<br />
(see (5.15)), we identified 〈T 〉 ∼ µ e γφ . These do not even agree in <strong>th</strong>e limit φ → −∞<br />
(l → 0). This confusion has plagued <strong>th</strong>e subject for several years now. As mentioned in <strong>th</strong>e<br />
paragraph following (4.11), it suggests <strong>th</strong>at we should really identify <strong>th</strong>e cosmological constant<br />
operator as 〈T 〉 ∼ µφ e γφ , which has <strong>th</strong>e µl 2 ln l behavior as φ → −∞. We comment<br />
fur<strong>th</strong>er on <strong>th</strong>e two possible cosmological constant operators at <strong>th</strong>e end of sec. 13.6 .<br />
11.7. Correlation functions of Vertex Operators<br />
As in <strong>th</strong>e c < 1 <strong>th</strong>eories, vertex operators are obtained by looking at <strong>th</strong>e small l<br />
expansion. In particular, using <strong>th</strong>e reasoning of <strong>th</strong>e exercise below (10.13) toge<strong>th</strong>er wi<strong>th</strong><br />
136
<strong>th</strong>e formula (4.12), we see <strong>th</strong>at <strong>th</strong>e coefficients of l |q| in <strong>th</strong>e small l expansion of <strong>th</strong>e<br />
macroscopic loop operators must be proportional to <strong>th</strong>e correlation functions of <strong>th</strong>e V q .<br />
We are fortunate <strong>th</strong>at for generic q <strong>th</strong>ere is only a one-dimensional BRST cohomology<br />
class. 49<br />
String amplitudes are asymptotic expansions in <strong>th</strong>e string coupling κ = 1/µ whose<br />
coefficients are integrals over moduli space. These can be calculated by considering <strong>th</strong>e<br />
l i → 0 expansion of <strong>th</strong>e matrix model loop operators:<br />
〈µ| ∏ W (l i , q i )|µ〉 = ∏ l |q i|<br />
i<br />
R n (q 1 , . . . q n ; µ) ( 1 + O(l 2 i ) ) + analytic in l i . (11.59)<br />
Then, as asymptotic expansions in κ = 1/µ, we have<br />
R n (q 1 , . . . q n ; µ) = A n (Ṽq 1<br />
· · · Ṽq n<br />
) , (11.60)<br />
where<br />
A n (Ṽq 1<br />
· · · Ṽq n<br />
) ∼ ∑ h≥0<br />
) ∑<br />
∫<br />
κ −χ A h,n<br />
(Ṽq1 · · · Ṽq n ≡ κ −χ<br />
h≥0<br />
M h,n<br />
〈Ṽq1 · · · Ṽq n<br />
〉<br />
(11.61)<br />
(<strong>th</strong>e CFT correlator 〈 Ṽ q1 · · · Ṽq n<br />
〉<br />
is interpreted as a differential form on moduli space<br />
M h,n , i.e. includes a product ∏ b¯b over ghost zero modes), and<br />
Ṽ q = Γ ( |q| ) c¯c e (iqX−|q|φ)/√2 e√<br />
2φ<br />
(11.62)<br />
is <strong>th</strong>e tachyon vertex operator of (4.12). The normalization is fixed by comparison of<br />
computations of <strong>th</strong>e right hand side performed by Di Francesco and Kutasov [48], as<br />
described in sec. 14.2 . We see from (11.61) <strong>th</strong>at <strong>th</strong>e S-matrix for <strong>th</strong>e spacetime tachyon<br />
can be extracted from correlation functions of <strong>th</strong>e macroscopic loop operator W , hence T<br />
and W are interpolating fields for <strong>th</strong>e same asymptotic states, as suggested in <strong>th</strong>e preceding<br />
section.<br />
49 Historically, <strong>th</strong>e first attempts at D = 1 correlation functions used powers of <strong>th</strong>e matrix field<br />
as scaling operators [122,92].<br />
137
Remarks:<br />
1) The definition (11.59) of R n is ambiguous if q i ∈ Z. The functions can defined for<br />
q i ∈ Z by continuity.<br />
2) The right hand side of (11.60) is by definition an asymptotic expansion in <strong>th</strong>e string<br />
coupling 1/µ. On <strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>e matrix model gives a nonperturbative completion<br />
since (in contrast to <strong>th</strong>e difficulties at c < 1) we may perform all manipulations<br />
wi<strong>th</strong> a potential giving a perfectly well-defined matrix model integral.<br />
As an example of <strong>th</strong>e use of (11.60) we may immediately extract from <strong>th</strong>e small l<br />
expansion of <strong>th</strong>e all-orders Wheeler–DeWitt wavefunction (11.50) <strong>th</strong>e two-point function<br />
of <strong>th</strong>e tachyon:<br />
∂<br />
∂µ 〈V q V −q 〉 = ( Γ(−|q|) ) (<br />
2<br />
Im e iπ|q|/2( Γ(|q| + 1 2 − iµ) Γ( 1 2<br />
Γ( 1 2 − iµ) −<br />
− iµ) ) )<br />
Γ(−|q| + 1 2 − iµ)<br />
∼ (qΓ(−|q|)) 2 µ |q|( 1<br />
|q| − (|q| − − |q| − 1)<br />
1)(q2 24<br />
3∏<br />
+ (|q| − r) (3q4 − 10|q| 3 − 5q 2 + 12|q| + 7)<br />
5760<br />
−<br />
r=1<br />
r=1<br />
µ −2<br />
µ −4<br />
5∏<br />
(|q| − r) (9q6 − 63|q| 5 + 42q 4 + 217|q| 3 − 205|q| − 93)<br />
)<br />
µ −6 + · · · .<br />
2903040<br />
(11.63)<br />
In sec. 13.5 below we will describe a much better way to compute tachyon correlators<br />
which easily yields <strong>th</strong>e generalization of (11.63) to arbitrary tachyon correlation functions.<br />
<br />
Special states in <strong>th</strong>e matrix model<br />
If one wishes to interpret <strong>th</strong>e integer powers of l in terms of an operator expansion<br />
it is necessary to introduce <strong>th</strong>e redundant operators B r,q corresponding to moments of λ r .<br />
It can already be seen from (11.45) <strong>th</strong>at to define such operators we must take<br />
( ∂<br />
) rW ∣ ∫<br />
∣∣l=0<br />
B r,q = lim (l, q) ∼<br />
l→0 ∂l<br />
dx dλ e iqx ψ † λ r ψ , (11.64)<br />
where q > r, o<strong>th</strong>erwise <strong>th</strong>e limit l → 0 diverges. We <strong>th</strong>en analytically continue to any q.<br />
The physical origin of <strong>th</strong>e divergence at l → 0, or, at λ → ∞, is in <strong>th</strong>e ultraviolet region<br />
of <strong>th</strong>e worldsheet integral, and is probably connected wi<strong>th</strong> <strong>th</strong>e fact <strong>th</strong>at <strong>th</strong>e special state<br />
operators are irrelevant operators.<br />
138
By upper triangular transformations of <strong>th</strong>e basis of operators, analogous to <strong>th</strong>e change<br />
of basis in sec. 10.3 relating σ j to ˆσ j , we obtain <strong>th</strong>e full operator expansion of <strong>th</strong>e macroscopic<br />
loop [91]:<br />
W in (l, p) = Ṽp Γ(|p| + 1) µ −|p|/2 I |p| (2 √ µl) −<br />
∞∑<br />
r=1<br />
ˆB r,p<br />
2(−1) r r<br />
r 2 − p 2 µ−r/2 I r (2 √ µl) . (11.65)<br />
12. Fermi Sea Dynamics and Collective Field Theory<br />
12.1. Time dependent Fermi Sea<br />
Ano<strong>th</strong>er source of intuition, very different from <strong>th</strong>e macroscopic loop approach, comes<br />
from <strong>th</strong>e motions of <strong>th</strong>e Fermi sea of <strong>th</strong>e upside-down oscillator and <strong>th</strong>e associated collective<br />
field <strong>th</strong>eory, also known as <strong>th</strong>e Das–Jevicki–Sakita <strong>th</strong>eory [76,77,93]. As we saw in our<br />
analysis of <strong>th</strong>e tree-level free energy (sec. 11.2), one is naturally lead to <strong>th</strong>ink about <strong>th</strong>e<br />
fermionic phase space. This point of view leads to a very beautiful description of tree-level<br />
c = 1 dynamics [95].<br />
In sec. 11.2, we studied <strong>th</strong>e ground state from <strong>th</strong>e point of view of a fluid in phase<br />
space. When describing dynamics, we have to perturb <strong>th</strong>e system so we are now looking at<br />
time-dependent Fermi seas resulting from <strong>th</strong>e disturbances produced by various operators.<br />
The possible dynamical solutions of <strong>th</strong>e system can be described in <strong>th</strong>e following way [95].<br />
Consider <strong>th</strong>e generating functional for correlators in <strong>th</strong>e <strong>th</strong>eory,<br />
∫<br />
Z[J] =<br />
dψ e − ∫ ψ † (id/dt + d 2 /dλ 2 + λ 2 )ψ + Jψ † ψ , (12.1)<br />
where we imagine J has been turned on and off during a finite time interval. (In <strong>th</strong>is section,<br />
<strong>th</strong>e c = 1 coordinate t will always be taken to be a Minkowskian time coordinate.). The<br />
source J acts as an external force on <strong>th</strong>e fermions. After it has been turned off, <strong>th</strong>e state<br />
evolves as some time-dependent solution of <strong>th</strong>e system. It is clear <strong>th</strong>at <strong>th</strong>e points simply<br />
move along trajectories in phase space appropriate to <strong>th</strong>e upside-down oscillator, <strong>th</strong>at is,<br />
<strong>th</strong>ey move along lines of constant p 2 − λ 2 .<br />
139
p<br />
λ<br />
Fig. 25: A generic initial configuration of <strong>th</strong>e Fermi sea.<br />
Thus to write down <strong>th</strong>e general time-dependent motion of <strong>th</strong>e system we imagine at<br />
time zero a generic Fermi sea as in fig. 25, which we may describe as a parametrized curve<br />
λ = (1 + a(σ)) cosh(σ) , p = (1 + a(σ)) sinh(σ) ,<br />
where a(σ) is a smoo<strong>th</strong> function subject to <strong>th</strong>e constraint <strong>th</strong>at <strong>th</strong>e initial Fermi surface<br />
(λ, p) be physically reasonable. Hamilton’s equations<br />
∂ t p = {H, p}<br />
= λ − p ∂ λ p .<br />
(12.2)<br />
<strong>th</strong>en give <strong>th</strong>e general solution for time evolution:<br />
λ = (1 + a(σ)) cosh(σ − t)<br />
p = (1 + a(σ)) sinh(σ − t) .<br />
(12.3a)<br />
(12.3b)<br />
12.2. Collective Field Theory<br />
We are now in a position to derive <strong>th</strong>e collective field <strong>th</strong>eory of c = 1. Consider <strong>th</strong>e<br />
case in which <strong>th</strong>e Fermi sea only has two branches p ± . The functions p ± (λ, t) may be<br />
<strong>th</strong>ought of as on-shell fields related by a boundary condition p + (λ ∗ , t) = p − (λ ∗ , t) where<br />
λ ∗ is <strong>th</strong>e leftmost point of <strong>th</strong>e sea. As in sec. 11.2, <strong>th</strong>e energy, or Hamiltonian, is given by<br />
∫ dp dλ<br />
H =<br />
2π<br />
ɛ θ(ɛ F − ɛ) + µ 2 N<br />
∫ ( (p<br />
3<br />
= dλ +<br />
6 − p λ2 + ) (p 3<br />
−<br />
−<br />
2 6 − p λ2 − ) ) + µ ∫<br />
2 2<br />
140<br />
dλ (p + − p − ) .<br />
(12.4)
To interpret (12.4) as a field <strong>th</strong>eory of <strong>th</strong>e eigenvalue density, define<br />
p ± = −κ 2 Π χ ± π∂ λ χ . (12.5)<br />
In terms of ζ, <strong>th</strong>e eigenvalue density is given by<br />
ρ(λ) = p + − p − = 2π ∂ λ χ . (12.6)<br />
After rescaling, <strong>th</strong>e Hamiltonian in <strong>th</strong>ese variables may be written as<br />
∫<br />
H =<br />
( κ<br />
2<br />
dλ<br />
2 χ′ πχ 2 + π2<br />
6κ 2 (χ′ ) 3 + v(λ) )<br />
κ 2 χ′ + µ ∫<br />
2κ 2<br />
dλ χ ′ , (12.7)<br />
where v(λ) is <strong>th</strong>e double-scaled matrix model potential − 1 2 λ2 . This Hamiltonian appeared<br />
from very different points of view in [76,77,93] as <strong>th</strong>e field <strong>th</strong>eory of an eigenvalue density<br />
field ρ(λ, t). The present derivation overcomes some of <strong>th</strong>e difficulties wi<strong>th</strong> understanding<br />
<strong>th</strong>e Jacobian for <strong>th</strong>e change ∏ dλ i → dρ(λ, t).<br />
Exercise.<br />
Derive <strong>th</strong>e Lagrangian corresponding to <strong>th</strong>e Hamiltonian (12.7).<br />
p<br />
λ<br />
Fig. 26: A configuration of <strong>th</strong>e Fermi sea wi<strong>th</strong> folds. p is a multivalued function<br />
of λ.<br />
Folds<br />
As pointed out in [95], <strong>th</strong>ere can be solutions to (12.3a, b) which have four or more<br />
branches p(λ, t) for a given λ (e.g. fig. 26). These solutions are perfectly sensible from<br />
<strong>th</strong>e free fermion point of view but are quite strange from <strong>th</strong>e collective field <strong>th</strong>eory point<br />
of view. Curiously, <strong>th</strong>e number of folds is not conserved in time. A surface wi<strong>th</strong> two<br />
141
anches can very well evolve into one wi<strong>th</strong> four or more branches, and vice versa. These<br />
fold-solutions are extremely interesting in <strong>th</strong>e context of collective field <strong>th</strong>eory as a model of<br />
string field <strong>th</strong>eory, for <strong>th</strong>ey show <strong>th</strong>at <strong>th</strong>e “obvious” string field might be a bad description<br />
of <strong>th</strong>e string degrees of freedom for some perfectly sensible backgrounds. It has been<br />
suggested <strong>th</strong>at if <strong>th</strong>e c = 1 model is equivalent to a model of 1 + 1-dimensional black holes,<br />
<strong>th</strong>en folding solutions will be an important piece of <strong>th</strong>e puzzle.<br />
12.3. Relation to 1+1 dimensional relativistic field <strong>th</strong>eory<br />
It is natural to rewrite <strong>th</strong>e collective field <strong>th</strong>eory as a relativistic <strong>th</strong>eory. Let us<br />
consider solutions such <strong>th</strong>at <strong>th</strong>ere are only two branches p ± of <strong>th</strong>e Fermi sea. Consider<br />
<strong>th</strong>e classical equations of motion for a particle in an upside-down oscillator:<br />
¨λ = λ , (12.8)<br />
solved by λ(t) = A cosh(t + B). Thus if we change our spatial coordinates to λ =<br />
2 √ µ cosh τ, motion in τ, t space is relativistic: τ(t) = t + B.<br />
Let us now explain <strong>th</strong>is point wi<strong>th</strong>in <strong>th</strong>e collective field <strong>th</strong>eory. We are interested in <strong>th</strong>e<br />
fluctuations δp ± of <strong>th</strong>e Fermi sea. Since <strong>th</strong>ere is a nonzero background field configuration<br />
corresponding to <strong>th</strong>e genus zero eigenvalue density,<br />
p ± = ± √ λ 2 − µ = ±(λ − µ 2λ ) + O(λ−2 ) , (12.9)<br />
we make a field redefinition<br />
p ± (λ, t) = ±λ ∓ µ 2λ χ ± , (12.10)<br />
so <strong>th</strong>at, up to a constant shift independent of χ ± , <strong>th</strong>e Hamiltonian (12.4) becomes<br />
H = + µ2<br />
8<br />
= µ2<br />
8<br />
∫<br />
∫ (<br />
√<br />
(1 − e<br />
dτ (χ 2 + + χ2 − ) −6τ )(1 − e −2τ )<br />
(1 + e −2τ ) 2<br />
− 1 )<br />
χ<br />
3<br />
6(<br />
+ + χ 3 − e<br />
−2τ 1 − e −2τ )<br />
(1 + e −2τ ) 3<br />
dτ<br />
((χ 2 + + χ 2 −) − e−2τ (<br />
χ<br />
3<br />
6 + + χ 3 ) ) (<br />
− 1 + O(e −2τ ) ) .<br />
(12.11)<br />
Far from <strong>th</strong>e edge of <strong>th</strong>e eigenvalue density, we may define χ ± = ±π S − ∂ τ S , where S is<br />
a free massless scalar field.<br />
142
Dirichlet boundary conditions imply <strong>th</strong>e free propagator for <strong>th</strong>e fermions χ ± :<br />
∫ ∞<br />
0<br />
dE cos Eτ 1 cos Eτ 2<br />
E 2 + p 2 . (12.12)<br />
The change of variables to (12.11) can be pursued more carefully and perturbation <strong>th</strong>eory<br />
calculations can be performed in <strong>th</strong>is formalism. See [76] and [16] for details.<br />
Remark: The one-loop free energy in <strong>th</strong>is 1+1 dimensional relativistic field <strong>th</strong>eory<br />
can be calculated [76] and goes as log µ, which is hence interpreted as <strong>th</strong>e volume of φ-space.<br />
Fur<strong>th</strong>er attempts to interpret <strong>th</strong>is result may be found in [15] (see also sec. 11.6).<br />
12.4. τ-space and φ-space<br />
Collective field <strong>th</strong>eory is a <strong>th</strong>eory of a massless boson <strong>th</strong>at represents fluctuations in<br />
<strong>th</strong>e eigenvalue density. On <strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>e massless boson of string <strong>th</strong>eory is a scalar<br />
field T (φ, t). In <strong>th</strong>is section we discuss <strong>th</strong>e nontrivial relation between <strong>th</strong>ese two bosonic<br />
fields [91]. 50<br />
As we have seen, <strong>th</strong>e macroscopic loop and tachyon field are essentially <strong>th</strong>e same. In<br />
turn, W and ρ are related by a Laplace-like transform,<br />
W (l, x) =<br />
∫ ∞<br />
2 √ µ<br />
dλ e −lλ ρ = 2√ µ<br />
K 1 (2 √ ∫ ∞<br />
µl) + dτ e −2l√ µ cosh τ ∂ τ ζ , (12.13)<br />
l<br />
0<br />
where in <strong>th</strong>e second equation we have shifted <strong>th</strong>e field ρ by its genus zero one-point function<br />
ρ = √ λ 2 − 4µ + ∂ λ χ, and changed variables to λ = 2 √ µ cosh τ.<br />
Using <strong>th</strong>is transformation on fields, we can understand <strong>th</strong>e relation between <strong>th</strong>e tree<br />
level propagators of <strong>th</strong>e W -<strong>th</strong>eory, (11.46, 11.47) and <strong>th</strong>ose of <strong>th</strong>e collective field <strong>th</strong>eory<br />
(12.12). The key relation is provided by <strong>th</strong>e kernel e −2l√ µ cosh τ , which satisfies <strong>th</strong>e<br />
differential equation:<br />
H φ e −2l√ µ cosh τ ≡<br />
(− ∂2<br />
∂φ 2 + 4µl2 )<br />
e −2l√ µ cosh τ = − ∂2<br />
∂τ 2 e−2l√ µ cosh τ . (12.14)<br />
50 Many au<strong>th</strong>ors continue to identify τ wi<strong>th</strong> <strong>th</strong>e Liouville coordinate φ. While bo<strong>th</strong> spaces<br />
share many qualitative features, <strong>th</strong>ey cannot be <strong>th</strong>e same. The matrix model coordinate λ has<br />
no (obvious) geometric meaning in <strong>th</strong>e discrete worldsheet sum. Ra<strong>th</strong>er, it is <strong>th</strong>e loop operator<br />
W (L) <strong>th</strong>at has a geometric meaning and is related to <strong>th</strong>e worldsheet metric, and <strong>th</strong>erefore to <strong>th</strong>e<br />
Liouville field φ.<br />
143
It follows <strong>th</strong>at if we define a nonlocal transformation of functions:<br />
ˆB(φ) =<br />
ˇB(τ) =<br />
∫ ∞<br />
0<br />
∫ ∞<br />
0<br />
dτ e −l cosh τ B(τ)<br />
dl<br />
l e−l cosh τ B(φ) ,<br />
(12.15)<br />
<strong>th</strong>en we have<br />
f(H φ ) ˆB(φ) =<br />
∫ ∞<br />
0<br />
dτ e −l cosh τ f(− ∂2<br />
∂τ 2 )B(τ) ,<br />
if B is such <strong>th</strong>at integration by parts is valid. In <strong>th</strong>is way we may establish <strong>th</strong>e classical<br />
function identities:<br />
K iE (2 √ µl) =<br />
∫ ∞<br />
0<br />
∫<br />
cos Eτ ∞<br />
E sinh πE =<br />
0<br />
dτ e −2√ µl cosh τ cos Eτ<br />
dl<br />
l e−2√ µl cosh τ K iE (2 √ µl) ,<br />
(12.16)<br />
relating eigenfunctions of <strong>th</strong>e Bessel and Laplace operators. Comparing, we now see <strong>th</strong>at<br />
(12.13) indeed maps (11.46, 11.47) to (12.12). As a fur<strong>th</strong>er check, <strong>th</strong>e tree level 3-point<br />
function of <strong>th</strong>e eigenvalue density has been calculated in [97] to be<br />
〈<br />
ρ(λ1 , q 1 ) ρ(λ 2 , q 2 ) ρ(λ 3 , q 3 ) 〉 = δ(q 1<br />
c 1 + q 2 + q 3 )<br />
8µ 3/2 sinh τ 1 sinh τ 2 sinh τ 3<br />
∫ ∞ ∏ (<br />
E<br />
)<br />
i<br />
· dE i cos E i τ i (E 1 + E 2 + E 3 ) co<strong>th</strong> ( π<br />
−∞ E i − iq i 2 (E 1 + E 2 + E 3 ) ) ,<br />
i<br />
(12.17)<br />
which is related to (11.48) by (12.13).<br />
The transform (12.13) is very subtle. While it is nonlocal, it can be shown to map<br />
exactly <strong>th</strong>e quadratic τ-space action (12.11) to <strong>th</strong>e φ-space WdW action (11.55). On<br />
<strong>th</strong>e o<strong>th</strong>er hand, <strong>th</strong>e interaction terms will not be locally related. The nonlocality of <strong>th</strong>e<br />
Lagrangian for T (φ, t) is not a surprise. It is present in <strong>th</strong>e covariant formulations of closed<br />
string field <strong>th</strong>eory [44] and has also been found wi<strong>th</strong>in <strong>th</strong>e context of 2D string <strong>th</strong>eory by Di<br />
Francesco and Kutasov using continuum me<strong>th</strong>ods (see below). (The detailed comparison<br />
of <strong>th</strong>e W -field <strong>th</strong>eory wi<strong>th</strong> <strong>th</strong>e above two formulations has not been carried out.)<br />
As a second application, <strong>th</strong>e origin of <strong>th</strong>e Wheeler–DeWitt equation from <strong>th</strong>e point<br />
of view of <strong>th</strong>e eigenvalue dynamics can be understood as follows:<br />
144
Exercise. Variations on WdW<br />
a) Derive <strong>th</strong>e WdW equation for tachyon wavefunctions using <strong>th</strong>e dynamical Fermi<br />
sea picture as follows. Write<br />
W (l, t) =<br />
∫<br />
dλ e lλ( p + (λ, t) − p − (λ, t) ) .<br />
Using <strong>th</strong>e flow equations, show <strong>th</strong>at<br />
∂ t W = 1 2 l ∫<br />
dλ e lλ (p + (λ, t) 2 − p − (λ, t) 2 ) .<br />
Then take ano<strong>th</strong>er derivative to obtain<br />
(∂ 2 t − (l ∂ ∫ )W<br />
∂l )2 = 2l 2 dλ e lλ H(l)<br />
( ) ( )<br />
1<br />
H(λ) =<br />
6 p +(λ, t) 3 − λ 2 1<br />
p + −<br />
6 p −(λ, t) 3 − λ 2 p − .<br />
Take <strong>th</strong>e variation of <strong>th</strong>e loop to get <strong>th</strong>e wavefunction of <strong>th</strong>e tachyon and from <strong>th</strong>is<br />
recover <strong>th</strong>e WdW equation:<br />
) (∂ 2 t − ∂2<br />
δW = 2µl 2 δW . (12.18)<br />
∂l 2<br />
b) Generalize part a) to arbitrary Hamiltonians of <strong>th</strong>e form H = p 2 + 1 V (λ), where<br />
2<br />
V is a polynomial, to obtain<br />
(∂ 2 t + l 2 V ( ∂ ∂l ) + 1 lV ′ ( ∂ )<br />
2<br />
∂l ) δW = 2µl 2 δW , (12.19)<br />
for fluctuations in W along <strong>th</strong>e Fermi surface, H(p, λ) = µ. This equation was derived<br />
differently in [121,123].<br />
It should be emphasized <strong>th</strong>at (12.18, 12.19) are only valid at genus zero.<br />
Remark: 51<br />
