arXiv:hep-th/9304011 v1 Apr 5 1993

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15.3. Future prospects and Open Problems Singularity is almost invariably a clue. — Sherlock Holmes Each paragraph in the text marked with the “dangerous bend sign” represents an opportunity. • The quantum Liouville theory remains unsolved, and is still needed to calculate answers to many physics questions, so major surprises remain possible. • We need a better understanding of backgrounds. At present, we seem to have an infinite dimensional manifold of solutions to string theory, and an infinitely large class of superselection sectors. Are all these solutions related by some symmetry? • Can we use these backgrounds to understand anything about time-dependence in string theory? • Natural nonperturbative definitions of 2D string theory and 2D gravity are still lacking! One might have hoped that imposing some physical criterion such as unitarity would strongly constrain the possible nonperturbative definitions of the theory, but this does not occur in the case of the c = 1 model coupled to gravity. There we found infinitely many nonperturbative completions all of which seem perfectly natural from the matrix model point of view, and we thus obtain little guidance in this regard. • Can the comprehensive picture of the c < 1 backgrounds, unified via the KP formalism, be generalized to the case of 2D string backgrounds? Is there e.g. a multiparameter space of theories which encompasses both the black hole and c = 1 spacetimes? Finding a unified picture of all 2D or c ≤ 1 backgrounds remains an interesting open problem. • We need to find new ways of cancelling the tachyonic divergences of string theory — i.e., of making sense of the integrals over moduli spaces. This is essentially the problem of going beyond the “c = 1 barrier.” • Does the c = 1 model teach us how to understand better the covariant closed string field theory of [44]? • One of the great open puzzles in the subject is the absence of backreaction on the metric and other “special state” degrees of freedom, and in particular, the role of 2D black holes in c = 1 string theory. • Are there interesting supersymmetric extensions of the theories we consider here (i.e. with potentially interesting spacetime properties such as the construction of [170])? Acknowledgements 187
We thank especially N. Seiberg for a long series of collaborative efforts on the subject of 2d gravity. For commentary on various portions of the manuscript we would like to thank N. Seiberg and M. Staudacher. We also would like to thank many people for discussions and for teaching us much of the above material. In particular we thank T. Banks, R. Dijkgraaf, M. Douglas, J. Horne, C. Itzykson, I. Klebanov, D. Kutasov, B. Lian, E. Martinec, R. Plesser, S. Ramgoolam, H. Saleur, G. Segal, N. Seiberg, R. Shankar, S. Shatashvili, S. Shenker, M. Staudacher, A.B. Zamolodchikov, G. Zuckerman, and B. Zwiebach. GM is supported by DOE grant DE-AC02-76ER03075 and by a Presidential Young Investigator Award, and PG by DOE contract W-7405-ENG-36. Appendix A. Special functions A.1. Parabolic cylinder functions Unfortunately, there are four notations commonly used for parabolic cylinder functions [171,172]. Our wavefunctions ψ ± (a, x) are the δ-function normalized even and odd solutions of ( d2 dx 2 + x2 4 )ψ = aψ. In terms of degenerate hypergeometric 1F 1 (α, β; x) and Whittaker functions M µ,ν (x), D a (x), we have even and odd parity wavefunctions: = ψ + 1 (a, x) = √ (W (a, x) + W (a, −x)) 4π(1 + e 2πa ) 1/2 ∣ 1 ∣∣∣ 1/2 Γ(1/4 + ia/2) √ 4π(1 + e 2πa ) 1/2 21/4 Γ(3/4 + ia/2) ∣ e −ix2 /4 1 F 1 (1/4 − ia/2; 1/2; ix 2 /2) = e−iπ/8 2π 1 e−aπ/4 |Γ(1/4 + ia/2)| √ M ia/2,−1/4 (ix 2 /2) , |x| (A.1) = ψ − 1 (a, x) = √ (W (a, x) − W (a, −x)) 4π(1 + e 2πa ) 1/2 ∣ 1 ∣∣∣ 1/2 Γ(3/4 + ia/2) √ 4π(1 + e 2πa ) 1/2 23/4 Γ(1/4 + ia/2) ∣ xe −ix2 /4 1 F 1 (3/4 − ia/2; 3/2; ix 2 /2) = e−3iπ/8 π e −aπ/4 x |Γ(3/4 + ia/2)| |x| M ia/2,1/4(ix 2 /2) . 3/2 (A.2) 188
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YCTP-P23-92 LA-UR-92-3479 hep-th/93
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7.1. Orthogonal polynomials . . . .
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contexts. The aim of these lectures
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1) As Quantum Gravity In this guise
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semiclassical, and quantum Liouvill
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The Lagrangian and Hamiltonian form
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mation, on an annulus. This is give
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Quantum mechanically, we try to und
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about the domain of integration. Th
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diffeomorphism invariance, (2.11) s
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quantum gravity [31], so it would b
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To determine ∆, we employ the sam
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The stress-energy tensor following
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Note that the uniformizing map (3.1
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In particular, passing from the pla
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Exercise. The wall analogy Show tha
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Here ν is a real number. An import
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5) As will be seen in sec. 5.4 belo
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where ∆ i is the conformal weight
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where j k ≡ −α k /γ. Strictly
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where |Σ〉 is a state created by
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equation with sources, (3.42). We s
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Exercise. Show that (3.70) is an id
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In view of these observations, the
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3.11. Surfaces with boundaries The
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4. 2D Euclidean Quantum Gravity II:
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4.3. KPZ states in 2D Quantum Gravi
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In the KP formalism of the matrix m
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In fact, the c = 1 model has much m
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We may plot the quantum numbers of
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On the RHS we have one, rather than
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5.1. Particles in D Dimensions: QFT
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where H is the Wheeler-DeWitt opera
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Euclidean signature spacetimes. The
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5.4. 2D String Theory: Minkowskian
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2) The only difference in degrees o
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Those who look for special states i
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Fig. 10: The case of the exploding
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Fig. 11: A piece of a random triang
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Since ∂ /2 eJ2 ∂J = Je J2 /2 ,
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(6.8). As familiar from field theor
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continuum, as reviewed in preceding
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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
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with α = 1 10 . The solution to (7
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parameters to result in a (2l−1)
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and the recursion relations for thi
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The formal expansion of Q l−1/2 =
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If we consider the higher operators
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of the symmetry factor for the Feyn
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1) The expansions in V and ζ −1
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V′ ( ) + + + . . . ∂ ∂L L =
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and the main observation is 〈∏
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2 λ Fig. 17: The Wigner semicircle
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Example. The Gaussian potential We
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Exercise. Using the asymptotics of
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The formulae (10.1) and (10.2), whi
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10.2. Loops to Local Operators By s
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This suggests that instead of the l
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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
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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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Page 197 and 198: Phys. B368 (1992) 671; S. Jain and
Page 199 and 200: [145] E. Martinec and S. Shatashvil
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arXiv:hep-th/9304011 v1 Apr 5 1993

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