arXiv:hep-th/9304011 v1 Apr 5 1993

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Contents 0. Introduction, Overview, and Purpose . . . . . . . . . . . . . . . . . . . . . . . 3 0.1. Philosophy and Diatribe . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2. 2D Gravity and 2D String theory . . . . . . . . . . . . . . . . . . . . . . . 5 0.3. Review of reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Loops and States in Conformal Field Theory . . . . . . . . . . . . . . . . . . . 8 1.1. Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2. Hamiltonian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3. Equivalence of states and operators . . . . . . . . . . . . . . . . . . . . . . 10 1.4. Gaussian Field with a Background Charge . . . . . . . . . . . . . . . . . . . 12 2. 2D Euclidean Quantum Gravity I: Path Integral Approach . . . . . . . . . . . . . 13 2.1. 2D Gravity and Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 13 2.2. Path integral approach to 2D Euclidean Quantum Gravity . . . . . . . . . . . . 14 3. Brief Review of the Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 22 3.1. Classical Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2. Classical Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3. Quantum Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4. Spectrum of Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5. Semiclassical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.6. Seiberg bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.7. Semiclassical Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.8. Operator Products in Liouville Theory . . . . . . . . . . . . . . . . . . . . 39 3.9. Liouville Correlators from Analytic Continuation . . . . . . . . . . . . . . . . 40 3.10. Quantum Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.11. Surfaces with boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4. 2D Euclidean Quantum Gravity II: Canonical Approach . . . . . . . . . . . . . . 50 4.1. Canonical Quantization of Gravitational Theories . . . . . . . . . . . . . . . 50 4.2. Canonical Quantization of 2D Euclidean Quantum Gravity . . . . . . . . . . . 51 4.3. KPZ states in 2D Quantum Gravity . . . . . . . . . . . . . . . . . . . . . 52 4.4. LZ states in 2D Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . 53 4.5. States in 2D Gravity Coupled to a Gaussian Field: more BRST . . . . . . . . . 54 5. 2D Critical String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1. Particles in D Dimensions: QFT as 1D Euclidean Quantum Gravity. . . . . . . . 62 5.2. Strings in D Dimensions: String Theory as 2D Euclidean Quantum Gravity . . . . 64 5.3. 2D String Theory: Euclidean Signature . . . . . . . . . . . . . . . . . . . . 66 5.4. 2D String Theory: Minkowskian Signature . . . . . . . . . . . . . . . . . . . 68 5.5. Heterodox remarks regarding the “special states” . . . . . . . . . . . . . . . . 69 5.6. Bosonic String Amplitudes and the “c > 1 problem” . . . . . . . . . . . . . . 72 6. Discretized surfaces, matrix models, and the continuum limit . . . . . . . . . . . . 75 6.1. Discretized surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2. Matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.3. The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4. A first look at the double scaling limit . . . . . . . . . . . . . . . . . . . . 83 7. Matrix Model Technology I: Method of Orthogonal Polynomials . . . . . . . . . . . 84 1
7.1. Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.2. The genus zero partition function . . . . . . . . . . . . . . . . . . . . . . 86 7.3. The all genus partition function . . . . . . . . . . . . . . . . . . . . . . . 88 7.4. The Douglas Equations and the KdV hierarchy . . . . . . . . . . . . . . . . 90 7.5. Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.6. Multi-Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.7. Continuum Solution of the Matrix Chains . . . . . . . . . . . . . . . . . . . 95 8. Matrix Model Technology II: Loops on the Lattice . . . . . . . . . . . . . . . . . 99 8.1. Lattice Loop Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2. Precise definition of the continuum limit . . . . . . . . . . . . . . . . . . 101 8.3. The Loop Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9. Matrix Model Technology III: Free Fermions from the Lattice . . . . . . . . . . . 