A fur<strong>th</strong>er interesting property of <strong>th</strong>e transform (12.13) not directly related<br />
to 2D gravity is <strong>th</strong>at it relates massive and massless field <strong>th</strong>eories in 2 spacetime<br />
dimensions. To see <strong>th</strong>is, consider <strong>th</strong>e Lagrangian of a massive Klein–Gordon field in 2<br />
Euclidean dimensions:<br />
∫<br />
S KG =<br />
)<br />
d 2 w<br />
(∂ w Ψ∂ ¯w Ψ + 4µ 2 Ψ 2 . (12.20)<br />
Making a change of variables w = e z , z = 1 (φ + iX), <strong>th</strong>e action becomes<br />
2<br />
∫<br />
)<br />
S KG = d 2 z<br />
(∂ z Ψ∂¯z Ψ + 4µ 2 |w| 2 Ψ 2<br />
∫ (<br />
)<br />
= dφ dX ∂ φ Ψ∂ φ Ψ + 4µ 2 e φ Ψ 2 + ∂ X Ψ∂ X Ψ .<br />
(12.21)<br />
It is precisely <strong>th</strong>is action which is mapped to a massless field on <strong>th</strong>e half-space τ ≥ 0 by<br />
(12.13).<br />
51 Based on conversations wi<strong>th</strong> A.B. Zamolodchikov.<br />
145
12.5. The w ∞ Symmetry of <strong>th</strong>e Harmonic Oscillator<br />
Collective field <strong>th</strong>eory reduces genus zero matrix-model dynamics to <strong>th</strong>e dynamics<br />
of a phase-space fluid under <strong>th</strong>e influence of an upside-down harmonic oscillator. This<br />
system has a very interesting symmetry algebra, following from <strong>th</strong>e existence of an infinitedimensional<br />
symmetry of <strong>th</strong>e harmonic oscillator. 52<br />
Consider <strong>th</strong>e functions a − = λ−p, a + = λ+p on phase-space. Under <strong>th</strong>e Hamiltonian<br />
flow defined by H = 1 2 (p2 − λ 2 ), we have a ± (t) = a(0) e ±t , so <strong>th</strong>e functions<br />
˜C n,m = (a + ) n (a − ) m e (m−n)t (12.22)<br />
are, in fact, time-independent. As functions on (phase space)×IR, under Hamiltonian flow<br />
<strong>th</strong>ey satisfy<br />
d ˜C n,m<br />
dt<br />
= ∂ ˜C n,m<br />
∂t<br />
+ {H, ˜C n,m } = 0 , (12.23)<br />
and should be considered as conserved charges wi<strong>th</strong> explicit time-dependence. It is also<br />
evident <strong>th</strong>at <strong>th</strong>ey form a closed algebra under Poisson brackets,<br />
{ ˜Cn,m , ˜C n ′ ,m ′ }<br />
= 2(m ′ n − mn ′ ) ˜C n+n ′ −1,m+m ′ −1 n, m ≥ 0 . (12.24)<br />
As we will see below, <strong>th</strong>is defines <strong>th</strong>e “wedge subalgebra” of w 1+∞ . Notice <strong>th</strong>at ˜C 1,1 is<br />
itself <strong>th</strong>e Hamiltonian. Upon quantization, we obtain a quantum W ∞ -type algebra which<br />
is in fact a spectrum-generating algebra. 53<br />
Exercise. Classical w 1+∞ and its subalgebras<br />
There is a bewildering choice of bases and algebras in <strong>th</strong>e literature all related to<br />
w 1+∞ but differing in slight, yet important, ways. In <strong>th</strong>is exercise we survey some of<br />
<strong>th</strong>em.<br />
Classical w 1+∞ [124] is <strong>th</strong>e algebra generated by basis vectors W s,n , s = 0, 1, 2, . . .,<br />
n ∈ Z, subject to <strong>th</strong>e relations<br />
[W s,n , W s ′ ,n ′] = (s′ n − sn ′ )W s+s ′ −1,n+n ′ . (12.25)<br />
52 This symmetry of <strong>th</strong>e harmonic oscillator appears to have been noticed first by matrix-model<br />
<strong>th</strong>eorists in 1991! Al<strong>th</strong>ough it was well-known <strong>th</strong>at one could construct a phase space realization<br />
of <strong>th</strong>e wedge subalgebra of w ∞ , <strong>th</strong>e important point <strong>th</strong>at <strong>th</strong>is is a dynamical symmetry of <strong>th</strong>e<br />
oscillator appears to have been overlooked.<br />
53 Note <strong>th</strong>at <strong>th</strong>e ordinary harmonic oscillator action is minus <strong>th</strong>e Euclidean action of an inverted<br />
oscillator. Thus <strong>th</strong>e above results apply to <strong>th</strong>e ordinary harmonic oscillator. The formulae differ<br />
in some factors of i arising from <strong>th</strong>e analytic continuation of Euclidean to Minkowskian time.<br />
146
The basis generators are parametrized by a semilattice of points (s, n) in IR 2 . Equivalent<br />
bases in <strong>th</strong>e literature are obtained by applying affine transformations to <strong>th</strong>is semilattice.<br />
For example, one could instead take generators V s,n , s = 1, 2, . . ., n ∈ Z, related by<br />
V s,n = W s−1,n to give<br />
[V s,n , V s ′ ,n ′ ] = ((s′ − 1)n − (s − 1)n ′ ) V s+s ′ −2,n+n ′ . (12.26)<br />
Several subalgebras are notable:<br />
• w ∞ : generated by V s,n but wi<strong>th</strong> s ≥ 2. In <strong>th</strong>e study of extended chiral algebras<br />
of rational conformal field <strong>th</strong>eories, one encounters <strong>th</strong>ese algebras where V s,n<br />
are <strong>th</strong>e<br />
modes of spin s currents generating <strong>th</strong>e algebra. It is <strong>th</strong>erefore hardly surprising to find<br />
<strong>th</strong>e next subalgebra:<br />
s = 2:<br />
• Witt algebra = Virasoro (c = 0): <strong>th</strong>e algebra generated by <strong>th</strong>e elements wi<strong>th</strong><br />
[V 2,n , V 2,n ′ ] = (n − n ′ ) V 2,n+n ′ . (12.27)<br />
• ∨w: <strong>th</strong>e Wedge subalgebra, is generated by Q j,m wi<strong>th</strong> j = 0, 1 , 1, . . ., m ∈<br />
2<br />
{−j, −j + 1, . . . , j − 1, j} wi<strong>th</strong> relations<br />
[Q j,m , Q j ′ ,m ′] = (j′ m − jm ′ ) Q j+j ′ −1,m+m ′ . (12.28)<br />
• ∨ 2 w: <strong>th</strong>e double-wedge subalgebra is <strong>th</strong>e subalgebra of <strong>th</strong>e wedge algebra generated<br />
by Q j,m wi<strong>th</strong> j = 1, 3 , 2, . . ., |m| ≤ j − 1. Clearly we can continue <strong>th</strong>e process and<br />
2<br />
form a filtration of wedge algebras ∨ n w, defined by restricting |m| < j − n + 1.<br />
• V ir + : <strong>th</strong>e Borel subalgebra of <strong>th</strong>e Virasoro algebra. V ir + may be embedded in<br />
<strong>th</strong>e ∨w algebra in many ways:<br />
L 2s = Q s+1,s s = 0, 1 2 , 1, . . .<br />
L 2s = Q s+1,−s s = 0, 1 2 , 1, . . .<br />
(12.29)<br />
L s = V s,2−s s = 1, 2, . . . .<br />
• w + : Borel subalgebra of w.<br />
This is generated by V s,n where s = 2, 3, . . . and n ≥ −s + 1, and is <strong>th</strong>e analog <strong>th</strong>e<br />
Borel of Virasoro. It plays a role in <strong>th</strong>e W -constraints of <strong>th</strong>e c < 1 models. For w + 1+∞<br />
include s = 1.<br />
a) Show <strong>th</strong>at (12.28) is a subalgebra of (12.25).<br />
b) Show <strong>th</strong>at ∨ 2 w/ ∨ 3 w contains bo<strong>th</strong> of <strong>th</strong>e V ir + algebras defined in <strong>th</strong>e first two<br />
lines of (12.29).<br />
147
The w ∞ symmetry of <strong>th</strong>e inverted oscillator was nicely reformulated in terms of symplectic<br />
geometry in [67]. The action of <strong>th</strong>e oscillator, in first order form, is S = ∫ dα,<br />
where α = p dq − Hdt is a 1-form on (phase space)×IR. A transformation on <strong>th</strong>is space<br />
<strong>th</strong>at takes α → α + dβ is a symmetry. Symmetries are <strong>th</strong>us transformations preserving<br />
<strong>th</strong>e 2-form ω = dα. For <strong>th</strong>e inverted oscillator, we may write<br />
ω = dα = dp dq − (p dp − q dq)dt = dp ′ dq ′<br />
p ′ = cosh tp − sinh tq<br />
(12.30)<br />
q ′ = − sinh tp + cosh tq .<br />
The symmetries are <strong>th</strong>us generated by <strong>th</strong>e Hamiltonian vector fields<br />
V g = ∂g(p′ , q ′ )<br />
∂q ′<br />
∂<br />
∂p − ∂g(p′ , q ′ )<br />
′ ∂p ′<br />
associated to <strong>th</strong>e charges g, where g is a polynomial in p ′ , q ′ .<br />
geometry:<br />
∂<br />
∂q ′ (12.31)<br />
By standard symplectic<br />
[V g1 , V g2 ] = V {g1 ,g 2 } , (12.32)<br />
so we may invariantly characterize <strong>th</strong>e wedge algebra as <strong>th</strong>e algebra of area-preserving<br />
polynomial vector fields on IR 2 .<br />
Exercise. Realizations of w-algebras<br />
Verify <strong>th</strong>at:<br />
a) The wedge algebra may be realized as a Poisson algebra by<br />
Q j,m = 2a j+m<br />
+ a j−m<br />
− . (12.33)<br />
b) The Borel algebra realization occurs naturally in phase space via<br />
V s,n = p n+s−1 λ s−1 . (12.34)<br />
c) Using (12.34), show <strong>th</strong>at V ir + corresponds under Poisson action to <strong>th</strong>e algebra<br />
of analytic coordinate changes in λ.<br />
12.6. The w ∞ Symmetry of Free Field Theory<br />
Classical Theory. Finally let us note <strong>th</strong>at what is true of harmonic oscillators is necessarily<br />
true of free field <strong>th</strong>eory: any free field <strong>th</strong>eory contains an infinite set of w ∞ algebras.<br />
Spacetime locality considerably limits <strong>th</strong>e set of interesting algebras. For example, if φ(x, t)<br />
is a free massless field in 1 + 1 dimensions, we can consider <strong>th</strong>e spin s currents:<br />
whose moments form a classical w 1+∞ algebra.<br />
V s (x, t) = 1 s (∂φ)s s = 1, 2, . . . , (12.35)<br />
148
Exercise. Poisson brackets<br />
Use <strong>th</strong>e Poisson brackets {∂φ(x), ∂φ(y)} = 2πδ ′ (x − y) to show <strong>th</strong>at <strong>th</strong>e modes of<br />
V s<br />
∑<br />
V s (x) = V s,n e inx 0 ≤ x < 2π (12.36)<br />
n∈ Z<br />
obey a classical w 1+∞ algebra:<br />
{V s,n , V s ′ ,n ′ } = i( (s ′ − 1)n − (s − 1)n ′) V s+s ′ −2,n+n ′ .<br />
Quantum Theory. There is a large literature on quantum extensions of w ∞ . One of<br />
particular interest to us is W 1+∞ which may be realized as <strong>th</strong>e algebra of modes of <strong>th</strong>e<br />
Fermion bilinears : ∂ k ψ(z) ∂ l ψ(z): where ψ, ψ comprise a Weyl fermion in 2 dimensions.<br />
By bosonization <strong>th</strong>is may be related to <strong>th</strong>e algebra generated by <strong>th</strong>e modes of <strong>th</strong>e currents<br />
V s = 1 s : e−φ(z) ∂ s e φ(z) :. The structure constants are very complicated and can be found in<br />
[125,126].<br />
Remark: It should be clear from <strong>th</strong>e above discussion <strong>th</strong>at w 1+∞ symmetry is generic,<br />
and occurs whenever <strong>th</strong>ere is a massless scalar field in <strong>th</strong>e problem. This symmetry is so<br />
robust <strong>th</strong>at its seeming presence in completely wrong or meaningless formulae has deceived<br />
many an au<strong>th</strong>or.<br />
12.7. w ∞ symmetry of Classical Collective Field Theory<br />
Let us apply <strong>th</strong>e results of <strong>th</strong>e previous section to collective field <strong>th</strong>eory. Bo<strong>th</strong> in<br />
φ-space and in τ-space, we have asymptotic conformal field <strong>th</strong>eories (=massless scalars) in<br />
spacetime. Thus, we expect on a priori grounds to find a spacetime w ∞ symmetry of <strong>th</strong>e<br />
S-matrix. (See sec. 13.9 below.)<br />
One approach, pursued by Avan and Jevicki [127], is to form <strong>th</strong>e charges<br />
∫<br />
Q j,m = dλ<br />
∫ p+<br />
p −<br />
dp (p + λ) j+m+1 (p − λ) j−m+1 , (12.37)<br />
interpreting p ± in terms of <strong>th</strong>e collective field as in (12.5). The integrals don’t converge<br />
so <strong>th</strong>e expression is somewhat formal, but, working formally, one can use <strong>th</strong>e Poisson<br />
bracket structure to show <strong>th</strong>at <strong>th</strong>e charges satisfy <strong>th</strong>e correct algebra. Al<strong>th</strong>ough collective<br />
field <strong>th</strong>eory is not a free <strong>th</strong>eory Avan and Jevicki show <strong>th</strong>at it has a spectrum generating<br />
algebra given by <strong>th</strong>ese charges. They go on to interpret <strong>th</strong>e collective field action in terms of<br />
149
coadjoint orbit quantization for a group of area-preserving diffeomorphisms [128]. Similar<br />
work has been undertaken in a series of papers by Wadia and collaborators [129].<br />
The w ∞ symmetry may also be seen in <strong>th</strong>e Fermi fluid picture [130,91], where <strong>th</strong>e<br />
charges have exactly <strong>th</strong>e realizations in terms of phase space coordinates described in<br />
<strong>th</strong>e previous section. In <strong>th</strong>e Fermi sea picture, <strong>th</strong>e wedge algebras ∨w, ∨ 2 w have pretty<br />
geometrical interpretations discussed in [67,131]. The phase space charges have associated<br />
Hamiltonian vector fields inducing diffeomorphisms of <strong>th</strong>e (λ, p) plane. The double-wedge<br />
algebra ∨ 2 w is <strong>th</strong>e algebra of area-preserving diffeomorphisms <strong>th</strong>at preserves <strong>th</strong>e hyperbola<br />
a + a − = 0. Therefore, by conjugating wi<strong>th</strong> an appropriate diffeomorphism, we can turn it<br />
into <strong>th</strong>e algebra preserving <strong>th</strong>e collective field ground state at µ > 0. Notice <strong>th</strong>at <strong>th</strong>ese<br />
diffeomorphisms fix <strong>th</strong>e Fermi level as a set, but not pointwise. Similarly <strong>th</strong>e triple wedge<br />
subalgebra ∨ 3 w preserves <strong>th</strong>e Fermi sea pointwise. The quotient ∨ 2 w/ ∨ 3 w contains two<br />
copies of <strong>th</strong>e Virasoro Borel, Vir + , corresponding to diffeomorphisms of <strong>th</strong>e upper and<br />
lower branches of <strong>th</strong>e Fermi sea.<br />
It was first proposed in [32] <strong>th</strong>at <strong>th</strong>e w ∞ symmetry of <strong>th</strong>e matrix model is related<br />
to <strong>th</strong>e extra complexity of <strong>th</strong>e BRST cohomology found in <strong>th</strong>e Liouville approach and<br />
discussed in sec. 4.5 above. The best evidence for <strong>th</strong>e connection is:<br />
1) Algebraic structures. As we will discuss in sec. 14.4 below, in <strong>th</strong>e continuum<br />
approach (at least, at µ = 0) one discovers very similar algebraic structures, in particular,<br />
a realization of <strong>th</strong>e ∨ 2 w algebra associated wi<strong>th</strong> <strong>th</strong>e charges A j,m discussed in sec. 4.5.<br />
However, in view of <strong>th</strong>e generic nature of such symmetries, and <strong>th</strong>e nontrivial relation<br />
between <strong>th</strong>e matrix model coordinate τ and <strong>th</strong>e Liouville field φ, we should be cautious<br />
about such identifications.<br />
2) Quantum numbers. From <strong>th</strong>e local operator expansion (11.65) we see <strong>th</strong>at <strong>th</strong>e<br />
operators ˆB, which are simply related to <strong>th</strong>e moments B r of λ r , have wavefunction<br />
〈 ˆB r,q W (l, −q)〉 = −rµ r/2 K r (2 √ µl) , (12.38)<br />
as one would expect for <strong>th</strong>e Wheeler–DeWitt wavefunctions of <strong>th</strong>e ghost number G = 2<br />
special operators of sec. 4.5. In particular, after <strong>th</strong>e transform to φ-space <strong>th</strong>ese operators<br />
have <strong>th</strong>e correct Liouville quantum numbers.<br />
3) Behavior of Redundant Operators. The transformations<br />
δ s,q λ = (λ + i ˙λ) (s+q)/2−1 (λ − i ˙λ) (s−q)/2−1 (qλ − is ˙λ) e iqt , (12.39)<br />
150
where s = 1, 2, . . . and q ∈ IR form a closed algebra if we interpret <strong>th</strong>e fractional powers by<br />
expanding in λ/ ˙λ and dropping nonpolynomial terms. The algebra of such transformations<br />
can be shown to be [91]:<br />
[δ s 1,q 1<br />
, δ s 2,q 2<br />
] = (q 1 s 2 − q 2 s 1 ) δ s 1+s 2 −2,q 1 +q 2<br />
. (12.40)<br />
For q /∈ {−s, −s + 2, . . . s}, <strong>th</strong>ese transformations are not symmetries of <strong>th</strong>e harmonic<br />
oscillator action<br />
but ra<strong>th</strong>er induce <strong>th</strong>e variation<br />
S = 1 4<br />
δ s,q S = − 1<br />
(s − 1)!<br />
∫<br />
dt ( ˙λ 2 − λ 2 ) , (12.41)<br />
s∏ ( ) ∫<br />
q − (s − 2r) dt λ s e iqx . (12.42)<br />
r=0<br />
In o<strong>th</strong>er words, <strong>th</strong>e operators B s (q) are redundant operators, wi<strong>th</strong> only contact term<br />
interactions if q /∈ {−s, −s + 2, . . . s}. For q ∈ {−s, −s + 2, . . . s}, <strong>th</strong>ey are not redundant<br />
and hence are bulk operators. The failure of <strong>th</strong>ese operators to be redundant in <strong>th</strong>e latter<br />
case is a signal of <strong>th</strong>e appearance of an extra cohomology class, as is indeed predicted by<br />
<strong>th</strong>e continuum formalism.<br />
One weakness of <strong>th</strong>e matrix-model approach to understanding <strong>th</strong>e special states is<br />
<strong>th</strong>at one cannot tell which of <strong>th</strong>e four cohomology classes at discrete values of (p φ , p X ) is<br />
represented by <strong>th</strong>e matrix model operators.<br />
One can try to use <strong>th</strong>e transformations (12.40) to obtain Ward identities for insertions<br />
of special state operators. This works nicely for s = 1 [91]. However, as shown by <strong>th</strong>e<br />
results of <strong>th</strong>e next chapter, for s ≥ 2 <strong>th</strong>e measure and ordering problems present serious<br />
obstacles to <strong>th</strong>is approach.<br />
13. String scattering in two spacetime dimensions<br />
13.1. Definitions of <strong>th</strong>e S-Matrix<br />
We are finally ready to calculate <strong>th</strong>e scattering of strings in two spacetime dimensions<br />
described physically in sec. 0.2 (i.e. fig. 2). Recall <strong>th</strong>at scattering takes place in Minkowski<br />
space. In <strong>th</strong>is chapter we study <strong>th</strong>e <strong>th</strong>eory of sec. 5.4.A: <strong>th</strong>e Liouville coordinate φ is<br />
151
egarded as space, <strong>th</strong>e time coordinate t is a negative signature c = 1 field obtained by<br />
analytically continuing X. The tachyon background<br />
〈<br />
T (φ, t)<br />
〉<br />
= µ e<br />
√<br />
2φ<br />
(13.1)<br />
acts as a repulsive wall for incoming bosons and <strong>th</strong>e dilaton background leads to a spatiallyvarying<br />
coupling<br />
κ eff (φ) = κ 0 e 1 2 Qφ . (13.2)<br />
Because <strong>th</strong>e S-matrix of massless bosons in two-dimensions is a subtle object, we<br />
begin wi<strong>th</strong> some precise ma<strong>th</strong>ematical definitions of what we are talking about. We begin<br />
wi<strong>th</strong> <strong>th</strong>e string definition. As explained in sec. 5.4, <strong>th</strong>e vertex operators are V ± ω<br />
(5.22). Using (11.61), we write<br />
given by<br />
Def 1: The connected string scattering matrix elements are asymptotic expansions in κ<br />
given by<br />
S ST<br />
c<br />
( k∑<br />
ω i →<br />
i=1<br />
l∑<br />
i=1<br />
ω ′ i<br />
)<br />
= A n (Vω − 1<br />
, . . . Vω − k<br />
, V + ω<br />
, . . . V +<br />
1<br />
′ ω<br />
) . (13.3)<br />
′<br />
l<br />
Ma<strong>th</strong>ematically it is easier to use a Euclidean signature boson X via <strong>th</strong>e analytic<br />
continuation |q| → −iω:<br />
V + ω → V q q > 0<br />
V − ω → V q q < 0 .<br />
(13.4)<br />
We’ll refer to <strong>th</strong>e S-matrix elements calculated wi<strong>th</strong> V q as <strong>th</strong>e “Euclidean S-matrix.”<br />
According to <strong>th</strong>e matrix model hypo<strong>th</strong>esis, <strong>th</strong>ese amplitudes may be calculated via<br />
<strong>th</strong>e c = 1 matrix model according to <strong>th</strong>e discussion of sec. 11.7. If one is interested<br />
in <strong>th</strong>e S-matrix and not in <strong>th</strong>e macroscopic loop amplitudes (which contain much more<br />
information), <strong>th</strong>en it is most efficient to calculate <strong>th</strong>e collective field S-matrix which we<br />
describe next. 54<br />
In collective field <strong>th</strong>eory we define <strong>th</strong>e S-matrix according to <strong>th</strong>e coordinate-space<br />
version of <strong>th</strong>e LSZ prescription, <strong>th</strong>at is, we isolate <strong>th</strong>e piece of <strong>th</strong>e large spacetime asymptotics<br />
of time-ordered Green’s functions which is proportional to <strong>th</strong>e product of on-shell<br />
incoming and outgoing wavefunctions.<br />
54 Indeed, defining <strong>th</strong>e S-matrix directly via asymptotics in τ-space [132], as presented below,<br />
was an important technical advance over <strong>th</strong>e original me<strong>th</strong>od [90] of calculating loop amplitudes<br />
and <strong>th</strong>en shrinking <strong>th</strong>e loops.<br />
152
An incoming or outgoing boson of energy ω > 0 has wavefunction ψ L ω (t, τ) = e −iω(t+τ) ,<br />
ψ R ω (t, τ) = e −iω(t−τ) , respectively. Therefore we define <strong>th</strong>e S-matrix according to<br />
Def 2: Consider <strong>th</strong>e asymptotic behavior of <strong>th</strong>e time-ordered, connected, Minkowskian<br />
collective field Green’s function:<br />
G(t 1 , τ 1 , . . . t n , τ n ) ≡ 〈0|T ∏ ∂ τ χ(t i , τ i )|0〉 , (13.5)<br />
as τ i → +∞, t i → −∞ (1 ≤ i ≤ k), t i → +∞ (k + 1 ≤ i ≤ n = k + l). Then we define <strong>th</strong>e<br />
connected S-matrix element for <strong>th</strong>e process |ω 1 , . . . ω k 〉 → |ω 1, ′ . . . ω<br />
l ′ 〉 to be <strong>th</strong>e function<br />
S CF<br />
c<br />
in <strong>th</strong>e asymptotic formula:<br />
G ∼<br />
∫ ∞<br />
0<br />
k∏<br />
dω i<br />
i=1<br />
l∏<br />
dω i ′ δ( ∑ ω i − ∑ ω i)<br />
′<br />
i=1<br />
· √ k! l! S CF<br />
c<br />
( k∑<br />
ω i →<br />
i=1<br />
l∑<br />
i=1<br />
ω ′ i<br />
k∏ l∏<br />
(ψω L i<br />
) ∗<br />
i=1<br />
i=1<br />
ψ R ω ′ i<br />
)<br />
+ off-shell terms .<br />
(13.6)<br />
The plane wave states are normalized such <strong>th</strong>at 〈ω|ω ′ 〉 = ω δ(ω −ω ′ ). An equivalent definition<br />
has been used in [96,97] to compute <strong>th</strong>e S-matrix from standard Feynman perturbation<br />
<strong>th</strong>eory applied to collective field <strong>th</strong>eory.<br />
While <strong>th</strong>is definition is physically satisfying, it is not <strong>th</strong>e best ma<strong>th</strong>ematical definition.<br />
An equivalent definition is obtained by continuing <strong>th</strong>e Minkowskian Green’s functions to<br />
Euclidean space ∆t → −i∆X. Fourier transforming <strong>th</strong>e Euclidean Green’s functions wi<strong>th</strong><br />
respect to X i , we obtain mixed Green’s functions<br />
∫ ∏<br />
G E (q 1 , τ 1 , . . . q n , τ n ) ≡<br />
i<br />
dX i e iq iX i<br />
G Euclidean (X 1 , τ 1 , . . . X n , τ n ) , (13.7)<br />
in terms of which we may define <strong>th</strong>e S-matrix via:<br />
Def2 ′ : The large τ i asymptotics<br />
G E (q 1 , τ 1 , . . . q n , τ n ) ∼ δ( ∑ q i ) ∏ i<br />
e −|q i|τ i<br />
R(q 1 , . . . q n ) ( 1 + O(e −τ i<br />
) ) (13.8)<br />
defines a function R n (q 1 , . . . q n ) from which we may obtain <strong>th</strong>e connected S CF -matrix<br />
elements via analytic continuation |q| → −iω, where q < 0 corresponds to <strong>th</strong>e incomers,<br />
and q > 0 corresponds to <strong>th</strong>e outgoers. Specifically, S c → − ik+l+1 √<br />
k! l!<br />
R.<br />
Remark: The non-obvious property <strong>th</strong>at <strong>th</strong>e function R n is independent of <strong>th</strong>e order<br />
in which <strong>th</strong>e τ i are taken to ∞ was demonstrated in [132].<br />
153
The equivalence of <strong>th</strong>e collective field <strong>th</strong>eory S-matrix and <strong>th</strong>e correlators defined by<br />
shrinking macroscopic loops is demonstrated using <strong>th</strong>e relation between τ-space and φ-<br />
space explained in sec. 12.4 above. In particular, transforming asymptotic wavefunctions<br />
according to (12.13), we relate l → 0 and τ → ∞ asymptotics via <strong>th</strong>e integral<br />
∫ ∞<br />
dτ e −l2√ µ cosh τ e −|q|τ ∼ (l √ µ) |q| Γ(−|q|) , (13.9)<br />
plus terms regular in l. Notice <strong>th</strong>at <strong>th</strong>e two prescriptions only make complete sense when<br />
q is nonintegral. O<strong>th</strong>erwise we must use <strong>th</strong>e full identity<br />
∫ ∞<br />
A<br />
dτ e −2l√ µ cosh τ e −|q|τ = −<br />
π<br />
sin π|q| I |q|(2 √ µl)<br />
− ∑ (−1) n (l √ µ) n<br />
n≥0<br />
n ∑<br />
m=0<br />
1 e A(m−n−|q|)<br />
(13.10)<br />
m!(n − m)! m − n − |q| .<br />
The pole in <strong>th</strong>e Γ–function in (13.9) is a warning <strong>th</strong>at we cannot unambiguously separate<br />
<strong>th</strong>e two terms in (13.10) via nonanalyticity in l.<br />
<br />
Leg Factors<br />
According to <strong>th</strong>e arguments of chapt. 11, we expect <strong>th</strong>at <strong>th</strong>e Euclidean S-matrices<br />
S ST and S CF should agree up to an overall normalization f(q) of <strong>th</strong>e vertex operators V q .<br />
This is because, for q /∈ Z, <strong>th</strong>e BRST cohomology wi<strong>th</strong> <strong>th</strong>e relevant quantum numbers is<br />
one-dimensional. Indeed, comparison wi<strong>th</strong> vertex operator calculations in Liouville <strong>th</strong>eory,<br />
which will be described in sec. 14.2 below, shows <strong>th</strong>at<br />
The factors<br />
A 0,n (V q1 , . . . V qn ) = (−i) n+1<br />
n ∏<br />
i=1<br />
f(q) = Γ(−|q|)<br />
Γ(|q|)<br />
Γ(−|q i |)<br />
Γ(|q i |) R n(q 1 , . . . q n ) . (13.11)<br />
(13.12)<br />
are called “leg factors.” Notice <strong>th</strong>at for <strong>th</strong>e Minkowskian S-matrix <strong>th</strong>ey are pure phases,<br />
but <strong>th</strong>e phases for incomers and outgoers are not complex conjugated. Comparison wi<strong>th</strong><br />
<strong>th</strong>e first quantized wavefunction for <strong>th</strong>e spacetime boson, described by <strong>th</strong>e Wheeler–DeWitt<br />
equation (5.21), indicates <strong>th</strong>at nei<strong>th</strong>er normalization in (13.11) is <strong>th</strong>e correct physical<br />
normalization since standard first-quantized scattering <strong>th</strong>eory predicts <strong>th</strong>at <strong>th</strong>e genus zero<br />
1 → 1 S-matrix is<br />
Γ(iE)<br />
Γ(−iE) .<br />
This suggests <strong>th</strong>at <strong>th</strong>e correct normalization of <strong>th</strong>e vertex<br />
operators is obtained by taking <strong>th</strong>e squareroot of (13.12). All <strong>th</strong>is needs to be clarified!<br />
154
13.2. On <strong>th</strong>e Violation of Folklore<br />
The c = 1 S-matrix violates several standard aspects of S-matrix folklore.<br />
commonly said, for example, <strong>th</strong>at one cannot define an S-matrix for massless bosons. For<br />
example, <strong>th</strong>e standard LSZ prescription appears to be problematic because if we make a<br />
field redefinition<br />
It is<br />
Φ → χ + a 2 χ 2 + a 3 χ 3 + · · · , (13.13)<br />
in <strong>th</strong>e massless case <strong>th</strong>ere is no gap between <strong>th</strong>e one-particle and two-particle <strong>th</strong>resholds, so<br />
S-matrix elements appear to depend on <strong>th</strong>e choice of interpolating field. 55 More physically,<br />
we cannot expect to tell <strong>th</strong>e difference between (say) a rightmoving boson of energy E<br />
and two rightmoving bosons of energy E/2.<br />
A related ma<strong>th</strong>ematical point is <strong>th</strong>at <strong>th</strong>e<br />
momentum-space Green’s functions should have cuts, not poles, so we can’t isolate an<br />
S-matrix element by extracting <strong>th</strong>e residues at poles.<br />
In <strong>th</strong>e present case we find <strong>th</strong>at <strong>th</strong>ere are no cuts, but <strong>th</strong>ere are instead kinematic<br />
regions, and <strong>th</strong>e momentum space Green’s functions are continuous, but not differentiable,<br />
across regions. The S-matrix will have a large symmetry group related to W ∞ , which is<br />
nonlinear in <strong>th</strong>e momentum and allows us to distinguish a rightmoving boson of energy E<br />
and two rightmoving bosons of energy E/2.<br />
Ano<strong>th</strong>er objection to massless S-matrices is <strong>th</strong>at by a simple conformal transformation<br />
one can fill <strong>th</strong>e vacuum wi<strong>th</strong> particles. 56 In our case, <strong>th</strong>e “defect” or wall at τ = 0 breaks<br />
conformal invariance enough to forbid such freedom.<br />
We have also violated folklore in ano<strong>th</strong>er way. The exactly solvable S-matrix presented<br />
below has particle production, yet at <strong>th</strong>e same time has a large W ∞ symmetry. Typically,<br />
exactly solvable S-matrices in field <strong>th</strong>eories wi<strong>th</strong> infinite numbers of conservation laws [135]<br />
do not have particle production.<br />
There are several related issues, connected wi<strong>th</strong> <strong>th</strong>e interpretation of <strong>th</strong>e wavefunction<br />
factors f(q). The resolution of <strong>th</strong>ese issues will probably require careful specification of<br />
how S-matrix elements are to be measured.<br />
Finally, we remark <strong>th</strong>at <strong>th</strong>e c = 1 S-matrix bears a great similarity to a number of<br />
o<strong>th</strong>er physical problems which have been of interest in recent years. These include <strong>th</strong>e<br />
55 For a discussion of <strong>th</strong>e independence of <strong>th</strong>e S-matrix from a choice of interpolating field, see<br />
[79,133].<br />
56 Indeed, calculations of Hawking radiation in <strong>th</strong>e CGHS <strong>th</strong>eory [134] are based on <strong>th</strong>is<br />
phenomenon.<br />
155
Kondo effect, <strong>th</strong>e Callan–Rubakov effect, Hawking radiation and particle scattering off a<br />
black hole (especially in <strong>th</strong>e CGHS model [134]) and 1 + 1 linear dilaton electrodynamics.<br />
Massless S-matrices have played a role in <strong>th</strong>e <strong>th</strong>eory of exactly solvable field <strong>th</strong>eories,<br />
for example <strong>th</strong>ey have appeared in past discussions of <strong>th</strong>e XXX and XYZ models [136]<br />
and more recently have begun to play a more central role in <strong>th</strong>e massless flows between<br />
conformal field <strong>th</strong>eories [137].<br />
13.3. Classical scattering in collective field <strong>th</strong>eory<br />
We now consider <strong>th</strong>e classical scattering problem for <strong>th</strong>e collective field using <strong>th</strong>e<br />
picture of <strong>th</strong>e time-dependent Fermi sea.<br />
Suppose <strong>th</strong>e solution is given by (12.3) and<br />
represents an incoming wavepacket which is dispersed as it travels in phase space. We will<br />
derive a functional relation between <strong>th</strong>e incoming and outgoing wavepackets [95].<br />
Let us return to <strong>th</strong>e general solution (12.3a, b), and assume <strong>th</strong>ere are no folds. 57<br />
We may solve <strong>th</strong>e first equation to obtain σ ± (λ, t).<br />
If <strong>th</strong>ere were no dispersion of <strong>th</strong>e<br />
wavepacket, we would find ¯σ ± (λ, t) = t ± τ. Denote <strong>th</strong>e difference by σ ± = ¯σ ± + δσ ± .<br />
From <strong>th</strong>e spacetime asymptotics of <strong>th</strong>e solution (12.3) above, we find<br />
(<br />
δσ ± (t ± τ) = ∓ log 1 + a ( t ± τ + δσ ± (t ± τ) )) , (13.14)<br />
and, in particular, δσ ± becomes a function of one variable.<br />
defining in- and out-waves is<br />
The asymptotic behavior<br />
where λ → +∞ holding t ∓ τ fixed.<br />
p ± (λ, t) → ±λ ∓ 1 (<br />
1 + ψ∓ (t ∓ τ) ) + O(1/λ 2 ) , (13.15)<br />
2λ<br />
Plugging <strong>th</strong>is into <strong>th</strong>e expression (p ± ∓ λ)λ and<br />
comparing wi<strong>th</strong> <strong>th</strong>e general solution, we find <strong>th</strong>at <strong>th</strong>e waves can be expressed in terms of<br />
<strong>th</strong>e function a as:<br />
1 + ψ ± =<br />
(<br />
1 + a ( t ± τ + δσ(t ± τ) )) 2<br />
. (13.16)<br />
Thus we can calculate <strong>th</strong>e time-delay, namely <strong>th</strong>e relation between t and t ′<br />
ψ + (x ′ ) = ψ + (t ′ + τ) = ψ − (t − τ) = ψ − (x):<br />
such <strong>th</strong>at<br />
x ′ + δσ + (x ′ ) = x + δσ − (x) =⇒ x ′ = x + 2δσ − (x) , (13.17)<br />
57 The conditions for <strong>th</strong>is are given in [138].<br />
156
since from (13.14) we see <strong>th</strong>at δσ + (x ′ ) = −δσ − (x). It follows from (13.14) and (13.16)<br />
<strong>th</strong>at <strong>th</strong>at we have <strong>th</strong>e functional relation between in- and out- waves:<br />
ψ − (x) = ψ + (x ′ )<br />
= ψ +<br />
(x + log ( 1 + ψ − (x) )) .<br />
(13.18)<br />
From <strong>th</strong>e derivation we see <strong>th</strong>e essential physics: different parts of <strong>th</strong>e wavepacket suffer<br />
different time delays.<br />
We now solve <strong>th</strong>e equation (13.18), <strong>th</strong>us solving <strong>th</strong>e classical field scattering and, in<br />
principle, <strong>th</strong>e tree level S-matrix of <strong>th</strong>e <strong>th</strong>eory. The solution of (13.18) was given in [138]<br />
and is derived as follows.<br />
Suppose Ψ ± constitute a solution of <strong>th</strong>e classical scattering<br />
equations (13.18), and suppose fur<strong>th</strong>er <strong>th</strong>at Ψ ± + γ ± is a nearby solution, where γ ± are<br />
small. To first order in <strong>th</strong>e variations, (13.18) becomes<br />
γ + (˜x) d˜x = γ − (x) dx , (13.19)<br />
where ˜x = x + log(1 + Ψ − (x)). Taking a Fourier transform of <strong>th</strong>is equation, wi<strong>th</strong><br />
γ ± (x) ≡<br />
∫ ∞<br />
−∞<br />
dξ γ ± (ξ) e iξx , (13.20)<br />
leads to<br />
γ + (ξ) = 1 ∫<br />
2π<br />
dx e −iξx γ − (x) ( 1 + Ψ − (x) ) −iξ<br />
. (13.21)<br />
This may be regarded as a first-order differential equation in function space. Integrating<br />
<strong>th</strong>is equation wi<strong>th</strong> <strong>th</strong>e boundary condition ψ + = 0 ⇒ ψ − = 0, we obtain <strong>th</strong>e general<br />
solution of Polchinski’s scattering equations:<br />
2πψ ± (ξ) = 1 ∫ ∞ ( (1<br />
dx e −iξx + ψ∓ (x) ) )<br />
1∓iξ<br />
− 1 . (13.22)<br />
1 ∓ iξ −∞<br />
In position space <strong>th</strong>is takes <strong>th</strong>e form<br />
ψ ± (x) = − ∑ p≥1<br />
Γ(±∂ x + p − 1)<br />
Γ(±∂ x )<br />
(−ψ ∓ (x)) p<br />
p!<br />
. (13.23)<br />
(The ratio of Γ-functions is interpreted as a polynomial in derivatives.)<br />
This completely solves <strong>th</strong>e classical scattering problem.<br />
157
13.4. Tree-Level Collective Field Theory S-Matrix<br />
From <strong>th</strong>e classical scattering matrix, we may derive <strong>th</strong>e tree-level quantum S-matrix<br />
by interpreting <strong>th</strong>e left- and right-moving fields as incoming and outgoing quantum fields:<br />
ψ ± → − √ π 1 µ (∂ t ± ∂ τ )χ ±<br />
χ + = i<br />
χ − = i<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
−∞<br />
dξ<br />
√<br />
4πξ<br />
α + (ξ) e iξ(t+τ)<br />
dξ<br />
√<br />
4πξ<br />
α − (ξ) e iξ(t−τ)<br />
[α ± (ξ), α ± (ξ ′ )] = −ξ δ(ξ + ξ ′ ) .<br />
(13.24)<br />
Now following Polchinski, we interpret <strong>th</strong>e relation (13.23) as a relation between incoming<br />
and outgoing Fourier modes:<br />
α ± (η) = ∑ ( 1 ∫<br />
Γ(1 ∓ iη) 1 ∞<br />
µ )p−1 d p ξ δ(η − ∑ ξ i ) : α ∓ (ξ 1 ) · · · α ∓ (ξ p ): . (13.25)<br />
Γ(2 ∓ iη − p) p!<br />
p≥1<br />
−∞<br />
Quantum mechanically, <strong>th</strong>e Fourier modes in (13.24) are creation and annihilation operators<br />
for left- and right-moving particles. Let us consider <strong>th</strong>e S-matrix element for one<br />
incoming left-mover of energy ω to decay to m outgoing particles of energies ω = ∑ ω i :<br />
S c (ω →<br />
m∑<br />
m∏<br />
ω i ) = 〈0|α − (−ω) α + (ω j )|0〉 c , (13.26)<br />
i=1<br />
j=1<br />
where <strong>th</strong>e vacuum is defined by α + (−ω)|0〉 = 0 for ω > 0. From (13.25) we may read off<br />
wi<strong>th</strong>out fur<strong>th</strong>er calculation <strong>th</strong>e result:<br />
Sc<br />
CF (ω →<br />
m∑<br />
ω i ) = −i( 1 µ )m−1 ω<br />
i=1<br />
The corresponding Euclidean S-matrix is<br />
m∏<br />
k=1<br />
ω k<br />
Γ(−iω)<br />
Γ(2 − m − iω) . (13.27)<br />
µ |q| R m+1 (q 1 , . . . q m , q) = ( 1 µ )m−1 i m |q| ∏ |q i |( ∂<br />
∂µ )m−2 µ |q|−1 , (13.28)<br />
a formula we will obtain in <strong>th</strong>e next chapter via continuum me<strong>th</strong>ods.<br />
O<strong>th</strong>er S-matrix amplitudes can be derived analogously [138]. The S-matrix is not<br />
analytic in <strong>th</strong>e energies ω i and does not satisfy crossing symmetry. In general we<br />
must divide momentum space into kinematic regions. These are defined as follows. (It<br />
158
is convenient to work in Euclidean space here).<br />
For any set S of momenta we let<br />
H(S) = {⃗q ∈ IR k | ∑ q∈S<br />
q = 0}. Then we take connected components of <strong>th</strong>e region<br />
{∑ [<br />
]<br />
qi = 0}<br />
∩ IR k − ∪ S H(S) = ∐ C α , (13.29)<br />
α<br />
where ∪ S is over proper subsets S of momenta, and <strong>th</strong>e C α are <strong>th</strong>e disjoint kinematic<br />
regions.<br />
From <strong>th</strong>e above formulae one can show <strong>th</strong>at S is continuous on { ∑ q i = 0} ∩ IR k , and<br />
indeed in each region C α , S-matrix elements are polynomials in <strong>th</strong>e q i . The polynomials<br />
change from region to region, however, so <strong>th</strong>e S-matrix elements are not differentiable<br />
across regions.<br />
Example: Four-point function.<br />
Bo<strong>th</strong> cases S c (ω 1 → ω 2 + ω 3 + ω 4 ) and S c (ω 1 + ω 2 → ω 3 + ω 4 ) are covered by <strong>th</strong>e<br />
formula<br />
i( 1 4∏ (<br />
µ )2 ω i 1 + i max{ωi } ) , (13.30)<br />
i=1<br />
where analyticity is lost due to <strong>th</strong>e appearance of <strong>th</strong>e maximal value max{ω i }.<br />
13.5. Nonperturbative S-matrices<br />
The tree level S-matrix can be extended to all orders of perturbation <strong>th</strong>eory, and<br />
can even be given an unambiguous nonperturbative definition by returning to <strong>th</strong>e eigenvalue/macroscopic<br />
loop correlators of chapt. 11. According to Def2 and (13.9) above we<br />
must isolate <strong>th</strong>e large λ asymptotics of (11.38). These in turn follow from <strong>th</strong>e large λ<br />
asymptotics of <strong>th</strong>e Euclidean fermion propagator (11.37), which we now describe.<br />
The function I can be written in terms of parabolic cylinder functions, whose asymptotics<br />
are well-known. In <strong>th</strong>is way we find <strong>th</strong>e asymptotics for λ i → +∞ to be:<br />
I(q, λ 1 , λ 2 ) ∼<br />
−i √<br />
λ1 λ 2<br />
(e −q|τ 1−τ 2 | e iµ|G(τ 1)−G(τ 2 )|<br />
I(q, λ 1 , λ 2 ) ∼<br />
+R q e iµ(G(τ 1)+G(τ 2 )) e −q(τ 1+τ 2 ) ) (<br />
1 + O(e<br />
−τ i<br />
) ) q > 0<br />
i<br />
√<br />
λ1 λ 2<br />
(e q|τ 1−τ 2 | e −iµ|G(τ 1)−G(τ 2 )|<br />
(13.31)<br />
+(R q ) ∗ e −iµ(G(τ 1)+G(τ 2 )) e q(τ 1+τ 2 )) (1<br />
+ O(e<br />
−τ i<br />
) ) q < 0 ,<br />
where G(τ) is <strong>th</strong>e WKB wavefunction factor. The two terms in (13.31) may be understood<br />
intuitively as <strong>th</strong>ose corresponding to direct and reflected propagation of <strong>th</strong>e fermions in <strong>th</strong>e<br />
159
presence of a wall. The function R q is a Euclidean continuation of <strong>th</strong>e fermion reflection<br />
factor R(E) for potential scattering wi<strong>th</strong> V (λ) ∼ −λ 2 . In particular, for scattering on a<br />
half-line λ ∈ [0, ∞), we have<br />
√ √<br />
R(E) = iµ iE 1 + ie −πE Γ( 1 2 − iE)<br />
1 − ie −πE Γ( 1 2 + iE)<br />
= µ iE √<br />
2<br />
π e3iπ/4 cos ( π<br />
2 ( 1 2 + iE)) Γ( 1 2 − iE) . (13.32)<br />
The corresponding Euclidean “bounce factor” is given by R q = R(µ + i|q|). Using <strong>th</strong>e<br />
rule |q| → −iω, we can pass easily back and for<strong>th</strong> from <strong>th</strong>e Euclidean to <strong>th</strong>e Minkowskian<br />
picture (keeping in mind <strong>th</strong>at q < 0 corresponds to incomers and q > 0 to outgoers).<br />
In order to obtain <strong>th</strong>e S-matrix from (11.38) we must substitute (13.31) into (11.38)<br />
and isolate only <strong>th</strong>e terms corresponding to <strong>th</strong>e coefficients of <strong>th</strong>e on-shell wavefunctions.<br />
In particular, we are only interested in <strong>th</strong>e terms where (1) <strong>th</strong>e factors of e iµG(τ) cancel, and<br />
(2) <strong>th</strong>e overall τ-dependence is proportional to ∏ e −|q i|τ i<br />
. The decomposition of I in terms<br />
of direct and reflected propagation is easily encapsulated in a diagrammatic formalism<br />
whose detailed derivation is given in [132]. The final result is sufficiently intuitive <strong>th</strong>at <strong>th</strong>e<br />
reader should be satisfied wi<strong>th</strong> our presentation here wi<strong>th</strong>out proof.<br />
t<br />
R*(µ − ω )<br />
1<br />
+ω 1<br />
ω − ω 1<br />
(ω , ω)<br />
R(µ + (ω − ω ) )<br />
1<br />
−ω<br />
1<br />
τ<br />
−(ω − ω )<br />
1<br />
(−ω , ω)<br />
Fig. 27: 1 → 1 scattering<br />
Consider <strong>th</strong>e case of 1 → 1 scattering, illustrated by fig. 27. We have depicted an<br />
incoming relativistic boson, which may be fermionized to a particle–hole pair. The particle<br />
and hole undergo potential scattering, and reflect back from <strong>th</strong>e wall. They may <strong>th</strong>en be<br />
160
ebosonized. The amplitude for <strong>th</strong>is process is simply an integral over possible particle–<br />
hole energies weighted by <strong>th</strong>e reflection factor for <strong>th</strong>e particle and hole, <strong>th</strong>at is, we have<br />
<strong>th</strong>e 1 → 1 S-matrix element:<br />
S(ω → ω) =<br />
∫ ω<br />
0<br />
dω 1 R ∗ (µ − ω 1 ) R ( µ + (ω − ω 1 ) ) . (13.33)<br />
I =<br />
q > 0<br />
+<br />
q<br />
I =<br />
q < 0<br />
+<br />
- q<br />
- q q<br />
(a)<br />
(b)<br />
Fig. 28: a) A pictorial version of <strong>th</strong>e integral I for positive momentum.<br />
pictorial version of <strong>th</strong>e integral I for negative momentum<br />
b) A<br />
q i<br />
q i<br />
Fig. 29: Incoming and outgoing vertices. The dotted line carrying negative (positive)<br />
momentum q i should be <strong>th</strong>ought of as an incoming (outgoing) boson wi<strong>th</strong><br />
energy |q i |. Momentum carried by lines is always conserved as time flows upwards.<br />
This intuitive description may be formalized by <strong>th</strong>e following set of general rules: 58<br />
To each incoming and outgoing boson associate a vertex in <strong>th</strong>e (t, τ) half-space. Connect<br />
points via line segments to form a one-loop graph. Since <strong>th</strong>e expression for I in (13.31) has<br />
two terms, we have bo<strong>th</strong> direct and reflected propagators as in fig. 28. Each line segment<br />