105 9.1. Lattice Fermi Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 105 9.2. Eigenvalue distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.3. Double–Scaled Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . 109 10. Loops and States in Matrix Model Quantum Gravity . . . . . . . . . . . . . . 113 10.1. Computation of Macroscopic Loops . . . . . . . . . . . . . . . . . . . . 113 10.2. Loops to Local Operators . . . . . . . . . . . . . . . . . . . . . . . . 116 10.3. Wavefunctions and Propagators from the Matrix Model . . . . . . . . . . . 117 10.4. Redundant operators, singular geometries and contact terms . . . . . . . . . 120 11. Loops and States in the c = 1 Matrix Model . . . . . . . . . . . . . . . . . . 120 11.1. Definition of the c = 1 Matrix Model . . . . . . . . . . . . . . . . . . . 120 11.2. Matrix Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 122 11.3. Double-Scaled Fermi Field Theory . . . . . . . . . . . . . . . . . . . . . 127 11.4. Macroscopic Loops at c = 1 . . . . . . . . . . . . . . . . . . . . . . . . 129 11.5. Wavefunctions and Wheeler–DeWitt Equations . . . . . . . . . . . . . . . 134 11.6. Macroscopic Loop Field Theory and c = 1 scaling . . . . . . . . . . . . . . 134 11.7. Correlation functions of Vertex Operators . . . . . . . . . . . . . . . . . 136 12. Fermi Sea Dynamics and Collective Field Theory . . . . . . . . . . . . . . . . 139 12.1. Time dependent Fermi Sea . . . . . . . . . . . . . . . . . . . . . . . . 139 12.2. Collective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 140 12.3. Relation to 1+1 dimensional relativistic field theory . . . . . . . . . . . . . 142 12.4. τ-space and φ-space . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 12.5. The w ∞ Symmetry of the Harmonic Oscillator . . . . . . . . . . . . . . . 146 12.6. The w ∞ Symmetry of Free Field Theory . . . . . . . . . . . . . . . . . . 148 12.7. w ∞ symmetry of Classical Collective Field Theory . . . . . . . . . . . . . . 149 13. String scattering in two spacetime dimensions . . . . . . . . . . . . . . . . . 151 13.1. Definitions of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . 151 13.2. On the Violation of Folklore . . . . . . . . . . . . . . . . . . . . . . . 155 13.3. Classical scattering in collective field theory . . . . . . . . . . . . . . . . 156 13.4. Tree-Level Collective Field Theory S-Matrix . . . . . . . . . . . . . . . . 158 13.5. Nonperturbative S-matrices . . . . . . . . . . . . . . . . . . . . . . . 159 13.6. Properties of S-Matrix Elements . . . . . . . . . . . . . . . . . . . . . 163 13.7. Unitarity of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . 165 2
Page 1: YCTP-P23-92 LA-UR-92-3479 hep-th/93
Page 5 and 6: contexts. The aim of these lectures
Page 7 and 8: 1) As Quantum Gravity In this guise
Page 9 and 10: semiclassical, and quantum Liouvill
Page 11 and 12: The Lagrangian and Hamiltonian form
Page 13 and 14: mation, on an annulus. This is give
Page 15 and 16: Quantum mechanically, we try to und
Page 17 and 18: about the domain of integration. Th
Page 19 and 20: diffeomorphism invariance, (2.11) s
Page 21 and 22: quantum gravity [31], so it would b
Page 23 and 24: To determine ∆, we employ the sam
Page 25 and 26: The stress-energy tensor following
Page 27 and 28: Note that the uniformizing map (3.1
Page 29 and 30: In particular, passing from the pla
Page 31 and 32: Exercise. The wall analogy Show tha
Page 33 and 34: Here ν is a real number. An import
Page 35 and 36: 5) As will be seen in sec. 5.4 belo
Page 37 and 38: where ∆ i is the conformal weight
Page 39 and 40: where j k ≡ −α k /γ. Strictly
Page 41 and 42: where |Σ〉 is a state created by
Page 43 and 44: equation with sources, (3.42). We s
Page 45 and 46: Exercise. Show that (3.70) is an id
Page 47 and 48: In view of these observations, the
Page 49 and 50: 3.11. Surfaces with boundaries The
Page 51 and 52: 4. 2D Euclidean Quantum Gravity II:
Page 53 and 54:
4.3. KPZ states in 2D Quantum Gravi
Page 55 and 56:
In the KP formalism of the matrix m
Page 57 and 58:
In fact, the c = 1 model has much m
Page 59 and 60:
We may plot the quantum numbers of
Page 61 and 62:
On the RHS we have one, rather than
Page 63 and 64:
5.1. Particles in D Dimensions: QFT
Page 65 and 66:
where H is the Wheeler-DeWitt opera
Page 67 and 68:
Euclidean signature spacetimes. The