carries a momentum and an arrow. Note <strong>th</strong>at <strong>th</strong>e reflected propagator in fig. 28, which we<br />
call simply a “bounce,” is composed of two segments wi<strong>th</strong> opposite arrows and momenta.<br />
These line segments are joined according to <strong>th</strong>e following rules:<br />
RH1. Lines wi<strong>th</strong> positive (negative) momenta slope upwards to <strong>th</strong>e right (left).<br />
RH2. At any vertex arrows are conserved and momentum is conserved as time flows<br />
upwards. In particular momentum q i is inserted at <strong>th</strong>e vertex as in fig. 29.<br />
RH3. Outgoing vertices at (t out , τ out ) all have later times <strong>th</strong>an incoming vertices (t in , τ in ):<br />
t out > t in .<br />
58 The following is paraphrased directly from [132].<br />
161
ω<br />
q<br />
R(µ + ω ; V )<br />
R q<br />
q > 0<br />
− ω<br />
− q<br />
ω<br />
− q<br />
R*(µ − ω ; V )<br />
R* q<br />
q < 0<br />
− ω<br />
Fig. 30: Bounce factors for reflected propagators. The Minkowskian factors are<br />
shown at <strong>th</strong>e left, and <strong>th</strong>eir Euclidean analogs are shown at <strong>th</strong>e right.<br />
q<br />
To each graph we associate an amplitude, wi<strong>th</strong> bounce factors R for reflected propagators<br />
as in fig. 30. and ±1 for upwards (downwards) sloping direct propagators. Finally, we<br />
sum over graphs and integrate over kinematically allowed momenta, <strong>th</strong>us getting a formula<br />
for <strong>th</strong>e Euclidean amplitudes R n which reads schematically:<br />
R = i ∑ ∫<br />
∏<br />
n ± dq R Q (−R Q ) ∗ . (13.34)<br />
See [132] for more details.<br />
graphs bounces<br />
t<br />
x<br />
ω − x<br />
ω − x<br />
2<br />
(ω , ω )<br />
1 1<br />
(ω , ω )<br />
2 2<br />
t<br />
x<br />
ω − x<br />
1<br />
ω − x<br />
(ω , ω )<br />
1 1<br />
(ω , ω )<br />
2 2<br />
−x<br />
τ<br />
−x<br />
τ<br />
−(ω − x)<br />
(−ω , ω)<br />
0 ≤ x ≤ ω<br />
2<br />
−(ω − x)<br />
(−ω , ω)<br />
ω 1 ≤ x ≤ ω 1<br />
+ ω 2<br />
Fig. 31: 1 → 2 scattering<br />
Exercise.<br />
Returning to Minkowski space, derive <strong>th</strong>e 1 → 2 scattering matrix by showing <strong>th</strong>at<br />
<strong>th</strong>e two diagrams in fig. 31 correspond to<br />
∫<br />
√ ω2<br />
2S(ω → ω1 + ω 2 ) = dx R(µ + ω − x) R ∗ (µ − x) −<br />
0<br />
∫ ω<br />
ω 1<br />
dx R(µ + ω − x) R ∗ (µ − x) . (13.35)<br />
162
Finally, we must relate <strong>th</strong>ese nonperturbative S-matrices to string perturbation <strong>th</strong>eory.<br />
By double-scaling, <strong>th</strong>e string perturbation series can be extracted by restoring <strong>th</strong>e string<br />
coupling µ → µ/κ and taking κ → 0 asymptotics. This is <strong>th</strong>e same as taking µ → ∞<br />
asymptotics holding p i or ω i fixed. Thus we need <strong>th</strong>e asymptotic behavior of <strong>th</strong>e bounce<br />
factors. To all orders of perturbation <strong>th</strong>eory, we can replace <strong>th</strong>e expression (13.32) by <strong>th</strong>e<br />
simpler expression for <strong>th</strong>e Euclidean bounce factor at p > 0:<br />
Here <strong>th</strong>e Q k are polynomials in p.<br />
R p = (−iµ) −p Γ( 1 2<br />
− iµ + p)<br />
Γ( 1 2 − iµ) ∼ 1 +<br />
∞∑<br />
k=1<br />
Q k (p)<br />
µ k . (13.36)<br />
13.6. Properties of S-Matrix Elements<br />
From <strong>th</strong>e above algori<strong>th</strong>m one can calculate any S-matrix element. Some general<br />
properties of <strong>th</strong>e S-matrix elements following from <strong>th</strong>e above construction are <strong>th</strong>e following.<br />
First let us define some notation. By KPZ scaling, <strong>th</strong>e Euclidean S-matrix elements<br />
〈 k∏ 〉<br />
T q + i<br />
i=1<br />
h<br />
= µ 2h+k−1− 1 2<br />
∑<br />
|qi | F h (q 1 , . . . q k ) (13.37)<br />
define certain functions F h (q 1 , . . . q k ) associated to <strong>th</strong>e moduli spaces M h,k of curves wi<strong>th</strong><br />
h handles and k punctures. Defining different kinematic regions C α as in eq. (13.29), one<br />
can <strong>th</strong>en show:<br />
Some Properties of perturbative amplitudes:<br />
i) F h is parity-invariant: F h (q i ) = F h (−q i ).<br />
ii) F h is continuous on { ∑ q i = 0} ∩ IR k<br />
iii) In C α , F h is a polynomial in <strong>th</strong>e momenta wi<strong>th</strong> rational coefficients. In general <strong>th</strong>e<br />
polynomial is different in different regions. That is, <strong>th</strong>e expressions are continuous<br />
but not continuously differentiable.<br />
iv) The degree of <strong>th</strong>e polynomial is 2k + 4h − 3.<br />
v) As any momentum goes to zero, we have<br />
〈 k∏<br />
i=1<br />
T + q i<br />
〉<br />
qi →0<br />
∼<br />
|q i | ∂ 〈∏<br />
∂µ<br />
j≠i<br />
T + q i<br />
〉<br />
. (13.38)<br />
vi) If q i ∈ Z, <strong>th</strong>en F h = 0 for sufficiently large genus, specifically, for 2h − 2 + k > ∑ |q i |.<br />
163
Property (i) follows from <strong>th</strong>e integral representations of macroscopic loops [90]. Properties<br />
(ii),(iii) and (v) are proved in [132]. (Property (iii) was first noted in [90,95,48].)<br />
Properties (iv) and (vi) are proved in [139].<br />
Properties (ii–v) have interesting physical interpretations: Properties (ii) and (iii)<br />
result from having derivatively coupled massless bosons. Usually, massless particles lead to<br />
cuts in <strong>th</strong>e S-matrix. In our case, <strong>th</strong>e cuts become simple discontinuities of <strong>th</strong>e derivatives<br />
wi<strong>th</strong> respect to energy. Property (iv) essentially says <strong>th</strong>at at large spacetime energies<br />
<strong>th</strong>e string coupling becomes effectively energy dependent, κ eff<br />
(ω) ∼ ω 2 /µ. This effective<br />
energy-dependence of <strong>th</strong>e string coupling has been discussed from <strong>th</strong>e continuum Liouville<br />
<strong>th</strong>eory point of view in [140].<br />
Exercise. Energy-Dependent Effective String Coupling<br />
Derive <strong>th</strong>e rule κ eff (ω) ∼ ω 2 from <strong>th</strong>e Liouville <strong>th</strong>eory as follows [140]: From <strong>th</strong>e<br />
formula for Liouville energy, compute <strong>th</strong>e turning point in φ. Plug into <strong>th</strong>e formula for<br />
<strong>th</strong>e spatially-dependent string coupling to find κ eff (ω) = κ 0 e 1 2 Qφ = κ 0 ω 2 /µ.<br />
A related phenomenon is <strong>th</strong>e inapplicability of <strong>th</strong>e string perturbation expansion for<br />
high energy scattering [90]. The asymptotic expansion for string perturbation amplitudes<br />
is an expansion at fixed ω i for µ → +∞. Ordinarily in physics we measure physical values<br />
of <strong>th</strong>e coupling constants (e.g. α = 1<br />
137<br />
) and we probe physical laws by building ever larger<br />
and more expensive accelerators, i.e., by increasing <strong>th</strong>e energies ω i . 59 In <strong>th</strong>e c = 1 model<br />
we would find, at fixed µ and sufficiently high energies, <strong>th</strong>at <strong>th</strong>e string perturbation series<br />
ceases even to be an asymptotic expansion. At such energies new physics must emerge<br />
and <strong>th</strong>e string approximation — which is now seen to be only a low energy approximation<br />
— breaks down. In <strong>th</strong>e present context <strong>th</strong>e “underlying physics” which we would discover<br />
would be <strong>th</strong>e spacetime matrix model fermions. It remains to be seen if <strong>th</strong>is situation is<br />
typical of nonperturbative string <strong>th</strong>eory.<br />
<br />
Decoupling of <strong>th</strong>e cosmological constant<br />
The low-energy <strong>th</strong>eorem, property (v), is probably related to <strong>th</strong>e decoupling of one<br />
of <strong>th</strong>e two cosmological constant operators, and plays a key role in <strong>th</strong>e analysis of [140].<br />
Taking a naive q → 0 limit of <strong>th</strong>e tachyon vertex operator, we obtain V q → e √ 2φ (1 +<br />
(iqX − |q|φ)/ √ 2 + O(q 2 )). One may <strong>th</strong>erefore try to interpret property (v) in terms of <strong>th</strong>e<br />
decoupling of <strong>th</strong>e cosmological constant operator e √ 2φ and as well <strong>th</strong>e “operator” Xe √ 2φ ,<br />
59 Ignore renormalization group flow of couplings, for <strong>th</strong>e sake of <strong>th</strong>is argument.<br />
164
since <strong>th</strong>e leading term in amplitude goes as |q| which multiplies <strong>th</strong>e “operator” φ e √ 2φ .<br />
This fits in well wi<strong>th</strong> <strong>th</strong>e Seiberg bound (3.40). By a limiting process, we may interpret<br />
e √ 2φ and φ e √ 2φ as <strong>th</strong>e two KPZ dressings of <strong>th</strong>e unit operator. We choose <strong>th</strong>e root of<br />
<strong>th</strong>e KPZ equation (2.19) so <strong>th</strong>at <strong>th</strong>e exponential grows at φ → −∞, <strong>th</strong>is being <strong>th</strong>e root<br />
we expect to correspond to a local operator. In <strong>th</strong>e present case we must choose <strong>th</strong>e root<br />
φ e √ 2φ , as anticipated in <strong>th</strong>e paragraph following (4.11), and in accord wi<strong>th</strong> <strong>th</strong>e argument<br />
given at <strong>th</strong>e end of sec. 11.6.<br />
Exercise. Spacetime interpretation of <strong>th</strong>e bounce factor<br />
Apply <strong>th</strong>e low energy <strong>th</strong>eorem, property (v), to <strong>th</strong>e two-point function to show<br />
<strong>th</strong>at <strong>th</strong>e “bounce factor” is <strong>th</strong>e one-point function of <strong>th</strong>e tachyon zeromode [141]:<br />
〈T 0 〉 = i log R(µ; V ) . (13.39)<br />
We regard property (vi) as intriguing: it strongly hints at a topological field <strong>th</strong>eory<br />
interpretation of c = 1.<br />
13.7. Unitarity of <strong>th</strong>e S-Matrix<br />
One immediate application of <strong>th</strong>e algori<strong>th</strong>m of sec. 13.5 is <strong>th</strong>at we can give a very<br />
simple and conceptual discussion of <strong>th</strong>e unitarity of <strong>th</strong>e S-matrix [132].<br />
3. Bosonization<br />
2. Free Fermion<br />
Scattering, S FF<br />
1. Fermionization<br />
Fig. 32: Composition of <strong>th</strong>ree maps: fermionization, free-fermion potential scattering,<br />
and rebosonization.<br />
165
The key observation is <strong>th</strong>at <strong>th</strong>e combinatorics of connecting lines according to <strong>th</strong>e<br />
diagrammatic rules of <strong>th</strong>e previous section is identical to <strong>th</strong>e combinatorics of bosonization.<br />
We can <strong>th</strong>en describe <strong>th</strong>e algori<strong>th</strong>m as a <strong>th</strong>ree-step process: fermionization, <strong>th</strong>en freefermion<br />
potential scattering, <strong>th</strong>en rebosonization, as shown in fig. 32.<br />
To be more precise, we describe in/out bosonic Fock spaces F in/out made from <strong>th</strong>e<br />
Heisenberg algebra of in/out massless bosons: α(η) in/out where η ∈ IR, [α(η), α(η ′ )] =<br />
η ′ δ(η + η ′ ) and <strong>th</strong>e in/out vacua are defined by α(η)|0〉 = 0 for η < 0. Now, <strong>th</strong>e Hilbert<br />
space of <strong>th</strong>e <strong>th</strong>eory may also be described in terms of <strong>th</strong>e Fermionic Fock space H FF<br />
defined by <strong>th</strong>e oscillators a(E) of sec. 11.3 (see (11.31, 11.32)). 60<br />
As is well-known, <strong>th</strong>e<br />
fermionization map<br />
ι b→f<br />
: α(η) →<br />
∫ ∞<br />
−∞<br />
dξ a(µ + ξ) a † (µ − (η − ξ)) (13.40)<br />
defines an isometry H 0 FF ∼ = F in/out , where <strong>th</strong>e superscript 0 indicates restriction to <strong>th</strong>e<br />
sector wi<strong>th</strong> <strong>th</strong>e difference #particles− #holes = 0. Thus, <strong>th</strong>e prescription of sec. 13.5 may<br />
be summarized by writing <strong>th</strong>e collective field S CF as a composition of <strong>th</strong>ree maps:<br />
S CF = ι f→b ◦ S FF ◦ ι b→f , (13.41)<br />
where ι b→f<br />
is <strong>th</strong>e fermionization map, ι f→b is <strong>th</strong>e inverse bosonization map, and S FF<br />
is <strong>th</strong>e free-fermion potential scattering S-matrix defined by (13.32). Al<strong>th</strong>ough standard<br />
bosonization is definitely not exact for <strong>th</strong>e nonrelativistic fermion systems, <strong>th</strong>e asymptotic<br />
bosonization is exact for fermions in a potential approaching V (λ) ∼ −λ 2 at infinity, and<br />
<strong>th</strong>is suffices for computation of <strong>th</strong>e S-matrix.<br />
From (13.41), we immediately deduce <strong>th</strong>at <strong>th</strong>e S-matrix is nonperturbatively unitary<br />
if and only if S FF is unitary. There are two immediate consequences of <strong>th</strong>is remark.<br />
1) In <strong>th</strong>eories wi<strong>th</strong> no infinite wall where <strong>th</strong>e reflection factors have absolute value<br />
smaller <strong>th</strong>an one, <strong>th</strong>e <strong>th</strong>eory will fail to be nonperturbatively unitary. This is not because<br />
bosons can tunnel, but because a single fermion in a particle–hole pair can tunnel, <strong>th</strong>us<br />
leaving a nontrivial soliton sector on ei<strong>th</strong>er side of <strong>th</strong>e world.<br />
Put ano<strong>th</strong>er way, if we<br />
insist <strong>th</strong>at <strong>th</strong>e (left and right) Hilbert space of <strong>th</strong>e <strong>th</strong>eory be H 0 FF<br />
(again <strong>th</strong>e sector wi<strong>th</strong><br />
#particles − #holes = 0) <strong>th</strong>en <strong>th</strong>e model will be non-unitary. If we allow nonzero #<br />
60 We are describing only one world so we drop <strong>th</strong>e ɛ label.<br />
166
particle − # hole number, i.e., nonzero soliton sectors, <strong>th</strong>en nonperturbative unitarity will<br />
be restored. A target space string interpretation of <strong>th</strong>e solitons would be quite interesting.<br />
2) By making small perturbations of <strong>th</strong>e matrix model potential fig. 21, we can produce<br />
infinitely many nonperturbatively unitary completions of <strong>th</strong>e string S-matrix [132]. In<br />
o<strong>th</strong>er words, <strong>th</strong>e requirement of nonperturbative unitarity is a very weak constraint on<br />
nonperturbative formulations of string <strong>th</strong>eory. Strangely, <strong>th</strong>e situation is opposite to <strong>th</strong>at of<br />
unitary c < 1 models coupled to gravity, where no satisfactory nonperturbative definitions<br />
exist. In ei<strong>th</strong>er case, we see <strong>th</strong>at matrix models have been somewhat disappointing as a<br />
source of nonperturbative physics.<br />
13.8. Generating functional for S-matrix elements<br />
The key formula (13.41) leads to a concise generating functional for all S-matrix<br />
elements [141]. A very intriguing aspect of <strong>th</strong>is formula is <strong>th</strong>at it involves <strong>th</strong>e asymptotic<br />
conformal field <strong>th</strong>eory in spacetime in a natural way.<br />
We have mentioned above <strong>th</strong>at <strong>th</strong>e collective field <strong>th</strong>eory, or equivalently <strong>th</strong>e spacetime<br />
tachyon <strong>th</strong>eory T (φ, t), is asymptotically a conformal field <strong>th</strong>eory. In fact <strong>th</strong>ere are two<br />
asymptotic conformal field <strong>th</strong>eories corresponding to <strong>th</strong>e two different null infinities I ±<br />
in <strong>th</strong>e past and <strong>th</strong>e future. According to (13.41), <strong>th</strong>e entire content of broken conformal<br />
invariance in <strong>th</strong>e interior is summarized by <strong>th</strong>e potential scattering of fermions:<br />
a(E) out = R(E)a(E) in = S −1 a(E) in S<br />
(∫ ∞<br />
S ≡ exp dE log ( R(E) )( a † (E) a(E) ) )<br />
in<br />
−∞<br />
.<br />
(13.42)<br />
As we have noted, unitarity of <strong>th</strong>e S-matrix is equivalent to <strong>th</strong>e identity R(E)R(E) ∗ = 1<br />
on <strong>th</strong>e reflection factors.<br />
We may use (13.42) to summarize <strong>th</strong>e entire S-matrix as follows. Define vertex operators<br />
wi<strong>th</strong> normalization<br />
Ṽ ± ω<br />
= Γ(−iω)<br />
Γ(iω) µ1+iω/2 V ± ω (13.43)<br />
relative to <strong>th</strong>e normalization of (5.22), and define <strong>th</strong>e generating functional<br />
µ 2 F [ t(ω), ¯t(ω) ] ≡<br />
〈〈<br />
e<br />
∫ ∞<br />
0<br />
dω t(ω)Ṽ + ω<br />
e<br />
∫ ∞<br />
0<br />
dω ¯t(ω)Ṽ − ω<br />
〉〉<br />
c<br />
, (13.44)<br />
where 〈〈. . .〉〉 indicates a sum over genus and integral over moduli space, ∑ h≥0 κ−χ ∫ M h,n<br />
(as in (11.61)), and <strong>th</strong>e subscript c indicates <strong>th</strong>e connected part. The genus expansion of<br />
167
(13.44) is given by F = F 0 + 1<br />
µ 2 F 1 + · · ·, and <strong>th</strong>us by KPZ scaling, combining (13.41) and<br />
(13.42) we have <strong>th</strong>e formula [141]:<br />
µ 2 F [ t(ω), ¯t(ω) ] ∫<br />
= 〈0|e µ ∞<br />
0<br />
∫<br />
dω t(ω)α(−ω) · S · e µ ∞<br />
dω ¯t(ω)α(ω)<br />
0 |0〉 c . (13.45)<br />
The expression (13.45) has a simple compactified Euclidean space analog. If we take<br />
<strong>th</strong>e Euclidean coordinate X to have finite radius β, <strong>th</strong>en from [142,143], we see <strong>th</strong>at <strong>th</strong>e<br />
only modification in (11.38) is <strong>th</strong>at bosonic momenta instead lie on a lattice q ∈ 1 β Z and<br />
<strong>th</strong>e fermions, now interpreted as being at finite temperature 1 β<br />
, have Matsubara frequencies<br />
1<br />
β ( Z + 1 2 ).<br />
τ = ∞<br />
τ = 0<br />
. (matrix model wall)<br />
Fig. 33: The Euclidean spacetime of <strong>th</strong>e matrix model in natural coordinates.<br />
Note <strong>th</strong>at <strong>th</strong>e asymptotic conformal field <strong>th</strong>eory on spacetime is concentrated in<br />
<strong>th</strong>e “ultraviolet” region at <strong>th</strong>e center of <strong>th</strong>e disk.<br />
The analytic continuation of <strong>th</strong>e asymptotically conformal collective field is given by<br />
<strong>th</strong>e standard c = 1 scalar field<br />
∂φ in/out (z) = ∑ n<br />
α in/out<br />
n z −n−1 , (13.46)<br />
where z = e −τ+iX , so <strong>th</strong>at <strong>th</strong>e Euclidean spacetime in <strong>th</strong>e z-plane looks as in fig. 33.<br />
In particular, <strong>th</strong>e bosonization becomes <strong>th</strong>e standard one wi<strong>th</strong> Weyl fermions in <strong>th</strong>e<br />
Neveu-Schwarz sector:<br />
ψ(z) = ∑ ψ m+<br />
1 z −m−1 ψ(z) = ∑ ψ<br />
2<br />
m+<br />
1 z−m−1<br />
2<br />
m∈ Z m∈ Z<br />
{ψ r , ψ s } = δ r+s,0 ∂φ = ψ(z) ¯ψ(z) .<br />
(13.47)<br />
The Euclidean analog of (13.42) is<br />
ψ in −(m+ 1 2 ) = R(µ + ip m) ψ out<br />
−(m+ 1 2 )<br />
¯ψ in −(m+ 1 2 ) = R(µ − ip m) ∗ ¯ψout −(m+ 1 2 ) . (13.48)<br />
168
where p m ≡ (m + 1 2 )/β.<br />
Thus defining Euclidean equivalents Ṽ q = µ 1−|q|/2<br />
Γ(−|q|) V q of (13.43), <strong>th</strong>e Euclidean<br />
analog of (13.44) becomes<br />
µ 2 F ≡<br />
Γ(|q|<br />
〈〈 ∑ ∑ 〉〉<br />
e<br />
t nṼn/β+ ¯t n Ṽ<br />
n≥1 n≥1 −n/β<br />
c<br />
= − 1 ∑ ∑ β 〈0|eiµ t nα<br />
n≥1 nS iµ ¯t n α<br />
e<br />
n≥1 −n|0〉c<br />
.<br />
(13.49)<br />
where |0〉 is <strong>th</strong>e standard SL(2,C) invariant vacuum and <strong>th</strong>e scattering operator is now<br />
given by<br />
( ∑<br />
S = : exp<br />
m∈ Z<br />
)<br />
log R pm ψ out<br />
−(m+ 1 2 )ψout m+ 1 : . (13.50)<br />
2<br />
The formulae (13.44, 13.49) are enormous simplifications over previous expressions for<br />
c = 1 amplitudes. They also clarify several ma<strong>th</strong>ematical properties of <strong>th</strong>e c = 1 S-matrix,<br />
in particular, its connections to integrable systems. 61<br />
13.9. Tachyon recursion relations<br />
From <strong>th</strong>e previous formulae we can obtain some interesting relations between tachyon<br />
amplitudes. We will restrict attention to genus zero wi<strong>th</strong> X uncompactified in <strong>th</strong>is section.<br />
We may interpret <strong>th</strong>e solution (13.22) or (13.25) to <strong>th</strong>e classical scattering problem in<br />
terms of operators in <strong>th</strong>e coherent state representation acting on <strong>th</strong>e generating functional<br />
Z of all amplitudes. This leads immediately to <strong>th</strong>e w 1+∞ flow equations for genus zero<br />
amplitudes. The equations are most elegantly stated at <strong>th</strong>e self-dual radius, or by working<br />
at infinite radius but restricting to integer momenta. In ei<strong>th</strong>er case we have:<br />
µ −2 ∂<br />
∂t(n) Z = ∮<br />
where we have <strong>th</strong>e coherent state representation:<br />
dw 1<br />
n + 1 : (¯∂φ(w)<br />
) n+1:<br />
Z , (13.51)<br />
∞∑<br />
¯∂φ = w −1 + k ¯t(k) w k−1 + 1 µ 2<br />
k=1<br />
∑<br />
∞ w −k−1<br />
k=1<br />
∂<br />
∂¯t(k) , (13.52)<br />
61 M. Green and T. Eguchi have pointed out some intriguing similarities between <strong>th</strong>e present<br />
discussion of <strong>th</strong>e c = 1 S-matrix and <strong>th</strong>e topological–antitopological fusion of [144]. Indeed, a<br />
picture of in- and out- disks joined along <strong>th</strong>e τ = 0 boundary of fig. 33 defines exactly <strong>th</strong>e same<br />
geometrical setup.<br />
169
and in (13.51) we only keep terms to leading order in <strong>th</strong>e 1/µ 2 expansion for any given<br />
correlator.<br />
In terms of explicit constraints on amplitudes, <strong>th</strong>ese flow equations lead to <strong>th</strong>e following<br />
relations between tachyon amplitudes [138]. The identities are most simply written in<br />
terms of<br />
T q =<br />
Γ(|q|)<br />
Γ(1 − |q|) V q . (13.53)<br />
Consider first <strong>th</strong>e insertion of a “special tachyon,” wi<strong>th</strong> q ∈ Z + . If we continue ω → i n<br />
wi<strong>th</strong> n ∈ Z + <strong>th</strong>en <strong>th</strong>e series (13.25) truncates after n + 1 terms. These terms have a<br />
“universal” effect in correlation functions. Specifically, an insertion of T n is given by<br />
∏<br />
m<br />
〈T q T qi 〉 =<br />
i=1<br />
·<br />
m∑<br />
k=2<br />
∑<br />
S 1 ,...S k−l<br />
k−l<br />
min(m − ,k−1)<br />
Γ(n) ∑<br />
Γ(2 + q − k)<br />
l=1<br />
∑<br />
|T |=l<br />
θ ( −q(T ) − q )<br />
∏<br />
(<br />
θ ( q(S j ) ) 〈 ∏ 〉 )<br />
q(S j ) T −q(Sj ) T qi ,<br />
j=1<br />
S j<br />
(13.54)<br />
where q > 0. 62<br />
The notation is as follows: Let S = {q 1 . . . q m }, and let S − denote <strong>th</strong>e<br />
subset of S of negative momenta. Denote m − = |S − |. The sum on T is over subsets of S − of<br />
order l. The subsequent sum is over distinct disjoint decompositions S 1 ∐. . .∐S k−l = S\T .<br />
q(T ) denotes <strong>th</strong>e sum of momenta in <strong>th</strong>e set T . The momenta q i are taken to be generic<br />
so <strong>th</strong>at <strong>th</strong>e step functions are unambiguous. This entails no loss of generality since <strong>th</strong>e<br />
amplitudes are continuous (but not differentiable) across kinematic boundaries [90].<br />
The first two examples of (13.54) are:<br />
n = 2 :<br />
+ ∑<br />
q i
Note <strong>th</strong>at in (13.56) <strong>th</strong>ere is a change in tachyon number by one in <strong>th</strong>e second line, and<br />
<strong>th</strong>e product of two correlators in <strong>th</strong>e <strong>th</strong>ird line. The pattern continues for higher n: <strong>th</strong>ere<br />
are terms wi<strong>th</strong> |T | = l = k − 1 removing l incoming tachyons, which are linear in <strong>th</strong>e<br />
correlators, and terms wi<strong>th</strong> a product of k −l correlators. Using <strong>th</strong>e representation (13.44)<br />
one can write <strong>th</strong>e analog of (13.51), which is valid to all orders of 1/µ perturbation <strong>th</strong>eory.<br />
Essentially, <strong>th</strong>e w 1+∞ algebra is replaced by <strong>th</strong>e W 1+∞ algebra [141].<br />
13.10. The many faces of c = 1<br />
In recent years many au<strong>th</strong>ors have tried to relate o<strong>th</strong>er interesting physical systems<br />
to <strong>th</strong>e c = 1 matrix model. These include:<br />
1) Two-dimensional black holes.<br />
It was originally proposed by Witten [32] <strong>th</strong>at <strong>th</strong>e SL(2, IR)/U(1) model of black holes<br />
would be, in some sense, equivalent to <strong>th</strong>e c = 1 model. This fascinating conjecture has<br />
inspired an enormous literature, but, despite all <strong>th</strong>e work, <strong>th</strong>e situation remains confused.<br />
Space does not allow a proper review here. A small sampling of <strong>th</strong>e vast literature includes<br />
<strong>th</strong>e following proposals:<br />
a) The models are equivalent after a non-local integral transform on <strong>th</strong>e field variables.<br />
In [145], a transform from <strong>th</strong>e Liouville equations of motion to <strong>th</strong>e SL(2, IR)/SO(2)<br />
equations of motion is proposed. See also [146]. In [147], <strong>th</strong>is transform was composed<br />
wi<strong>th</strong> <strong>th</strong>e τ-space to φ-space transform described in sec. 12.4. The results so far have<br />
been limited to transforms of <strong>th</strong>e tachyon equations of motion and, when treated<br />
nonperturbatively, have some difficulties wi<strong>th</strong> singularities at <strong>th</strong>e horizon and/or <strong>th</strong>e<br />
singularity. There have been many variants of <strong>th</strong>ese proposals in <strong>th</strong>e literature. See,<br />
for example, [148].<br />
b) The models have different operators turned on corresponding to non-normalizable<br />
modes. Consequently <strong>th</strong>e (Euclidean) black hole and <strong>th</strong>e c = 1 model are in different<br />
“superselection sectors” [140].<br />
c) The 2D black hole and <strong>th</strong>e c = 1 model are equivalent; <strong>th</strong>e c = 1 S-matrix includes<br />
black hole formation and evaporation as an intermediate process, but <strong>th</strong>e black hole<br />
physics is difficult to recognize because of <strong>th</strong>e exact solubility of <strong>th</strong>e model. Specifically<br />
<strong>th</strong>e w ∞ symmetry of <strong>th</strong>e <strong>th</strong>eory makes black holes difficult to recognize in <strong>th</strong>e c = 1<br />
S-matrix. This has been advocated in [149].<br />
d) The 2D black hole and <strong>th</strong>e c = 1 model are related, but are different cosets of SL(2, IR)<br />
current algebra [150].<br />
171
e) The c = 1 model is equivalent to a twisted N = 2 supersymmetric SL(2, IR)/U(1)<br />
model [151].<br />
f) The models are not equivalent and we will learn no<strong>th</strong>ing about black holes from <strong>th</strong>e<br />
c = 1 model. Several physical arguments may be advanced in favor of <strong>th</strong>is viewpoint.<br />
2) Topological Field Theory.<br />
In [152,151], a relation between <strong>th</strong>e c = 1 model wi<strong>th</strong> X compactified on a selfdual<br />
radius and a certain topological field <strong>th</strong>eory has been proposed. This is potentially<br />
significant because <strong>th</strong>e c = 1 model has, as we have seen, local physics and a nontrivial<br />
S-matrix. This topological field <strong>th</strong>eory is a twisted SL(2, IR)/U(1) Kazama–Suzuki model<br />
at level k = −3 coupled to topological gravity. Among o<strong>th</strong>er <strong>th</strong>ings, <strong>th</strong>is interpretation<br />
would equate <strong>th</strong>e tachyon S-matrix for T ki wi<strong>th</strong> <strong>th</strong>e Euler character of <strong>th</strong>e vector bundle<br />
V → M g,n whose fiber at a Riemann surface Σ is<br />
V| Σ = H 0( Σ; K 2 ⊗ n i=1 O(z i ) 1−k i ) , (13.57)<br />
where K is <strong>th</strong>e canonical bundle of Σ. This conjecture has been checked for <strong>th</strong>e free energy<br />
[153] and for <strong>th</strong>e four-point function [152,151]. Checking <strong>th</strong>is in o<strong>th</strong>er cases appears to be<br />
quite nontrivial.<br />
3) 2D QCD.<br />
Very recently [154], a connection wi<strong>th</strong> two-dimensional QCD has been advocated.<br />
14. Vertex Operator Calculations and Continuum Me<strong>th</strong>ods<br />
Matrix model reasoning is extremely indirect. It is <strong>th</strong>erefore important to verify matrix<br />
model results directly via vertex operator calculations. Aside from logical consistency, it<br />
is useful to see how matrix model results are explained by standard string-<strong>th</strong>eoretic ideas<br />
(for example in terms of operator product expansions, etc.). Moreover, vertex operator<br />
calculations are <strong>th</strong>e only known approach to <strong>th</strong>e supersymmetric models.<br />
14.1. Review of <strong>th</strong>e Shapiro-Virasoro Amplitude<br />
Many of <strong>th</strong>e important ideas of string perturbation <strong>th</strong>eory are nicely summarized in<br />
one of its oldest results: <strong>th</strong>e Shapiro-Virasoro amplitude for 4-point scattering of string<br />
tachyons. Al<strong>th</strong>ough <strong>th</strong>is material is completely standard, it is good to review it before<br />
plunging into <strong>th</strong>e bizarre world of 2D string <strong>th</strong>eory.<br />
172
The relevant density on moduli space for <strong>th</strong>e scattering of four on-shell closed string<br />
tachyons is<br />
where z = z 13 z 24 /z 12 z 34 and 1 2 p2 i = 1.<br />
Ω(V p1 , V p2 , V p3 , V p4 ) = dz ∧ d¯z |z| 2p 1p 3<br />
|z − 1| 2p 2p 3<br />
, (14.1)<br />
In <strong>th</strong>is case, <strong>th</strong>e integral over moduli space M 0,4 can be done using <strong>th</strong>e formula<br />
∫<br />
C d2 z |z| 2a |z − 1| 2b = π ∆(1 + a) ∆(1 + b) ∆(−a − b − 1) , (14.2)<br />
and hence<br />
A 0,4 (V p1 , V p2 , V p3 , V p4 ) = π Γ(1 + p 1 · p 3 )<br />
Γ(−p 1 · p 3 )<br />
Γ(1 + p 2 · p 3 )<br />
Γ(−p 2 · p 3 )<br />
Γ(1 + p 3 · p 4 )<br />
Γ(−p 3 · p 4 )<br />
. (14.3)<br />
The left hand side of (14.2) converges when <strong>th</strong>e arguments of all <strong>th</strong>e Γ-functions are<br />
positive. Amplitudes in o<strong>th</strong>er kinematic regimes, obtained by analytic continuation in <strong>th</strong>e<br />
external momenta, have an infinite set of poles at <strong>th</strong>e values:<br />
1<br />
2 (p 1 + p 3 ) 2 = 1, 0, −1, . . .<br />
1<br />
2 (p 2 + p 3 ) 2 = 1, 0, −1, . . .<br />
1<br />
2 (p 3 + p 4 ) 2 = 1, 0, −1, . . .<br />
(14.4)<br />
These poles have bo<strong>th</strong> spacetime and worldsheet interpretations:<br />
Spacetime interpretation. The poles signal <strong>th</strong>e existence of new particles in <strong>th</strong>e<br />
<strong>th</strong>eory. At <strong>th</strong>e above values of t, u, s, <strong>th</strong>ere is an on-shell particle in <strong>th</strong>e respective channel.<br />
This is <strong>th</strong>e first signal of <strong>th</strong>e infinite tower of string states of arbitrarily large target space<br />
spin.<br />
Worldsheet interpretation. The poles arise from <strong>th</strong>e terms in <strong>th</strong>e operator product<br />
expansion. The poles in <strong>th</strong>e t channel, for example, are best understood by considering<br />
<strong>th</strong>e operator product expansion of operators V p1 wi<strong>th</strong> V p3 . Then we have:<br />
c¯c e ip3·X (z, ¯z) c¯c e ip1·X (0) ∼ ∑ Φ s (0) 〈 Φ s (∞) c¯c e ip3·X 〉<br />
(z, ¯z) c¯c|p 1 (14.5)<br />
where Φ s has ghost number 4. Thus we can interpret <strong>th</strong>e expansion of Ω in powers of z as<br />
a statement about <strong>th</strong>e factorization properties of <strong>th</strong>e correlator:<br />
〈V V V V 〉 → ∑ s<br />
〈V V Φ s 〉〈Φ s V V 〉 . (14.6)<br />
Exercise. Gaussian OPE<br />
Express <strong>th</strong>e operators in (14.5) in terms of Schur polynomials of ∂ k c, ∂ k X.<br />
173
The expansion is only convergent for |z| < 1, so we must separate <strong>th</strong>e integral over<br />
moduli space into two parts: |z| < ρ and |z| > ρ, where ρ < 1. In <strong>th</strong>e first integral we may<br />
use <strong>th</strong>e OPE and integrate term by term to get:<br />
A 0,4 = 2π ∑ n≥1<br />
ρ 2n+2+2p 1·p 3<br />
∫<br />
〈V 1 V 3 Φ s 〉 〈Φ s V 2 V 4 〉 + Ω . (14.7)<br />
2n + 2 + 2p 1 · p 3 |z|≥ρ<br />
Thus we see <strong>th</strong>at <strong>th</strong>e poles in <strong>th</strong>e t-channel come from <strong>th</strong>e contribution of operators in<br />
<strong>th</strong>e V 1 , V 3 OPE <strong>th</strong>at satisfy 1 2 (p 1 + p 3 ) 2 = 1, 0, −1, . . . . Fur<strong>th</strong>ermore, <strong>th</strong>e Fock space for<br />
<strong>th</strong>e Gaussian conformal field <strong>th</strong>eory (tensored wi<strong>th</strong> ghosts) may be decomposed into states<br />
which are BRST cohomology representatives, unphysical states, and trivial states, in a<br />
manner invariant under <strong>th</strong>e conjugation Φ s → Φ s . That is, <strong>th</strong>e factorization behaves<br />
schematically like:<br />
∑<br />
|phys〉〈phys| +<br />
∑<br />
|unphys〉〈trivial| +<br />
∑<br />
|trivial〉〈unphys| . (14.8)<br />
Since <strong>th</strong>e V i are BRST invariant, only <strong>th</strong>e BRST invariant operators can contribute to <strong>th</strong>e<br />
sum in (14.7). Thus we finally conclude <strong>th</strong>at <strong>th</strong>e infinite sum of poles in <strong>th</strong>e scattering<br />
amplitude stem from <strong>th</strong>e BRST cohomology classes in <strong>th</strong>e operator product expansion.<br />
Similarly, <strong>th</strong>e poles in <strong>th</strong>e s, u channels arise from <strong>th</strong>e o<strong>th</strong>er two boundaries of moduli<br />
space.<br />
Note, in particular, <strong>th</strong>at it would be inconsistent wi<strong>th</strong> unitarity to truncate <strong>th</strong>e string<br />
spectrum to lowest lying states.<br />
Exercise. BRST puzzle<br />
When p 1 +p 3 is not on-shell, every term in (14.5) is a BRST commutator so V p1 V p3<br />
is BRST trivial. Explain why <strong>th</strong>is does not imply <strong>th</strong>at <strong>th</strong>e four-point function is zero.<br />
14.2. Resonant Amplitudes and <strong>th</strong>e “Bulk S-Matrix”<br />
Unfortunately <strong>th</strong>e Liouville <strong>th</strong>eory is incalculable: we cannot even write <strong>th</strong>e density<br />
on moduli space in general, much less integrate it. We can of course calculate in <strong>th</strong>e free<br />
<strong>th</strong>eory at µ = 0. This has led to a large literature on <strong>th</strong>e “Bulk scattering matrix” to<br />
be contrasted wi<strong>th</strong> <strong>th</strong>e “Wall scattering matrix” or W -matrix discussed in <strong>th</strong>e previous<br />
chapter.<br />
Bulk scattering is scattering in <strong>th</strong>e µ = 0 <strong>th</strong>eory wi<strong>th</strong> <strong>th</strong>e condition s = 0, where s is<br />
<strong>th</strong>e KPZ exponent (3.44), is imposed as a kinematical condition. This makes best physical<br />
174
sense if we rotate φ → it (as we may when µ = 0), and regard X as a spatial variable. We<br />
are <strong>th</strong>erefore discussing <strong>th</strong>eory B of sec. 5.4. As explained <strong>th</strong>ere, <strong>th</strong>e vertex operators are<br />
given by (5.24) and we have energy and momentum conservation laws for <strong>th</strong>e amplitude<br />
〈 ∏ T + ∏<br />
k i<br />
T<br />
−<br />
pi<br />
〉 given by<br />
∑<br />
ki + ∑ p i = 0<br />
s = 2 − ∑ (1 + 1 2 k i) − ∑ (1 − 1 2 p i) = 0 .<br />
(14.9)<br />
Standard vertex operator calculations now give<br />
〈∏ ∏ 〉 ∫<br />
T<br />
+<br />
k i<br />
T<br />
−<br />
pi<br />
=<br />
N<br />
∏<br />
i=4<br />
∏<br />
d 2 z i |z ij | −2s ij<br />
, (14.10)<br />
where we take <strong>th</strong>e <strong>th</strong>ree points at 0, 1, ∞ as usual, s ij = β i β j − 1 2 k ik j , β = √ 2 + k/ √ 2<br />
for T + k , and β = √ 2 − p/ √ 2 for T − p .<br />
i 0, and p i + p j > 1, <strong>th</strong>en <strong>th</strong>e integral (14.10) is convergent and well-defined, and<br />
results in<br />
〈T + k<br />
N∏<br />
Tp − i<br />
〉 =<br />
i=1<br />
N∏ π N−2<br />
∆(m i )<br />
(N − 2)! . (14.11)<br />
i=1<br />
This has been shown in [49,48] by analytic arguments and in [155] by an elegant algebraic<br />
technique. Note in particular <strong>th</strong>at:<br />
1) p i , k i ∈ IR. We have put µ = 0 so <strong>th</strong>ere is no longer any rationale to impose <strong>th</strong>e<br />
Seiberg bound (3.40).<br />
2) As in 26 dimensions, we can continue to o<strong>th</strong>er momenta for which <strong>th</strong>e integral representation<br />
does not converge. Then <strong>th</strong>ere are poles, but in <strong>th</strong>is case <strong>th</strong>ey occur for<br />
p i = 1, 2, . . . . These are known as <strong>th</strong>e “leg poles.”<br />
3) We already see a remarkable difference between D = 2 and D > 2 strings since in<br />
general <strong>th</strong>ere is no simple closed formula for (14.10) for N > 4.<br />
Let us now consider o<strong>th</strong>er combinations of chiralities. We find a new surprise. Because<br />
of kinematic “coincidences”, one cannot define <strong>th</strong>e integrals, even by analytic continuation,<br />
since one is always sitting on top of a Γ-function pole or zero. Indeed it has been argued<br />
in [48,49] <strong>th</strong>at <strong>th</strong>ese amplitudes are zero, at least for generic external momenta.<br />
175
Example. Let us consider <strong>th</strong>e most general four-point function. In string <strong>th</strong>eory,<br />
we apply <strong>th</strong>e fundamental identity (14.2). Usually we use <strong>th</strong>is expression in conjunction<br />
wi<strong>th</strong> analytic continuation in <strong>th</strong>e momenta to define <strong>th</strong>e scattering matrix in all kinematic<br />
regimes. Let us apply <strong>th</strong>is to <strong>th</strong>e amplitude 〈 T + k 1<br />
T + 〉<br />
k 2<br />
Tp − 1<br />
Tp − 2 . The kinematic constraints<br />
(14.9) force p 1 + p 2 = −(k 1 + k 2 ) = 2. Thus <strong>th</strong>e <strong>th</strong>ird factor of (14.2) becomes ∆(2) = 0,<br />
while <strong>th</strong>e first two factors remain nonsingular for generic momenta.<br />
t<br />
Fig. 34: Several nonvanishing bulk processes. One can also take <strong>th</strong>e parity conjugate<br />
of each <strong>th</strong>e above processes.<br />
x<br />
The mixed chirality amplitudes are put to zero by some au<strong>th</strong>ors [49,48,156] and argued<br />
(on <strong>th</strong>e basis of unitarity equations) to be proportional to δ-functions in momenta by o<strong>th</strong>ers<br />
[157,158]. Taken toge<strong>th</strong>er <strong>th</strong>ese amplitudes define <strong>th</strong>e “Bulk S-matrix,” or B-matrix for<br />
short. Bulk scattering is quite peculiar, some examples of processes are drawn in fig. 34.<br />
The existence of particle creation/annihilation in some processes is not surprising given <strong>th</strong>e<br />
time-variation of <strong>th</strong>e background, and in particular of <strong>th</strong>e coupling constant. The existence<br />
of δ-function singularities in <strong>th</strong>e o<strong>th</strong>er S-matrix elements suggests <strong>th</strong>at <strong>th</strong>e spacetime<br />
background wi<strong>th</strong> µ = 0 is highly unstable.<br />
<br />
Factorization on discrete states<br />
It should be emphasized <strong>th</strong>at <strong>th</strong>e simplicity of <strong>th</strong>e formula for N-tachyon scattering<br />
amplitudes (14.11) is extremely remarkable. The analogous singularity structure for <strong>th</strong>e 26-<br />
dimensional string would be vastly more complicated. Physically <strong>th</strong>is arises because in 2D<br />
<strong>th</strong>ere is only one propagating degree of freedom. Never<strong>th</strong>eless, since <strong>th</strong>e amplitudes (14.11)<br />
were calculated using free-field operator products, <strong>th</strong>e standard discussion of sec. 14.1<br />
applies here as well, wi<strong>th</strong> some small modifications implied by <strong>th</strong>e kinematic laws (14.9).<br />
This has been carried out in detail in [157,156]. Using <strong>th</strong>e free field OPE as in sec. 14.1,<br />
176
in [156] it is shown <strong>th</strong>at <strong>th</strong>e “leg poles” in (14.11) may be interpreted in terms of on-shell<br />
intermediate discrete states. The vanishing of <strong>th</strong>e mixed chirality amplitudes is important<br />
in <strong>th</strong>eir discussion. While <strong>th</strong>is makes sense from <strong>th</strong>e worldsheet point of view, <strong>th</strong>e existence<br />
in spacetime of (normalizable!) modes which are only physical at discrete momenta is quite<br />
peculiar and has not been adequately interpreted.<br />
14.3. Wall vs. Bulk Scattering<br />
We finally discuss <strong>th</strong>e relation of <strong>th</strong>e B-matrix to <strong>th</strong>e W -matrix, <strong>th</strong>at is, we compare<br />
amplitudes at µ = 0 wi<strong>th</strong> amplitudes at µ > 0. Since we cannot expand in µ we have no<br />
right to expect a simple relation. Moreover, <strong>th</strong>e perturbative W -matrix does not have a<br />
good µ → 0 limit. Never<strong>th</strong>eless, <strong>th</strong>ere is an interesting series of conjectures explored in<br />
[47,48,49] on <strong>th</strong>e relation between <strong>th</strong>ese S-matrices. We describe <strong>th</strong>ese here.<br />
To compare, we must continue back to Euclidean space and impose <strong>th</strong>e Seiberg bound<br />
(3.40). Thus we only consider processes wi<strong>th</strong> T + k , k > 0, and T − k<br />
, k < 0. The chirality<br />
rule <strong>th</strong>us becomes <strong>th</strong>e rule <strong>th</strong>at amplitudes are generically zero unless all but one of <strong>th</strong>e<br />
momenta k i have <strong>th</strong>e same sign. Wi<strong>th</strong>out loss of generality, we take k 1 , . . . k N < 0, hence<br />
s = 0 implies <strong>th</strong>at k N+1 = N − 1. We now try to relate <strong>th</strong>e µ = 0 and µ > 0 <strong>th</strong>eories by<br />
integrating over <strong>th</strong>e Liouville zero mode as in sec. 3.9, splitting φ = φ 0 + ˜φ. This gives:<br />
〈V · · · V 〉 = µ s Γ(−s) 〈V · · · V 〉 ˜φ,µ=0<br />