Page 69 and 70:
5.4. 2D String Theory: Minkowskian
Page 71 and 72:
2) The only difference in degrees o
Page 73 and 74:
Those who look for special states i
Page 75 and 76:
Fig. 10: The case of the exploding
Page 77 and 78:
Fig. 11: A piece of a random triang
Page 79 and 80:
Since ∂ /2 eJ2 ∂J = Je J2 /2 ,
Page 81 and 82:
(6.8). As familiar from field theor
Page 83 and 84:
continuum, as reviewed in preceding
Page 85 and 86:
7. Matrix Model Technology I: Metho
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with r n a scalar coefficient indep
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gravity. It is natural that pure gr
Page 91 and 92:
with α = 1 10 . The solution to (7
Page 93 and 94:
parameters to result in a (2l−1)
Page 95 and 96:
and the recursion relations for thi
Page 97 and 98:
The formal expansion of Q l−1/2 =
Page 99 and 100:
If we consider the higher operators
Page 101 and 102:
of the symmetry factor for the Feyn
Page 103 and 104:
1) The expansions in V and ζ −1
Page 105 and 106:
V′ ( ) + + + . . . ∂ ∂L L =
Page 107 and 108:
and the main observation is 〈∏
Page 109 and 110:
2 λ Fig. 17: The Wigner semicircle
Page 111 and 112:
Example. The Gaussian potential We
Page 113 and 114:
Exercise. Using the asymptotics of
Page 115 and 116:
The formulae (10.1) and (10.2), whi
Page 117 and 118:
10.2. Loops to Local Operators By s
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This suggests that instead of the l
Page 121 and 122:
Fig. 20: Two loops on a continuum s
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Using Feynman diagrams to obtain th
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continuous and particle states are
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Here λ 1,2 are the two turning poi
Page 129 and 130:
As we have seen from the tree-level
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Laplace transform the eigenvalue de
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The perturbative expansion of the H
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11.5. Wavefunctions and Wheeler-DeW
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x Fig. 24: The function x 2 K 0 (x)
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Remarks: 1) The definition (11.59)
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p λ Fig. 25: A generic initial con
Page 143 and 144:
anches can very well evolve into on
Page 145 and 146:
It follows that if we define a nonl
Page 147 and 148:
12.5. The w ∞ Symmetry of the Har
Page 149 and 150:
The w ∞ symmetry of the inverted
Page 151 and 152:
coadjoint orbit quantization for a
Page 153 and 154:
egarded as space, the time coordina
Page 155 and 156:
The equivalence of the collective f
Page 157 and 158:
Kondo effect, the Callan-Rubakov ef
Page 159 and 160:
13.4. Tree-Level Collective Field T
Page 161 and 162:
presence of a wall. The function R
Page 163 and 164:
ω q R(µ + ω ; V ) R q q > 0 −
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Property (i) follows from the integ
Page 167 and 168:
The key observation is that the com
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(13.44) is given by F = F 0 + 1 µ
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and in (13.51) we only keep terms t
Page 173 and 174:
e) The c = 1 model is equivalent to
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The expansion is only convergent fo
Page 177 and 178:
Example. Let us consider the most g
Page 179 and 180:
free-field techniques. But this cal
Page 181 and 182:
Quite generally, the operator produ
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x p−1 ∼ y q−1 ∼ 1 (where C(
Page 185 and 186:
spin j even at the self-dual radius
Page 187 and 188:
15.2. Disappointments From the quan
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We thank especially N. Seiberg for
Page 191 and 192:
References [1] M. B. Green and J. S
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[36] G. Moore, N. Seiberg, and M. S
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[76] S.R. Das and A. Jevicki, “St
Page 197 and 198:
Phys. B368 (1992) 671; S. Jain and
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[145] E. Martinec and S. Shatashvil
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arXiv:hep-th/9304011 v1 Apr 5 1993

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