. (14.12)<br />
Since s = 0, <strong>th</strong>e RHS is ill-defined, but <strong>th</strong>e pole of <strong>th</strong>e Γ-function has a nice physical<br />
interpretation. Returning to fixed area correlators we see <strong>th</strong>at it arises from an ultraviolet<br />
A → 0 divergence. That is, in spacetime terms, a divergence from an integration over <strong>th</strong>e<br />
volume of <strong>th</strong>e φ-coordinate. We may <strong>th</strong>erefore regulate <strong>th</strong>e <strong>th</strong>eory and consider log µ <strong>th</strong>e<br />
regularized volume of <strong>th</strong>e world,<br />
µ s Γ(−s)| s=0 →<br />
∫ ∞<br />
ɛ<br />
dA<br />
A e−µA → log(µ ɛ) . (14.13)<br />
To extract <strong>th</strong>e residue of <strong>th</strong>e s = 0 pole, we divide by <strong>th</strong>e volume of φ-space. As mentioned<br />
in sec. 3.9, <strong>th</strong>e Liouville interaction is effectively zero in “most” of φ-space so we should be<br />
able to treat φ as a free field in <strong>th</strong>is regime and calculate <strong>th</strong>e residue of <strong>th</strong>e s = 0 pole wi<strong>th</strong><br />
177
free-field techniques. But <strong>th</strong>is calculation just leads to <strong>th</strong>e B-matrix. Using <strong>th</strong>e free-field<br />
result (14.11) gives (k i < 0, i = 1, N):<br />
〈V k1 · · · V kN+1 〉 = µ s Γ(−s)| s=0 〈V · · · V 〉 ˜φ,µ=0<br />
∏<br />
N<br />
= µ s π N−2<br />
Γ(−s)| s=0 ∆(m i )<br />
(N − 2)!<br />
i=1<br />
∏<br />
N<br />
= π N−2 Γ(1 − (N − 1) + ɛ)<br />
(14.14)<br />
∆(m i ) (N − 2)!<br />
Γ(N − 1)<br />
i=1<br />
N+1<br />
∏<br />
= ∆(m i ) ( ∂ ) N−2µ s+N−2 | s=0<br />
∂µ<br />
i=1<br />
where now m i = 1 − |k i | and |k N+1 | = N − 1 and we regulate by taking s = ɛ → 0. Di<br />
Francesco and Kutasov [48] have generalized <strong>th</strong>is result to positive integer values for s by<br />
carefully taking limits k i → 0, and find<br />
〈V · · · V 〉 ∝<br />
N+1<br />
∏<br />
i=1<br />
∆(m i ) ( ∂ ) N−2µ s+N−2 . (14.15)<br />
∂µ<br />
The equation (14.15) has an obvious “continuation” to s /∈ Z + wi<strong>th</strong> s = 1 − N + |k N+1 |,<br />
where <strong>th</strong>e RHS becomes well-defined and finite. Remarkably, comparison wi<strong>th</strong> <strong>th</strong>e matrix<br />
model result (13.27) shows <strong>th</strong>at we obtain an identical amplitude (13.27, 13.28) differing<br />
only by “wavefunction renormalization factors” f(q) (13.12), and <strong>th</strong>e continuation |p| →<br />
−iω appropriate to <strong>th</strong>e W -matrix.<br />
In order for <strong>th</strong>is story to be consistent, we should<br />
understand from <strong>th</strong>e W -matrix why <strong>th</strong>e mixed chirality amplitudes vanish. The reason is<br />
<strong>th</strong>at in <strong>th</strong>ese kinematic regimes, <strong>th</strong>e KPZ exponent is typically fractional. For example,<br />
for 2 → n scattering wi<strong>th</strong> p 1 + p 2 = k 1 + · · · k n we have s = 2 − n + p 1 + p 2 , even when leg<br />
factors blow up µ s is fractional, <strong>th</strong>ere is no log µ dependence and hence no “bulk” piece<br />
proportional to <strong>th</strong>e volume of <strong>th</strong>e world.<br />
Di Francesco and Kutasov [48] have argued <strong>th</strong>at is is never<strong>th</strong>eless possible to use <strong>th</strong>e<br />
data of <strong>th</strong>e B-matrix to obtain <strong>th</strong>e remaining W -matrix elements at µ ≠ 0. The crucial<br />
point is <strong>th</strong>at one must use spacetime reasoning [48]. First, note <strong>th</strong>at (14.15) depends (up<br />
to wavefunction factors) on <strong>th</strong>e momenta only <strong>th</strong>rough a polynomial. Therefore, we can<br />
construct a local spacetime field <strong>th</strong>eory of <strong>th</strong>e tachyon field analogous to <strong>th</strong>e macroscopic<br />
loop field <strong>th</strong>eory of sec. 11.6. 63 In [48], it is shown <strong>th</strong>at (14.15) uniquely fixes all <strong>th</strong>e interactions<br />
in <strong>th</strong>e Lagrangian and <strong>th</strong>at one can proceed to calculate <strong>th</strong>e amplitudes in o<strong>th</strong>er<br />
63 “Local” means we have an finite set of local finite derivative interactions for each interaction<br />
involving n fields. In total, <strong>th</strong>e Lagrangian involves an infinite set of interactions.<br />
178
kinematic regimes at µ ≠ 0 using <strong>th</strong>is field <strong>th</strong>eory. In all cases where <strong>th</strong>e procedure has<br />
been checked (five- and six-point functions), <strong>th</strong>e amplitudes obtained from <strong>th</strong>is procedure<br />
agree wi<strong>th</strong> <strong>th</strong>e matrix model amplitudes. Thus, <strong>th</strong>e B-matrix element (14.11) at µ = 0<br />
completely determines <strong>th</strong>e µ ≠ 0 W -matrix. These arguments have not been extended to<br />
higher genus and <strong>th</strong>e equality of S-matrices <strong>th</strong>us remains conjectural (al<strong>th</strong>ough physically<br />
plausible).<br />
Remarks:<br />
1) As we discussed in <strong>th</strong>e previous section <strong>th</strong>e leg factors of <strong>th</strong>e B-matrix give poles<br />
corresponding to on-shell intermediate discrete tachyons.<br />
On <strong>th</strong>e o<strong>th</strong>er hand, for <strong>th</strong>e<br />
Euclidean W -matrix <strong>th</strong>e analogous factors are of <strong>th</strong>e form (13.12), and have poles at<br />
|q| ∈ Z + . In <strong>th</strong>e physical regime of <strong>th</strong>e W -matrix, <strong>th</strong>ese correspond to phase factors<br />
Γ(iE)<br />
Γ(−iE)<br />
which do not have poles in <strong>th</strong>e physical regime. This makes perfectly good sense.<br />
we have discussed, <strong>th</strong>e poles of <strong>th</strong>e leg-factors correspond to non-normalizable states wi<strong>th</strong><br />
imaginary momentum, <strong>th</strong>ey cannot appear in intermediate channels for physical scattering.<br />
Thus, we see <strong>th</strong>at at µ > 0 <strong>th</strong>e different nature of <strong>th</strong>e Liouville OPE essentially changes<br />
<strong>th</strong>e physics and alters <strong>th</strong>e standard discussion of sec. 14.1. In particular, <strong>th</strong>e W -matrix<br />
has <strong>th</strong>e peculiar property, unique among string <strong>th</strong>eories, <strong>th</strong>at <strong>th</strong>e tachyon S-matrix is a<br />
unitary scattering matrix in <strong>th</strong>e absence of all o<strong>th</strong>er string states.<br />
2) These calculations have been extended to <strong>th</strong>e open string in [159], in which case<br />
<strong>th</strong>e amplitudes have a pole structure much more complicated <strong>th</strong>an <strong>th</strong>e closed string bulk<br />
amplitudes above. Explaining <strong>th</strong>ese amplitudes remains an important challenge for <strong>th</strong>e<br />
matrix model approach.<br />
14.4. Algebraic Structures of <strong>th</strong>e 2D String: Chiral Cohomology<br />
We have seen <strong>th</strong>at, at least at µ = 0, <strong>th</strong>e 2D string has a rich spectrum of cohomology.<br />
As mentioned in sec. 5.5, <strong>th</strong>is may be taken as an indication <strong>th</strong>at <strong>th</strong>e D = 2 string<br />
background is a much more symmetric background for string <strong>th</strong>eory.<br />
As<br />
By contrast, <strong>th</strong>e<br />
Minkowski background of standard critical strings would seem to be a very asymmetric<br />
background, not at all a good place to look for underlying symmetries and principles of<br />
string <strong>th</strong>eory. Wi<strong>th</strong> <strong>th</strong>ese motivations in mind, several groups have intensively investigated<br />
<strong>th</strong>e algebraic structures defined by <strong>th</strong>e BRST cohomology of D = 2 string <strong>th</strong>eory<br />
[66,69,67,131,155,160,161].<br />
179
Quite generally, <strong>th</strong>e operator product algebra of <strong>th</strong>e chiral operators in a conformal<br />
field <strong>th</strong>eory defines an example of a ma<strong>th</strong>ematical object known as a vertex operator algebra<br />
[162]. Indeed much of <strong>th</strong>e work on conformal field <strong>th</strong>eory (especially RCFT) has been an<br />
investigation of <strong>th</strong>ese algebraic structures [163,164].<br />
In string <strong>th</strong>eory, where <strong>th</strong>ere is a<br />
BRST operator Q, additional structures arise. This is nicely illustrated in <strong>th</strong>e example of<br />
<strong>th</strong>e operator product algebra of <strong>th</strong>e 2D string.<br />
First let us consider <strong>th</strong>e absolute chiral cohomology at <strong>th</strong>e self-dual radius. As we<br />
have described in chapt. 5, <strong>th</strong>is is spanned by operators at ghost numbers G = 0, 1, 2 for<br />
<strong>th</strong>e (+)-states:<br />
G = 0<br />
O j,,m<br />
G = 1 aO j,,m Y +<br />
j,m<br />
(14.16)<br />
G = 2 aY +<br />
j,m ,<br />
toge<strong>th</strong>er wi<strong>th</strong> <strong>th</strong>e (−)-states at ghost numbers 3,2,1, which are dual via <strong>th</strong>e tildeconjugation.<br />
The operator product of <strong>th</strong>e ground ring operators O ∈ G can be restricted to <strong>th</strong>e<br />
BRST cohomology:<br />
O 1 (x) O 2 (y) ∼ O 3 (y) mod{Q, ∗} , (14.17)<br />
since <strong>th</strong>e operator product is nonsingular and and ghost number is additive. Thus, <strong>th</strong>e<br />
ground ring operators form a ring. One can show <strong>th</strong>at <strong>th</strong>e BRST reduction of <strong>th</strong>e operator<br />
product algebra is [67]:<br />
O j1 ,m 1<br />
(x) O j2 ,m 2<br />
(y) = O j1 +j 2 ,m 1 +m 2<br />
(y) mod{Q, ∗} . (14.18)<br />
This result almost follows simply from consideration of ghost and momentum quantum<br />
numbers. The fact <strong>th</strong>at <strong>th</strong>e structure constant is unity requires more detailed analysis<br />
[67]. Wi<strong>th</strong> <strong>th</strong>e identifications x ≡ O 1/2,1/2 , y ≡ O 1/2,−1/2 , we identify <strong>th</strong>e chiral ground<br />
ring wi<strong>th</strong> <strong>th</strong>e algebra of polynomials in x and y, denoted C[x, y].<br />
Of course, <strong>th</strong>e existence of a ring in <strong>th</strong>e OPA of <strong>th</strong>e BRST cohomology does not require<br />
us to restrict to ghost number G = 0. To describe <strong>th</strong>e full operator product algebra, we<br />
first introduced some geometry.<br />
Geometry of <strong>th</strong>e BRST operator product algebra<br />
An old observation [165] is <strong>th</strong>at <strong>th</strong>e BRST cohomology of string <strong>th</strong>eory resembles<br />
cohomological structures of manifolds. The operator product algebra of <strong>th</strong>e 26-dimensional<br />
string has proven too complicated to pursue <strong>th</strong>is line of <strong>th</strong>ought very far, but <strong>th</strong>e 2D string<br />
180
example has provided some very interesting realizations of <strong>th</strong>at idea [67,66]. From (14.18),<br />
we see <strong>th</strong>at <strong>th</strong>e ground ring is <strong>th</strong>e ring of polynomial functions on <strong>th</strong>e x, y plane. Witten<br />
and Zwiebach [66] show <strong>th</strong>at <strong>th</strong>e remaining cohomology can be identified wi<strong>th</strong> polynomial<br />
vectors and bi-vectors via <strong>th</strong>e introduction of an area-form ω = dx ∧ dy. Indeed we have<br />
<strong>th</strong>e correspondence:<br />
O j,m ↔ f j,m ≡ x j+m y j−m<br />
Y +<br />
j,m<br />
↔ V j,m ≡ ∂f j,m<br />
∂y<br />
∂<br />
∂x − ∂f j,m<br />
∂x<br />
∂<br />
∂y<br />
aO j,m ↔ X j,m = x j+m y j−m (<br />
x ∂<br />
∂x + y ∂<br />
∂y<br />
aY +<br />
j,m<br />
↔ f j,m (x, y) ∂<br />
∂x ∧ ∂<br />
)<br />
∂y . (14.19)<br />
In <strong>th</strong>e <strong>th</strong>ird line, <strong>th</strong>e vector field X is an area non-preserving diffeomorphism and satisfies<br />
L Xj,m ω = f j,m ω, or, ∂ i X i = f j,m . Wi<strong>th</strong> <strong>th</strong>ese identifications, we can elegantly summarize<br />
<strong>th</strong>e operator product algebra as <strong>th</strong>e ring structure on Λ ∗ T = ⊕ 2 i=0T , where T is <strong>th</strong>e<br />
polynomial tangent bundle on <strong>th</strong>e x, y plane [66,166].<br />
Since we are working at µ = 0 we must also consider <strong>th</strong>e (−) states.<br />
These may<br />
be nicely incorporated into <strong>th</strong>e <strong>th</strong>eory. The full structure has been elucidated by Lian<br />
and Zuckerman [166] in terms of an algebraic structure <strong>th</strong>ey call a Gerstenhaber algebra.<br />
Related algebraic structures have also figured prominently in several recent works on string<br />
field <strong>th</strong>eory and topological string <strong>th</strong>eory. See [167,168].<br />
Exercise. Explicit Ring Structure<br />
Show <strong>th</strong>at <strong>th</strong>e ring structure in <strong>th</strong>e natural basis is<br />
O j1 ,m 1<br />
· O j2 ,m 2<br />
= O j1 +j 2 ,m 1 +m 2<br />
O j1 ,m 1<br />
· Y +<br />
j 2 ,m 2<br />
= α Y +<br />
j 1 +j 2 ,m 1 +m 2<br />
+ βa O j1 +j 2 +1,m 1 +m 2<br />
Y +<br />
j 1 ,m 1 · Y +<br />
j 2 ,m 2<br />
= a Y +<br />
j 1 +j 2 −1,m 1 +m 2<br />
.<br />
(14.20)<br />
Remarks:<br />
1) There is a dual interpretation replacing polyvectors by differential forms. In <strong>th</strong>is<br />
formulation, b 0 essentially plays <strong>th</strong>e role of an exterior derivative. See [66].<br />
2) It is natural to ask for <strong>th</strong>e analog of <strong>th</strong>e ground ring at c < 1. This has been<br />
discussed in [131,169]. The operator product ring is C((w)) ⊗ C[x, y] wi<strong>th</strong> relations<br />
181
x p−1 ∼ y q−1 ∼ 1 (where C((w)) designates <strong>th</strong>e ring of Laurent series wi<strong>th</strong> finite order<br />
poles).<br />
Lie Algebra of Derivations<br />
Let us investigate more closely some consequences of <strong>th</strong>e above assertions. The operator<br />
product algebra of <strong>th</strong>e ghost number G = 1 operators is <strong>th</strong>e Lie algebra of vector fields.<br />
When restricted to <strong>th</strong>e area-preserving vector fields Y +<br />
j,m<br />
, <strong>th</strong>is may be identified wi<strong>th</strong> <strong>th</strong>e<br />
∨w Lie algebra as follows. The operator Y has <strong>th</strong>e structure Y j,m = cW j,m ,where W is<br />
a ghost-free operator of dimension one, so applying <strong>th</strong>e descent equations to <strong>th</strong>e BRST<br />
invariant zero-form Ω (0) = Y j,n gives a dimension one operator Ω (1) = W . The associated<br />
Lie algebra can be deduced by direct calculations of <strong>th</strong>e operator products to be [160]<br />
W j1 ,m 1<br />
(z)W j2 ,m 2<br />
(0) ∼ 2(j 1m 2 − j 2 m 1 )<br />
W j1 +j<br />
z<br />
2 −1,m 1 +m 2<br />
(0) . (14.21)<br />
Again, much of <strong>th</strong>is formula is fixed simply by considering <strong>th</strong>e quantum numbers. The<br />
expression (14.21) is in agreement wi<strong>th</strong> <strong>th</strong>e commutator of polynomial vector fields.<br />
Associated wi<strong>th</strong> <strong>th</strong>e Lie algebra of currents are <strong>th</strong>e charges Q(Y +<br />
j,m ) = ∮ W j,m . These<br />
act on <strong>th</strong>e ground ring as derivations. To prove <strong>th</strong>is, let O 1 (P ), O 2 (Q) be two ground ring<br />
operators, and let C be a contour surrounding points P, Q, and C 1 , C 2 surround only P and<br />
Q, respectively. We have:<br />
∮<br />
C<br />
) (∮<br />
)<br />
(∮<br />
)<br />
W j,m<br />
(O 1 (P )O 2 (Q) = W j,m O 1 (P ) O 2 (Q)+O 1 (P ) W j,m O 2 (Q) . (14.22)<br />
C 1 C 2<br />
Since <strong>th</strong>e BRST invariant contribution to <strong>th</strong>e operator product is independent of <strong>th</strong>e<br />
difference z(P ) − z(Q), <strong>th</strong>e action of <strong>th</strong>e charges descends to a derivation on <strong>th</strong>e ground<br />
ring. In <strong>th</strong>e geometrical interpretation <strong>th</strong>is is just <strong>th</strong>e action of polynomial vector fields<br />
on polynomial functions.<br />
Exercise. Two viewpoints<br />
Show <strong>th</strong>at <strong>th</strong>e second description of <strong>th</strong>e operator algebra of ghost number G = 0<br />
and G = 1 states is equivalent to <strong>th</strong>e ring structure on Λ ∗ T . Use <strong>th</strong>e fact <strong>th</strong>at if<br />
L W ω = 0 is area preserving and L V ω = fω, <strong>th</strong>en L [V,W ] ω = W (f)ω.<br />
182
Tachyon Modules: Away from <strong>th</strong>e self-dual radius, <strong>th</strong>ere are new BRST cohomology<br />
classes V q<br />
= c e iqX/√2 e<br />
√<br />
2(1−<br />
1<br />
2 |q|)φ , wi<strong>th</strong> q /∈ Z. The ring of BRST operators acts on<br />
<strong>th</strong>ese new cohomology classes via operator products. Since <strong>th</strong>e position-dependence of <strong>th</strong>e<br />
operators is a BRST commutator, <strong>th</strong>e tachyon operators form a module representing <strong>th</strong>e<br />
ring Λ ∗ T [131].<br />
First let us determine <strong>th</strong>e action of <strong>th</strong>e ground ring. An easy free-field calculation,<br />
using <strong>th</strong>e explicit formulae for x, y given in chapt. 5, shows<br />
O 1/2,1/2 · V q = q V q+1 q > 0<br />
O 1/2,−1/2 · V q = 0 q > 0<br />
O 1/2,1/2 · V q = 0 q > 0<br />
(14.23)<br />
O 1/2,−1/2 · V q = q V q−1 q > 0 .<br />
So irreducible representations are classified by Sign(q) and q mod 1. The remaining ring<br />
action is somewhat complicated, but can be largely obtained by considering <strong>th</strong>e X, φ<br />
quantum numbers. For example, Y +<br />
j,m·V p is a state wi<strong>th</strong> p X<br />
= (p+2m)/ √ 2 and −ip φ<br />
/ √ 2 =<br />
|p|+2j −2. Thus <strong>th</strong>e resulting state can only lie on <strong>th</strong>e tachyon dispersion line if |p+2m| =<br />
|p| + 2j − 2. Therefore, for example, we can immediately conclude <strong>th</strong>at<br />
Y +<br />
j,m · V p = 0 , (14.24)<br />
for p /∈ 1 2 Z + if p + 2m < 0, p > 0, or if p + 2m > 0, p < 0. If p + 2m and p have <strong>th</strong>e<br />
same sign, <strong>th</strong>en we still require |m| = j − 1 for a nonzero product. In <strong>th</strong>e latter case, <strong>th</strong>e<br />
nonvanishing product is most simply described as<br />
∮<br />
W j,j−1 V p = (−1)2j−1<br />
(2j − 1)! (p) 2j−1V p+2(j−1) (14.25)<br />
for p > 0, wi<strong>th</strong> a similar formula for W j,1−j for p < 0 (and (p) m = Γ(p + m)/Γ(p) is <strong>th</strong>e<br />
Pochammer symbol). Thus, when <strong>th</strong>e ∨w algebra generated by <strong>th</strong>e currents W j,m acts on<br />
<strong>th</strong>e tachyon module, only <strong>th</strong>e Vir + subalgebra ∨ 2 w/ ∨ 3 w acts nontrivially on V p .<br />
14.5. Algebraic Structures of <strong>th</strong>e 2D String: Closed String Cohomology<br />
The algebraic structures for <strong>th</strong>e closed string case are quite similar. The only subtlety<br />
occurs in combining left and right-moving structures.<br />
Consider first <strong>th</strong>e ground ring for <strong>th</strong>e self-dual compactification. The ghost number<br />
G = 0 cohomology classes are spanned by R j,m,m ′<br />
183<br />
= O j,m Ō j,m ′. We must use <strong>th</strong>e same
spin j even at <strong>th</strong>e self-dual radius, since left- and right-moving Liouville momenta must<br />
match. The geometrical interpretation of <strong>th</strong>is ring emerges when one writes ground ring<br />
elements as x n y m¯x n ȳ m . Equating left and right Liouville-momenta we have n+m = n+m.<br />
The ground ring is <strong>th</strong>erefore always generated by polynomials in <strong>th</strong>e expressions a 1 = x¯x,<br />
a 2 = yȳ, a 3 = xȳ, a 4 = y¯x. Note <strong>th</strong>at <strong>th</strong>e a’s obey <strong>th</strong>e relation a 1 a 2 = a 3 a 4 , defining a<br />
<strong>th</strong>ree-dimensional quadric cone Q. At infinite radius we only have ground ring generators<br />
R j,m,m and <strong>th</strong>e ground ring again becomes <strong>th</strong>e ring of polynomial functions on <strong>th</strong>e x, y<br />
plane.<br />
In a manner analogous to <strong>th</strong>e previous section, one can consider <strong>th</strong>e o<strong>th</strong>er algebraic<br />
structures and <strong>th</strong>eir geometrical interpretations in terms of <strong>th</strong>e cone Q. For example, <strong>th</strong>e<br />
symmetries associated to <strong>th</strong>e ghost number G = 1 cohomology are <strong>th</strong>e volume preserving<br />
diffeomorphisms of Q. Fur<strong>th</strong>er results may be found in [66].<br />
As in <strong>th</strong>e chiral case, <strong>th</strong>e ground ring and discrete charges act on <strong>th</strong>e tachyon operators<br />
V p . Indeed, recall from sec. 4.5 <strong>th</strong>at we may apply <strong>th</strong>e descent equations to <strong>th</strong>e ghost<br />
number one BRST classes J j,m = Y +<br />
j,mŌj−1,m and its holomorphic conjugate. The first<br />
step in <strong>th</strong>e descent equations gives a current<br />
Ω (1)<br />
j,m = W + j,mŌj−1,m dz − cW + j,m ¯X d¯z , (14.26)<br />
where | ¯X〉 = ¯b −1 |Ōj−1,m〉. This is an unusual current: al<strong>th</strong>ough it has dimension (1, 0),<br />
it is not purely holomorphic. Moreover, its charge is only conserved up to BRST exact<br />
states. Never<strong>th</strong>eless, we can let <strong>th</strong>ese discrete currents act on tachyons. The story is very<br />
similar to <strong>th</strong>e chiral case. In BRST cohomology, <strong>th</strong>e only nonzero actions occur for p > 0,<br />
J j,j−1 or p < 0, J j,1−j . In <strong>th</strong>is case we have A j,j−1 = (−1)2j<br />
(2j+1)! L 2j we find <strong>th</strong>at, for p > 0:<br />
So, again Vir + (see (12.29)) acts.<br />
[L n , Ṽ p ] = p Ṽ p+n . (14.27)<br />
String <strong>th</strong>eory Ward identities as applied to 2D string <strong>th</strong>eory have been described in<br />
[66,69,131,161,155].<br />
Fur<strong>th</strong>er extensions of <strong>th</strong>is formalism and likely directions for future progress, including<br />
applications in physical contexts, are deferred to [19].<br />
15. Achievements, Disappointments, Future Prospects<br />
Quantum gravity has been a <strong>th</strong>eoretical challenge for 70 years. String <strong>th</strong>eory has been<br />
evolving for 25 years. In <strong>th</strong>e past 3–4 years, some new ideas have been applied to <strong>th</strong>ese<br />
old problems. It is time to assess <strong>th</strong>e harvest of <strong>th</strong>is recent effort.<br />
184
Exercise. Missing lessons<br />
Determine which of <strong>th</strong>e lessons below are covered quite elegantly in portions of text<br />
<strong>th</strong>at have been omitted from <strong>th</strong>ese lecture notes [0] but will be restored for <strong>th</strong>e book<br />
version [19].<br />
15.1. Lessons<br />
From <strong>th</strong>e quantum gravity point of view, <strong>th</strong>e main lessons we have learned from <strong>th</strong>e matrix<br />
model are:<br />
• Euclidean Quantum Gravity makes sense, at least in two dimensions.<br />
• The nature of quantum states in Euclidean quantum gravity, and <strong>th</strong>eir interpretation<br />
wi<strong>th</strong>in <strong>th</strong>e quantum mechanical framework is surprising, and requires <strong>th</strong>e introduction<br />
of non-normalizable wavefunctions as well as normalizable wavefunctions.<br />
• The Wheeler–DeWitt constraint is violated in topology-changing processes.<br />
• The contributions of singular geometries to <strong>th</strong>e pa<strong>th</strong> integral of quantum gravity are<br />
important.<br />
• There is a phase of topological gravity which can be connected to phases of nontopological<br />
gravity.<br />
From <strong>th</strong>e string <strong>th</strong>eory point of view, <strong>th</strong>e main lessons we have learned from <strong>th</strong>e matrix<br />
model are:<br />
• Nonperturbative definitions of string physics, at least in some target spaces, exist.<br />
• There are backgrounds wi<strong>th</strong> large unbroken symmetries, e.g., w 1+∞ and volume preserving<br />
diffeomorphism algebras.<br />
• The large order behavior of perturbation <strong>th</strong>eory at order g has <strong>th</strong>e typically “stringy”<br />
(2g)! grow<strong>th</strong>.<br />
• In solvable string <strong>th</strong>eories, <strong>th</strong>ere is a beautiful ma<strong>th</strong>ematical framework (KP flow,<br />
W -constraints, etc.) <strong>th</strong>at relates string physics in different backgrounds.<br />
• Wi<strong>th</strong> current understanding, it is fundamentally impossible to achieve complete background<br />
independence: There is always dependence on boundary and initial conditions<br />
associated wi<strong>th</strong> non-normalizable states.<br />
• There is a phase of string <strong>th</strong>eory which is topological, and can be connected to nontopological<br />
phases wi<strong>th</strong> local physics (such as string scattering in two dimensions).<br />
185
15.2. Disappointments<br />
From <strong>th</strong>e quantum gravity point of view, our main disappointments <strong>th</strong>us far are:<br />
• It is not yet obvious how to apply our new insights into quantum gravity in two<br />
dimensions to treat <strong>th</strong>e case of quantum gravity in four dimensions.<br />
• Even in two dimensions, <strong>th</strong>e matrix model results have not yet provided solutions to<br />
fundamental problems of quantum gravity, such as <strong>th</strong>e ultimate nature of singularities,<br />
whe<strong>th</strong>er Hawking radiation violates fundamental principles of quantum mechanics,<br />
and related paradoxes.<br />
• Some nonperturbative aspects of gravity have been investigated, but no clear lessons<br />
have been drawn and <strong>th</strong>ere remain many important open problems.<br />
From <strong>th</strong>e string <strong>th</strong>eory point of view, our main disappointments <strong>th</strong>us far are:<br />
• The spacetime physics for c < 1 conformal matter coupled to quantum gravity, while<br />
not fully elucidated, seems ra<strong>th</strong>er uneventful due to <strong>th</strong>e lack of a time dimension, i.e.<br />
due to <strong>th</strong>e lack of fully developed spacetime field <strong>th</strong>eory.<br />
• Spacetime physics of <strong>th</strong>e c = 1 matter coupled to quantum gravity is essentially <strong>th</strong>at of<br />
a free boson. We have as yet no understanding of <strong>th</strong>e infinite tower of string states or<br />
of backreaction. It may be <strong>th</strong>at strings propagating in two target space directions, i.e.<br />
wi<strong>th</strong> no transverse dimensions, is not representative of strings propagating in higher<br />
dimensions. Even for strings in two target space dimensions, we have not progressed<br />
so far beyond <strong>th</strong>e σ-model point of view to a conceptually new formulation.<br />
• The biggest disappointments have been from <strong>th</strong>e standpoint of nonperturbative<br />
physics:<br />
• There are stable non-perturbative solutions for <strong>th</strong>e minimal (2,5) model (Yang–<br />
Lee edge singularity), and higher non-unitary models coupled to quantum gravity,<br />
but again <strong>th</strong>e dynamics is limited due to <strong>th</strong>e lack of time coordinate and consequent<br />
lack of spacetime interpretation.<br />
• For <strong>th</strong>e c < 1 unitary models coupled to quantum gravity, <strong>th</strong>ere is no nonperturbative<br />
<strong>th</strong>eory.<br />
• For c = 1 matter coupled to quantum gravity, we have <strong>th</strong>e opposite problem:<br />
<strong>th</strong>ere are infinitely many nonperturbative completions of <strong>th</strong>e c = 1 S-matrix, i.e.,<br />
<strong>th</strong>ere are infinitely many “θ-parameters.”<br />
• Our lessons on background dependence are sobering: <strong>th</strong>ere are infinitely many superselection<br />
sectors.<br />
186
15.3. Future prospects and Open Problems<br />
Singularity is almost invariably a clue. — Sherlock Holmes<br />
Each paragraph in <strong>th</strong>e text marked wi<strong>th</strong> <strong>th</strong>e “dangerous bend sign” represents an<br />
opportunity.<br />
• The quantum Liouville <strong>th</strong>eory remains unsolved, and is still needed to calculate answers<br />
to many physics questions, so major surprises remain possible.<br />
• We need a better understanding of backgrounds. At present, we seem to have an<br />
infinite dimensional manifold of solutions to string <strong>th</strong>eory, and an infinitely large class<br />
of superselection sectors. Are all <strong>th</strong>ese solutions related by some symmetry?<br />
• Can we use <strong>th</strong>ese backgrounds to understand any<strong>th</strong>ing about time-dependence in<br />
string <strong>th</strong>eory?<br />
• Natural nonperturbative definitions of 2D string <strong>th</strong>eory and 2D gravity are still lacking!<br />
One might have hoped <strong>th</strong>at imposing some physical criterion such as unitarity<br />
would strongly constrain <strong>th</strong>e possible nonperturbative definitions of <strong>th</strong>e <strong>th</strong>eory, but<br />
<strong>th</strong>is does not occur in <strong>th</strong>e case of <strong>th</strong>e c = 1 model coupled to gravity. There we found<br />
infinitely many nonperturbative completions all of which seem perfectly natural from<br />
<strong>th</strong>e matrix model point of view, and we <strong>th</strong>us obtain little guidance in <strong>th</strong>is regard.<br />
• Can <strong>th</strong>e comprehensive picture of <strong>th</strong>e c < 1 backgrounds, unified via <strong>th</strong>e KP formalism,<br />
be generalized to <strong>th</strong>e case of 2D string backgrounds? Is <strong>th</strong>ere e.g. a multiparameter<br />
space of <strong>th</strong>eories which encompasses bo<strong>th</strong> <strong>th</strong>e black hole and c = 1 spacetimes?<br />
Finding a unified picture of all 2D or c ≤ 1 backgrounds remains an interesting open<br />
problem.<br />
• We need to find new ways of cancelling <strong>th</strong>e tachyonic divergences of string <strong>th</strong>eory<br />
— i.e., of making sense of <strong>th</strong>e integrals over moduli spaces. This is essentially <strong>th</strong>e<br />
problem of going beyond <strong>th</strong>e “c = 1 barrier.”<br />
• Does <strong>th</strong>e c = 1 model teach us how to understand better <strong>th</strong>e covariant closed string<br />
field <strong>th</strong>eory of [44]?<br />
• One of <strong>th</strong>e great open puzzles in <strong>th</strong>e subject is <strong>th</strong>e absence of backreaction on <strong>th</strong>e<br />
metric and o<strong>th</strong>er “special state” degrees of freedom, and in particular, <strong>th</strong>e role of 2D<br />
black holes in c = 1 string <strong>th</strong>eory.<br />
• Are <strong>th</strong>ere interesting supersymmetric extensions of <strong>th</strong>e <strong>th</strong>eories we consider here (i.e.<br />
wi<strong>th</strong> potentially interesting spacetime properties such as <strong>th</strong>e construction of [170])?<br />
Acknowledgements<br />
187
We <strong>th</strong>ank especially N. Seiberg for a long series of collaborative efforts on <strong>th</strong>e subject<br />
of 2d gravity. For commentary on various portions of <strong>th</strong>e manuscript we would like to <strong>th</strong>ank<br />
N. Seiberg and M. Staudacher. We also would like to <strong>th</strong>ank many people for discussions and<br />
for teaching us much of <strong>th</strong>e above material. In particular we <strong>th</strong>ank T. Banks, R. Dijkgraaf,<br />
M. Douglas, J. Horne, C. Itzykson, I. Klebanov, D. Kutasov, B. Lian, E. Martinec, R.<br />
Plesser, S. Ramgoolam, H. Saleur, G. Segal, N. Seiberg, R. Shankar, S. Shatashvili, S.<br />
Shenker, M. Staudacher, A.B. Zamolodchikov, G. Zuckerman, and B. Zwiebach. GM is<br />
supported by DOE grant DE-AC02-76ER03075 and by a Presidential Young Investigator<br />
Award, and PG by DOE contract W-7405-ENG-36.<br />
Appendix A. Special functions<br />
A.1. Parabolic cylinder functions<br />
Unfortunately, <strong>th</strong>ere are four notations commonly used for parabolic cylinder functions<br />
[171,172]. Our wavefunctions ψ ± (a, x) are <strong>th</strong>e δ-function normalized even and odd<br />
solutions of ( d2<br />
dx 2<br />
+ x2<br />
4 )ψ = aψ. In terms of degenerate hypergeometric 1F 1 (α, β; x) and<br />
Whittaker functions M µ,ν (x), D a (x), we have even and odd parity wavefunctions:<br />
=<br />
ψ + 1<br />
(a, x) = √ (W (a, x) + W (a, −x))<br />
4π(1 + e<br />
2πa<br />
)<br />
1/2<br />
∣ 1<br />
∣∣∣<br />
1/2<br />
Γ(1/4 + ia/2)<br />
√<br />
4π(1 + e 2πa ) 1/2 21/4 Γ(3/4 + ia/2) ∣ e −ix2 /4<br />
1 F 1 (1/4 − ia/2; 1/2; ix 2 /2)<br />
= e−iπ/8<br />
2π<br />
1<br />
e−aπ/4 |Γ(1/4 + ia/2)| √ M ia/2,−1/4 (ix 2 /2) ,<br />
|x|<br />
(A.1)<br />
=<br />
ψ − 1<br />
(a, x) = √ (W (a, x) − W (a, −x))<br />
4π(1 + e 2πa )<br />
1/2<br />
∣ 1<br />
∣∣∣<br />
1/2<br />
Γ(3/4 + ia/2)<br />
√<br />
4π(1 + e<br />
2πa<br />
) 1/2 23/4 Γ(1/4 + ia/2) ∣ xe −ix2 /4 1 F 1 (3/4 − ia/2; 3/2; ix 2 /2)<br />
= e−3iπ/8<br />
π<br />
e −aπ/4 x<br />
|Γ(3/4 + ia/2)|<br />
|x| M ia/2,1/4(ix 2 /2) .<br />
3/2<br />
(A.2)<br />
188
A.2. Asymptotics<br />
Define<br />
Φ(µ) ≡ π 4 + 1 2 arg Γ( 1 2 + iµ)<br />
k(µ) = √ 1 + e 2πµ − e πµ = O(e −πµ )<br />
(A.3)<br />
k(µ) −1 = √ 1 + e 2πµ + e πµ = 2e πµ + O(e −πµ ) .<br />
The asymptotic properties of <strong>th</strong>e wavefunctions [172] are:<br />
ψ + (µ, λ) ∼ e−πµ/2<br />
(2π) 1/2 µ cosh( √ )<br />
µλ 1/4<br />
1) µ ≫ λ 2<br />
ψ − (µ, λ) ∼<br />
2) −µ ≫ λ 2 ψ + (µ, λ) ∼<br />
3) λ ≫ |µ|<br />
ψ ± (µ, λ) ∼<br />
ψ − (µ, λ) ∼<br />
e−πµ/2<br />
(2π) 1/2 µ 1/4 sinh( √ µλ<br />
)<br />
.<br />
1<br />
(4π) 1/2 |µ| cos(√ −µλ)<br />
1/4<br />
1<br />
(4π) 1/2 |µ| sin(√ −µλ) .<br />
1/4<br />
1<br />
(√<br />
(2πλ √ (<br />
k(µ) cos λ 2 /4 − µ log λ + Φ(µ) )<br />
1 + e 2πµ ) 1/2<br />
± 1/ √ k(µ) sin ( λ 2 /4 − µ log λ + Φ(µ) )) .<br />
(A.4)<br />
(A.5)<br />
(A.6)<br />
4) X ≡ √ λ 2 − 4µ ≫ 1<br />
ψ ± 1<br />
(√ ( 1<br />
(µ, λ) ∼<br />
(2πX √ k(µ) cos<br />
1 + e 2πµ ) 1/2 4 λX − µτ(λ, µ) + π )<br />
4<br />
± 1/ √ ( 1<br />
k(µ) sin<br />
4 λX − µτ(λ, µ) + π ) ) .<br />
4<br />
(A.7)<br />
189
References<br />
[1] M. B. Green and J. Schwarz, Phys. Lett. 149B (1984) 117.<br />
[2] A. M. Polyakov, Phys. Lett. 103B (1981) 207, 211.<br />
[3] A. M. Polyakov, lecture at Nor<strong>th</strong>eastern Univ., spring 1990.<br />
[4] M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 635.<br />
[5] E. Brézin and V. Kazakov, Phys. Lett. B236 (1990) 144.<br />
[6] D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127; Nucl. Phys. B340 (1990) 333.<br />
[7] N. Seiberg, “Notes on Quantum Liouville Theory and Quantum Gravity,” in Common<br />
Trends in Ma<strong>th</strong>ematics and Quantum Field Theory, Proc. of <strong>th</strong>e 1990 Yukawa<br />
International Seminar, Prog. Theor. Phys. Suppl 102, and in Random surfaces and<br />
quantum gravity, proceedings of 1990 Cargèse workshop, edited by O. Alvarez, E.<br />
Marinari, and P. Windey, Plenum (1991).<br />
[8] J. Polchinski, “Remarks on <strong>th</strong>e Liouville Field Theory,” UTTG-19-90, published in<br />
Strings ’90, Texas A&M, Coll. Station Wkshp (1990) 62.<br />
[9] V. A. Kazakov and A. A. Migdal, Nucl. Phys. B311 (1988) 171.<br />
[10] L. Alvarez-Gaumé, “Random surfaces, statistical mechanics, and string <strong>th</strong>eory”, Lausanne<br />
lectures, winter 1990.<br />
[11] P. Ginsparg, “Matrix models of 2d gravity,” Trieste Lectures (July, 1991), LA-UR-91-<br />
4101 (<strong>hep</strong>-<strong>th</strong>/9112013).<br />
[12] A. Bilal, “2d gravity from matrix models,” Johns Hopkins Lectures, CERN TH5867/90.<br />
[13] E. Brézin, “Large N limit and discretized two-dimensional quantum gravity”, in Two<br />
dimensional quantum gravity and random surfaces, proceedings of Jerusalem winter<br />
school (90/91), edited by D. Gross, T. Piran, and S. Weinberg;<br />
D. Gross, “The c = 1 matrix models”, in proceedings of Jerusalem winter school<br />
(90/91);<br />
J. Mañes and Y. Lozano, “Introduction to Nonperturbative 2d quantum gravity”,<br />
Barcelona preprint UB-ECM-PF3/91.<br />
[14] P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, “2D Gravity and Random Matrices,”<br />
Physics Reports, to appear (<strong>1993</strong>).<br />
[15] For a recent review see V. Kazakov, “Bosonic strings and string field <strong>th</strong>eories in onedimensional<br />
target space,” LPTENS 90/30, published in proceedings of 1990 Cargèse<br />
workshop.<br />
[16] I. Klebanov, “String <strong>th</strong>eory in two dimensions”, Trieste lectures, spring 1991, Princeton<br />
preprint PUPT–1271 (<strong>hep</strong>-<strong>th</strong>/9108019).<br />
[17] E. Martinec, “An Introduction to 2d Gravity and Solvable String Models” (<strong>hep</strong><strong>th</strong>/9112019),<br />
lectures at 1991 Trieste spring school, Rutgers preprint RU-91-51.<br />
[18] F. David, “Simplicial quantum gravity and random lattices” (<strong>hep</strong>-<strong>th</strong>/9303127), Lectures<br />
given at Les Houches Summer School, July 1992, Saclay T93/028;<br />
A. Morozov, “Integrability and Matrix Models” (<strong>hep</strong>-<strong>th</strong>/9303139), ITEP-M2/93.<br />
190
[19] P. Ginsparg and G. Moore, “Lectures on 2d gravity and 2d string <strong>th</strong>eory, <strong>th</strong>e book,”<br />
Cambridge University Press, to appear later in <strong>1993</strong>.<br />
[20] P. Ginsparg, “Applied conformal field <strong>th</strong>eory,” Les Houches Session XLIV, 1988, in<br />
Fields, Strings, and Critical Phenomena, ed. by E. Brézin and J. Zinn-Justin, Nor<strong>th</strong><br />
Holland (1989), and references <strong>th</strong>erein.<br />
[21] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241 (1984)<br />
333.<br />
[22] D. Friedan, E. Martinec, and S. Shenker, Nucl. Phys. B271 (1986) 93.<br />
[23] E. Witten, Comm. Ma<strong>th</strong>. Phys. 113 (1988) 529;<br />
L. Alvarez-Gaume, C. Gomez, G. Moore, and C. Vafa, Nucl. Phys. B303 (1988) 455;<br />
C. Vafa, Phys. Lett. 206B (1988) 421;<br />
G. Segal, “The definition of conformal field <strong>th</strong>eory,” unpublished.<br />
[24] A. Chodos and C. Thorn, “Making <strong>th</strong>e massless string massive,” Nucl. Phys. B72<br />
(1974) 509.<br />
[25] Vl. S. Dotsenko and V. A. Fateev, “Conformal algebra and multipoint correlation<br />
functions in 2d statistical models,” Nucl. Phys. B240[FS12] (1984) 312;<br />
Vl. S. Dotsenko and V. A. Fateev, “Four-point correlation functions and <strong>th</strong>e operator<br />
algebra in 2d conformal invariant <strong>th</strong>eories wi<strong>th</strong> central charge c ≤ 1,” Nucl. Phys.<br />
B251[FS13] (1985) 691.<br />
[26] D. Friedan, Les Houches lectures summer 1982, in Recent Advances in Field Theory<br />
and Statistical Physics, J.-B. Zuber and R. Stora eds, (Nor<strong>th</strong> Holland, 1984).<br />
[27] O. Alvarez, in Unified String Theories, M. Green and D. Gross, eds., (World Scientific,<br />
Singapore, 1986).<br />
[28] F. David, Mod. Phys. Lett. A3 (1988) 1651;<br />
J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509.<br />
[29] N. E. Mavromatos and J. L. Miramontes, Mod.Phys.Lett. A4 (1989) 1847;<br />
E. d’Hoker and P. S. Kurzepa, Mod.Phys.Lett. A5 (1990) 1411.<br />
[30] T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; E. Braaten, T.<br />
Curtright and C. Thorn, Phys. Lett. 118B (1982) 115; Ann. Phys. 147 (1983) 365;<br />
E. Braaten, T. Curtright, G. Ghandour and C. Thorn, Phys. Rev. Lett. 51 (1983) 19;<br />
Ann. Phys. 153 (1984) 147.<br />
[31] J. Polchinski, “A two-dimensional model for quantum gravity,” Nucl. Phys. B324<br />
(1989) 123.<br />
[32] E. Witten, “On String <strong>th</strong>eory and black holes,” Phys. Rev. D44 (1991) 314.<br />
[33] V. G. Knizhnik, A. M. Polyakov, and A. B. Zamolodchikov, Mod. Phys. Lett. A3<br />
(1988) 819.<br />
[34] M. R. Douglas, Phys. Lett. B238 (1990) 176.<br />
[35] P. Ginsparg, M. Goulian, M. R. Plesser, and J. Zinn-Justin, “(p, q) string actions,”<br />
Nucl. Phys. B342 (1990) 539.<br />
191
[36] G. Moore, N. Seiberg, and M. Staudacher, Nucl. Phys. 362 (1991) 665.<br />
[37] E. D’Hoker and R. Jackiw, Phys. Rev. Lett. 50 (1983) 1719; Phys. Rev. D26 (1982)<br />
3517;<br />
E. D’Hoker, D. Freedman and R. Jackiw, Phys. Rev. D28 (1983) 2583.<br />
[38] L. Takhtadjan and P.G. Zograf, Funct. Anal. Appl. 19 (1985) 67;<br />
L. Takhtadjan, Proc. Symp. Pure Ma<strong>th</strong>. 49 (1989) 581.<br />
[39] J.-L. Gervais and A. Neveu, Nucl. Phys. 199 (1982) 59; B209 (1982) 125; B224 (1983)<br />
329; 238 (1984) 125, 396; Phys. Lett. 151B (1985) 271; J.-L. Gervais, LPTENS 89/14;<br />
90/4.<br />
[40] R. Gunning, Lectures on Riemann Surfaces, Princeton University Press (1966).<br />
[41] H. M. Farkas and I. Kra, Riemann Surfaces, Springer (1980).<br />
[42] M. Reed and B. Simon, Me<strong>th</strong>ods of Modern Ma<strong>th</strong>ematical Physics, Academic Press<br />
(1972) vol 1.<br />
[43] J.A. Hempel, Bull. Lond. Ma<strong>th</strong>. Soc. 20 (1988) 97.<br />
[44] B. Zwiebach, “Closed string field <strong>th</strong>eory: quantum action and <strong>th</strong>e B-V master equation”<br />
(<strong>hep</strong>-<strong>th</strong>/9206084), Nucl. Phys. B390 (<strong>1993</strong>) 33.<br />
[45] A. Gupta, S. Trivedi and M. Wise, Nucl. Phys. B340 (1990) 475.<br />
[46] M. Goulian and M. Li, Phys. Rev. Lett. 66 (1991) 2051.<br />
[47] Y. Kitazawa, Phys. Lett. B265 (1991) 262;<br />
N. Sakai and Y. Tanii, Prog. Theor. Phys. 86 (1991) 547;<br />
V. Dotsenko, Mod. Phys. Lett. A6 (1991) 3601.<br />
[48] P. Di Francesco and D. Kutasov, Phys. Lett. 261B (1991) 385;<br />
P. Di Francesco and D. Kutasov, “World sheet and space time physics in two dimensional<br />
(super) string <strong>th</strong>eory” (<strong>hep</strong>-<strong>th</strong>/9109005), Nucl. Phys. B375 (1992) 119.<br />
[49] A. M. Polyakov, “Self-tuning fields and resonant correlations in 2-d gravity,” Mod.<br />
Phys. Lett. A6 (1991) 635.<br />
[50] P. Di Francesco and D. Kutasov, Nucl. Phys. B342 (1990) 589; and Princeton preprint<br />
PUPT-1206 (1990) published in proceedings of Cargèse workshop (1990).<br />
[51] M. Bershadsky and I. Klebanov, Phys. Rev. Lett. 65 (1990) 3088.<br />
[52] J.-L. Gervais, “Gravity-Matter Couplings from Liouville Theory,” LPTENS-91/22<br />
(<strong>hep</strong>-<strong>th</strong>/9205034).<br />
[53] H. Poincaré, Papers on Fuchsian Functions, J. Stillwell, transl., Springer Verlag 1985.<br />
[54] Z. Nehari, Conformal Mapping. McGraw-Hill 1952.<br />
[55] E. Martinec, G. Moore, and N. Seiberg, “Boundary operators in 2-d gravity” (<strong>hep</strong><strong>th</strong>/9109055),<br />
Phys. Lett. 263B (1991) 190.<br />
[56] J. Cardy, “Conformal invariance and surface critical behavior,” Nucl. Phys. B240<br />
(1984) 514;<br />
J. Cardy, “Boundary conditions, fusion rules and <strong>th</strong>e Verlinde formula,” Nucl. Phys.<br />
B324 (1989) 581.<br />
192
[57] I. Kostov and M. Staudacher, “Multicritical phases of <strong>th</strong>e O(N) model on a random<br />
lattice” (<strong>hep</strong>-<strong>th</strong>/9203030), Nucl. Phys. B384 (1992) 459.<br />
[58] C. W. Misner, K. S. Thorne, and J. Wheeler, Gravitation, W.H. Freeman and Co.<br />
(1973).<br />
[59] R. Wald, General Relativity, Univ. of Chicago Press (1984).<br />
[60] J. A. Wheeler, “Geometrodynamics and <strong>th</strong>e issue of <strong>th</strong>e final state” in Relativity,<br />
Groups, and Topology, C. M. DeWitt and B. S. DeWitt, eds., Gordon and Breach,<br />
N.Y. (1964).<br />
[61] B.S. DeWitt, “Quantum Theory of Gravity, I: Canonical Theory,” Phys. Rev. 160<br />
(1967) 1113.<br />
[62] See R. Laflamme, “Introduction and Applications of Quantum Cosmology,” 1991 Gift<br />
lectures for a recent review of quantum cosmology.<br />
[63] B. Lian and G. Zuckerman, Phys. Lett. B254 (1991) 417; Phys. Lett. B266 (1991) 21;<br />
Comm. Ma<strong>th</strong>. Phys. 135 (1991) 547; Comm. Ma<strong>th</strong>. Phys. 145 (1992) 561.<br />
[64] B. Lian, “Semi-infinite homology and 2d quantum gravity,” PhD <strong>th</strong>esis, Yale Univ.<br />
(1991).<br />
[65] P. Bouwknegt, J. McCar<strong>th</strong>y and K. Pilch, “Fock space resolutions of <strong>th</strong>e virasoro<br />
highest weight modules wi<strong>th</strong> c ≤ 1,” Lett. Ma<strong>th</strong>. Phys. 23 (1991) 193; “BRST analysis<br />
of physical states for 2-d gravity coupled to c ≤ 1 matter,” Comm. Ma<strong>th</strong>. Phys. 145<br />
(1992) 541; and reviews CERN-TH-6646-92 (<strong>hep</strong>-<strong>th</strong>/9209034), CERN-TH-6645-92,<br />
CERN-TH-6279-91 (<strong>hep</strong>-<strong>th</strong>/9110031).<br />
[66] E. Witten and B. Zwiebach, “Algebraic Structures and Differential Geometry in 2D<br />
String Theory” (<strong>hep</strong>-<strong>th</strong>/9201056), Nucl. Phys. B377 (1992) 55.<br />
[67] E. Witten, Ground ring of two-dimensional string <strong>th</strong>eory” (<strong>hep</strong>-<strong>th</strong>/9108004), Nucl.<br />
Phys. B373 (1992) 187.<br />
[68] V. Kac, Infinite Dimensional Lie Algebras, Cambridge (1985).<br />
[69] E. Verlinde, Nucl. Phys. B381 (1992) 141.<br />
[70] M. Green, J. Schwarz, and E. Witten, Superstring <strong>th</strong>eory, Cambridge Univ. Press<br />
(1987).<br />
[71] T. Banks, “The tachyon potential in string <strong>th</strong>eory,” Nucl. Phys. B361 (1991) 166.<br />
[72] C.G. Callan, E.J. Martinec, M.J. Perry, D. Friedan, “Strings in background fields,”<br />
Nucl. Phys. B262 (1985) 593.<br />
[73] D. Kutasov and N. Seiberg, “Number of degrees of freedom, density of states, and<br />
tachyons in string <strong>th</strong>eory and CFT,” Nucl. Phys. B358 (1991) 600.<br />
[74] R. Myers, “New dimensions for old strings,” Phys. Lett. 199B (1987) 371;<br />
I. Antoniadis, C. Bachas, John Ellis, and D.V. Nanopoulos, “An expanding universe<br />
in string <strong>th</strong>eory,” Nucl. Phys. B328 (1989) 117.<br />
[75] J. Polchinski, “Critical Behavior of Random Surfaces in One Dimension,” Nucl. Phys.<br />
B346 (1990) 253.<br />
193
[76] S.R. Das and A. Jevicki, “String Field Theory and Physical Interpretation of D=1<br />
Strings,” Mod. Phys. Lett. A5 (1990) 1639.<br />
[77] A.M. Sengupta and S.R. Wadia, “Excitations and interactions in d = 1 string <strong>th</strong>eory,”<br />
Int. Jour. Mod. Phys. A6 (1991) 1961.<br />
[78] L. Faddeev, in Me<strong>th</strong>ods in Field Theory, Les Houches Summer 1975, ed. by R. Balian<br />
and J. Zinn-Justin, Nor<strong>th</strong> Holland/World Scientific (1976/1981).<br />
[79] S. Coleman, Aspects of Symmetry, Cambridge (1985).<br />
[80] F. David, Nucl. Phys. B257[FS14] (1985) 45, 543;<br />
J. Ambjørn, B. Durhuus and J. Fröhlich, Nucl. Phys. B257[FS14] (1985) 433; J.<br />
Fröhlich, in: Lecture Notes in Physics, Vol. 216, ed. L. Garrido (Springer, Berlin,<br />
1985);<br />
V. A. Kazakov, I. K. Kostov and A. A. Migdal, Phys. Lett. 157B (1985) 295; D.<br />
Boulatov, V. A. Kazakov, I. K. Kostov and A. A. Migdal, Phys. Lett. B174 (1986) 87;<br />
Nucl. Phys. B275[FS17] (1986) 641.<br />
[81] G. ’t Hooft, Nucl. Phys. B72 (1974) 461.<br />
[82] E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm. Ma<strong>th</strong>. Phys. 59 (1978) 35.<br />
[83] G. Harris and E. Martinec, Phys. Lett. B245 (1990) 384;<br />
E. Brezin and H. Neuberger, Phys. Rev. Lett. 65 (1990) 2098; Nucl.Phys. B350 (1991)<br />
513.<br />
[84] D. Bessis, C. Itzykson, and J.-B. Zuber, Adv. Appl. Ma<strong>th</strong>. 1 (1980) 109.<br />
[85] I. K. Kostov, Nucl. Phys. B326 (1989) 583.<br />
[86] E. Brézin, V. A. Kazakov, and Al. B. Zamolodchikov, Nucl. Phys. B338 (1990) 673.<br />
[87] P. Ginsparg and J. Zinn-Justin, “2D gravity + 1d matter,” Phys. Lett. 240B (1990)<br />
333.<br />
[88] G. Parisi, Phys. Lett. 238B (1990) 209,213; Europhys. Lett. 11 (1990) 595.<br />
[89] D. J. Gross and N. Miljkovic, Phys. Lett. 238B (1990) 217.<br />
[90] G. Moore, “Double-scaled field <strong>th</strong>eory at c = 1,” Nucl. Phys. B368 (1992) 557.<br />
[91] G. Moore and N. Seiberg, “From loops to fields in 2d gravity,” Int. Jour. Mod. Phys.<br />
A7 (1992) 2601.<br />
[92] D. Gross and I. Klebanov, Nucl. Phys. B359 (1991) 3;<br />
D. Gross, I. Klebanov, and M. Newman, Nucl. Phys. B350 (1991) 621.<br />
[93] D. Gross and I. Klebanov, “Fermionic String Field Theory of c = 1 2D Quantum<br />
Gravity,” Nucl. Phys. B352 (1991) 671.<br />
[94] I. Kostov, “Strings embedded in Dynkin Diagrams”, SACLAY-SPHT-90-133 (1990),<br />
published in proceedings of Cargèse Workshop (1990); Phys. Lett. B266 (1991) 42.<br />
[95] J. Polchinski, “Classical limit of 1+1 Dimensional String Theory,” Nucl. Phys. B362<br />
(1991) 125.<br />
[96] G. Mandal, A. Sengupta, and S. Wadia, Mod. Phys. Lett. A6 (1991) 1465;<br />
K. Demeterfi, A. Jevicki, and J.P. Rodrigues, Nucl. Phys. B362 (1991) 173.<br />
194
[97] K. Demeterfi, A. Jevicki, and J. Rodrigues, “Scattering Amplitudes and Loop Corrections<br />
in Collective String Field Theory (II),” Nucl. Phys. B365 (1991)499.<br />
[98] C. Crnković, P. Ginsparg, and G. Moore, Phys. Lett. B237 (1990) 196.<br />
[99] V. Kazakov, Mod. Phys. Lett A4 (1989) 2125.<br />
[100] P. Ginsparg and J. Zinn-Justin, “Action principle and large order behavior of nonperturbative<br />
gravity”, LA-UR-90-3687 / SPhT/90-140 (1990), published in proceedings<br />
of 1990 Cargèse workshop;<br />
P. Ginsparg and J. Zinn-Justin, “Large order behavior of nonperturbative gravity,”<br />
Phys. Lett. B255 (1991) 189.<br />
[101] F. David, “Nonperturbative effects in 2D gravity and matrix models,” Saclay-SPHT-<br />
90-178, published in proceedings of Cargèse workshop (1990).<br />
[102] M. Staudacher, “The Yang–Lee edge singularity on a dynamical planar random surface,”<br />
Nucl. Phys. B336 (1990) 349.<br />
[103] E. Brézin, M. Douglas, V. Kazakov, and S. Shenker, Phys. Lett. B237 (1990) 43;<br />
D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 717.<br />
[104] T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. 238B (1990) 279.<br />
[105] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “Notes on topological string <strong>th</strong>eory and<br />
2D quantum gravity”, Princeton preprint PUPT-1217, published in proceedings of<br />
Cargèse workshop (1990);<br />
R. Dijkgraaf, “Topological field <strong>th</strong>eory and 2d quantum gravity”, in proceedings of<br />
Jerusalem winter school (90/91).<br />
[106] R. Dijkgraaf, lectures in <strong>th</strong>is volume.<br />
[107] V. Kazakov, Phys. Lett. 119A (1986) 140;<br />
D. Boulatov and V. Kazakov, Phys. Lett. 186B (1987) 379.<br />
[108] M. L. Mehta, Comm. Ma<strong>th</strong>. Phys. 79 (1981) 327;<br />
S. Chadha, G. Mahoux and M. L. Mehta, J. Phys. A14 (1981) 579;<br />
C. Itzykson and J.B. Zuber, J. Ma<strong>th</strong>. Phys. 21 (1980) 411.<br />
[109] V. G. Drinfel’d and V. V. Sokolov, Jour. Sov. Ma<strong>th</strong>. (1985) 1975;<br />
G. Segal and G. Wilson, Pub. Ma<strong>th</strong>. I.H.E.S. 61 (1985) 5.<br />
[110] I. M. Gel’fand and L. A. Dikii, Russian Ma<strong>th</strong>. Surveys 30:5 (1975) 77;<br />
I. M. Gel’fand and L. A. Dikii, Funct. Anal. Appl. 10 (1976) 259.<br />
[111] M. Douglas, “The two-matrix model”, published in proceedings of 1990 Cargèse workshop.<br />
[112] T. Tada, Phys. Lett. B259 (1991) 442.<br />
[113] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, “Unification<br />
of All String Models wi<strong>th</strong> c < 1” (<strong>hep</strong>-<strong>th</strong>/9111037), Phys. Lett. B275 (1992) 311.<br />
[114] M. E. Agishtein and A. A. Migdal, Int. J. Mod. Phys. C1 (1990) 165; Nucl. Phys.<br />
B350 (1991) 690;<br />
F. David, “What is <strong>th</strong>e intrinsic geometry of two-dimensional quantum gravity?” Nucl.<br />
195
Phys. B368 (1992) 671;<br />
S. Jain and S. Ma<strong>th</strong>ur, “World sheet geometry and baby universes in 2-d quantum<br />
gravity,” Phys. Lett. B286 (1992) 239;<br />
H. Kawai, N. Kawamoto, T. Mogami and Y. Watabiki, “Transfer Matrix Formalism for<br />
Two-Dimensional Quantum Gravity and Fractal Structures of Space-time,” INS-969<br />
(<strong>hep</strong>-<strong>th</strong>/9302133).<br />
[115] F. David, Loop equations and non-perturbative effects in two dimensional quantum<br />
gravity,” Mod. Phys. Lett. A5 (1990) 1019.<br />
[116] F. David, “Phases of <strong>th</strong>e large N matrix model and non-perturbative effects in 2d<br />
gravity,” Nucl. Phys. B348 (1991) 507.<br />
[117] M. Fukuma, H. Kawai, R. Nakayama, “Continuum schwinger-dyson equations and universal<br />
structures in two-dimensional quantum gravity,” Int. J. Mod. Phys. A6 (1991)<br />
1385;<br />
M. Bowick, A. Morozov, Danny Shevitz, “Reduced unitary matrix models and <strong>th</strong>e<br />
hierarchy of tau functions,” Nucl. Phys. B354 (1991) 496;<br />
By Yu. Makeenko, A. Marshakov, A. Mironov, A. Morozov, “Continuum versus discrete<br />
virasoro in one matrix models,” Nucl. Phys. B356 (1991) 574.<br />
[118] E. Brézin, E. Marinari, and G. Parisi, Phys. Lett. B242 (1990) 35.<br />
[119] J. Ambjorn, J. Jurkiewicz, Yu.M. Makeenko, “Multiloop correlators for two-dimensional<br />
quantum gravity,” Phys. Lett. B251 (1990) 517.<br />
[120] T. Banks, unpublished. In <strong>th</strong>is work Banks showed how to derive <strong>th</strong>e linear equation for<br />
<strong>th</strong>e cosmological constant wavefunction in pure gravity from <strong>th</strong>e nonlinear equation.<br />
[121] U. Danielsson, “Symmetries and special states in two-dimensional string <strong>th</strong>eory” (<strong>hep</strong><strong>th</strong>/9112061),<br />
Nucl. Phys. B380 (1992) 83.<br />
[122] Ivan Kostov, “Two point correlator for <strong>th</strong>e d = 1 closed bosonic string,” Phys. Lett.<br />
(1988) B215;<br />
S. Ben-Menahem, “Two and <strong>th</strong>ree point functions in <strong>th</strong>e d = 1 matrix model,” Nucl.<br />
Phys. B364 (1991) 681.<br />
[123] U. Danielssohn, “A Study of Two Dimensional String Theory” (<strong>hep</strong>-<strong>th</strong>/9205063) PhD<br />
Thesis, Princeton Univ (1992).<br />
[124] I. Bakas, Phys. Lett. B228 (1989) 57.<br />
[125] C.N. Pope, L.J. Romans, and X. Shen, “A brief history of W ∞ ,” CTP TAMU-89/90,<br />
published in Coll. Station Wkshp (1990) 287.<br />
[126] M. Fukuma, H. Kawai, and R. Nakayama, “Infinite Dimensional Grassmannian Structure<br />
of Two-Dimensional Quantum Gravity,” Comm. Ma<strong>th</strong>. Phys. 143 (1992) 371.<br />
[127] J. Avan and A. Jevicki, “Classical integrability and higher symmetries of collective field<br />
<strong>th</strong>eory,” Phys. Lett. 266B (1991) 35; “Quantum integrability and exact eigenstates of<br />
<strong>th</strong>e collective string field <strong>th</strong>eory,” Phys. Lett. 272B (1991) 17; “Algebraic Structures<br />
196
and Eigenstates for Integrable Collective Field Theories” (<strong>hep</strong>-<strong>th</strong>/9202065); “Interacting<br />
Theory of Collective and Topological Fields in 2 Dimensions” (<strong>hep</strong>-<strong>th</strong>/9209036).<br />
[128] J. Avan and A. Jevicki,“String field actions from W-infinity” (<strong>hep</strong>-<strong>th</strong>/9111028), Mod.<br />
Phys. Lett. A7 (1992) 357.<br />
[129] S.R. Das, A. Dhar, G. Mandal, S. R. Wadia, “Gauge <strong>th</strong>eory formulation of <strong>th</strong>e c=1<br />
matrix model: symmetries and discrete states” (<strong>hep</strong>-<strong>th</strong>/9110021) Int. J. Mod. Phys.<br />
A7 (1992) 5165; “Bosonization of nonrelativistic fermions and W ∞ algebra,” Mod.<br />
Phys. Lett. A7 (1992) 71;<br />
A. Dhar, G. Mandal, S. R. Wadia, “Classical Fermi fluid and geometric action for<br />
c = 1” (<strong>hep</strong>-<strong>th</strong>/9204028) IASSNS-HEP-91-89; “Non-relativistic fermions, coadjoint<br />
orbits of w ∞ and string field <strong>th</strong>eory at c = 1” (<strong>hep</strong>-<strong>th</strong>/9207011), TIFR-TH-92-40.<br />
[130] D. Minic, J. Polchinski, and Z. Yang, “Translation-invariant backgrounds in 1+1 dimensional<br />
string <strong>th</strong>eory,” Nucl. Phys. B362 (1991) 125.<br />
[131] D. Kutasov, E. Martinec, and N. Seiberg, “Ground Rings and Their Modules in 2D<br />
Gravity wi<strong>th</strong> c ≤ 1 Matter” (<strong>hep</strong>-<strong>th</strong>/9111048), Phys. Lett. B276 (1992) 437.<br />
[132] G. Moore, R. Plesser, and S. Ramgoolam, “Exact S-Matrix for 2D String Theory”<br />
(<strong>hep</strong>-<strong>th</strong>/9111035) Nucl. Phys. B377(1992)143.<br />
[133] B. Lee in “Me<strong>th</strong>ods in Field Theory,” Les Houches, 1975, section 6.3.<br />
[134] C. Callan, S. Giddings, J. Harvey, and A. Strominger, “Evanescent Black Holes” (<strong>hep</strong><strong>th</strong>/9111056),<br />
Phys. Rev. D45 (1992) 1005.<br />
[135] Al. B. Zamolodchikov and A.B. Zamolodchikov, “Factorized S matrices in twodimensions<br />
as <strong>th</strong>e exact solutions of certain relativistic quantum field models,” Ann.<br />
Phys. 120 (1979) 253.<br />
[136] N. Andrei and J. Loewenstein, Phys. Lett. 91B (1980) 401;<br />
V. Korepin, Theor. and Ma<strong>th</strong>. Physics 41 (1979) 953.<br />
[137] A.B. Zamolodchikov and Al.B. Zamolodchikov, “Massless factorized scattering and<br />
sigma models wi<strong>th</strong> topological terms,” Nucl. Phys. B379 (1992) 602.<br />
[138] Moore and Plesser, “Classical Scattering in 1 + 1 Dimensional String Theory” (<strong>hep</strong><strong>th</strong>/9203060),<br />
to appear in Phys. Rev. D.<br />
[139] G. Moore, “Gravitational Phase Transitions and <strong>th</strong>e Sine-Gordon Model,” Yale<br />
preprint YCTP-P1-92, <strong>hep</strong>-<strong>th</strong>/9203061.<br />
[140] N. Seiberg and S. Shenker, “A note on background independence” (<strong>hep</strong>-<strong>th</strong>/9201017),<br />
Phys. Rev. D45 (1992) 4581.<br />
[141] R. Dijkgraaf, G. Moore, and R. Plesser, “The Partition Function of 2D String Theory,”<br />
<strong>hep</strong>-<strong>th</strong>/9208031, submitted to Nucl. Phys. B<br />
[142] D.J. Gross and I. Klebanov, Nucl. Phys. B344 (1990) 475; Nucl. Phys. B354 (1991)<br />
459.<br />
[143] I. Klebanov and D. Lowe, Nucl. Phys. B363 (1991) 543.<br />
[144] S. Cecotti and C. Vafa, Nucl. Phys. B367 (1991) 359.<br />
197
[145] E. Martinec and S. Shatashvili, “Black hole physics and liouville <strong>th</strong>eory,” Nucl. Phys.<br />
B368 (1992) 338.<br />
[146] M. Bershadsky and D. Kutasov, “Comment on gauged WZW <strong>th</strong>eory,” Phys. Lett.<br />
B266 (1991) 345.<br />
[147] S. Das, “Matrix models and black holes” (<strong>hep</strong>-<strong>th</strong>/9210107), Mod. Phys. Lett. A8<br />
(<strong>1993</strong>) 69;<br />
A. Dhar, G. Mandal, S. Wadia, “Stringy quantum effects in two-dimensional black<br />
hole” (<strong>hep</strong>-<strong>th</strong>/9210120), Mod. Phys. Lett. A7 (1992) 3703;<br />
T. Yoneya, “Matrix models and 2-d critical string <strong>th</strong>eory: 2-D black hole by c = 1<br />
matrix model” (<strong>hep</strong>-<strong>th</strong>/9211079), UT-KOMABA-92-13.<br />
[148] Jorge G. Russo, “Black hole formation in c = 1 string field <strong>th</strong>eory” (<strong>hep</strong>-<strong>th</strong>/9211057),<br />
Phys. Lett. B300 (<strong>1993</strong>) 336.<br />
[149] Edward Witten, “Two-dimensional string <strong>th</strong>eory and black holes” (<strong>hep</strong>-<strong>th</strong>/9206069),<br />
Lecture given at Conf. on Topics in Quantum Gravity, Cincinnati, OH, <strong>Apr</strong> 3–4, 1992.<br />
[150] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “String propagation in a black hole geometry,”<br />
Nucl. Phys. B371 (1992) 269.<br />
[151] S. Mukhi and C. Vafa, “Two dimensional black hole as a topological coset model of<br />
c = 1 string <strong>th</strong>eory” (<strong>hep</strong>-<strong>th</strong>/9301083).<br />
[152] Edward Witten, “The n matrix model and gauged WZW models,” Nucl. Phys. B371<br />
(1992) 191.<br />
[153] J. Distler and C. Vafa, in proceedings of Cargèse 1990 workshop; “A critical matrix<br />
model at c = 1,” Mod. Phys. Lett. A6 (1991) 259.<br />
[154] J. Minahan and A. Polychronakos, “Equivalence of Two Dimensional QCD and <strong>th</strong>e<br />
c = 1 Matrix Model” (<strong>hep</strong>-<strong>th</strong>/9303153);<br />
M. Douglas, “Conformal Field Theory Techniques for Large N Group Theory” (<strong>hep</strong><strong>th</strong>/9303159).<br />
[155] I. Klebanov and A. Pasquinucci, “Correlation functions from two-dimensional string<br />
Ward identities” (<strong>hep</strong>-<strong>th</strong>/9204052) PUPT-1313.<br />
[156] N. Sakai and Y. Tanii, “Operator product expansion and topological states in c = 1<br />
Matter Coupled to 2D Gravity” (<strong>hep</strong>-<strong>th</strong>/9111049), Prog. Theor. Phys. Supp. 110<br />
(1992) 117; “Factorization and topological states in c = 1 matter coupled to 2D<br />
gravity” (<strong>hep</strong>-<strong>th</strong>/9108027), Phys. Lett. B276 (1992) 41.<br />
[157] G. Minic and Z. Yang, “Is S = 1 for c = 1?” Phys. Lett. B274 (1992) 27.<br />
[158] D. Lowe, “Unitarity Relations in c = 1 Liouville Theory” (<strong>hep</strong>-<strong>th</strong>/9204084), Mod.<br />
Phys. Lett. A7 (1992) 2647.<br />
[159] M. Bershadsky and D. Kutasov, Phys. Lett. B274 (1992) 331-337; Nucl. Phys. B382<br />
(1992) 213.<br />
[160] I.R. Klebanov and A.M. Polyakov, “Interactions of Discrete States in Two-Dimensional<br />
String Theory,” Mod. Phys. Lett. A6 (1991) 3273.<br />
198
[161] I.R. Klebanov, “Ward Identities in Two-Dimensional String Theory,” Mod. Phys. Lett.<br />
A7 (1992) 723.<br />
[162] I. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and <strong>th</strong>e Monster<br />
Group, Academic Press (1988).<br />
[163] G. Moore and N. Seiberg, “Lectures on RCFT,” RU-89-32-mc, lectures at Trieste<br />
Spring School 1989, published in Trieste Superstrings, edited by A. Strominger and<br />
M. Green, World Scientific (1989), also in Banff NATO ASI (1989) 236.<br />
[164] P. Bouwknegt and K. Schoutens, “W -Symmetry in Conformal Field Theory” (<strong>hep</strong><strong>th</strong>/9210010),<br />
CERN-TH-6583/92.<br />
[165] E. Witten, “Some remarks about string field <strong>th</strong>eory,” Princeton preprint 86-1188<br />
(1986), Physica Scripta T15 (1987) 70, and in Marstrand Nobel Sympos. (1986) 70.<br />
[166] B. Lian and G. Zuckerman, “New perspectives on <strong>th</strong>e brst algebraic structure of string<br />
<strong>th</strong>eory,” (<strong>hep</strong>-<strong>th</strong>/9211072) TORONTO-9211072.<br />
[167] E. Getzler, “Batalin-Vilkovisky Algebras and Two-Dimensional Toplogical Field Theories”<br />
(<strong>hep</strong>-<strong>th</strong>/9212043);<br />
P. Horava, “Spacetime Diffeomorphisms and Topological w ∞ Symmetry in Two Dimensional<br />
Topological String Theory” (<strong>hep</strong>-<strong>th</strong>/9302020); “Two Dimensional String<br />
Theory and <strong>th</strong>e Topological Torus” (<strong>hep</strong>-<strong>th</strong>/9202008), Nucl. Phys. B386 (1992) 383.<br />
[168] G. Segal, Lectures at <strong>th</strong>e Isaac Newton Institute, August 1992, and lectures at Yale<br />
University, March <strong>1993</strong>.<br />
[169] M.H. Sarmadi, “The ring structure of chiral operators for minimal models coupled to<br />
2D gravity,” IC/92/301<br />
[170] D. Kutasov and N. Seiberg, Phys. Lett. B251 (1990) 67.<br />
[171] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products, Academic<br />
Press (1980).<br />
[172] M. Abramowitz and I. Stegun, Handbook of Ma<strong>th</strong>ematical Functions, Dover (1968).<br />
199