arXiv:hep-th/9304011 v1 Apr 5 1993

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and Eigenstates for Integrable Collective Field Theories” (hep-th/9202065); “Interacting Theory of Collective and Topological Fields in 2 Dimensions” (hep-th/9209036). [128] J. Avan and A. Jevicki,“String field actions from W-infinity” (hep-th/9111028), Mod. Phys. Lett. A7 (1992) 357. [129] S.R. Das, A. Dhar, G. Mandal, S. R. Wadia, “Gauge theory formulation of the c=1 matrix model: symmetries and discrete states” (hep-th/9110021) Int. J. Mod. Phys. A7 (1992) 5165; “Bosonization of nonrelativistic fermions and W ∞ algebra,” Mod. Phys. Lett. A7 (1992) 71; A. Dhar, G. Mandal, S. R. Wadia, “Classical Fermi fluid and geometric action for c = 1” (hep-th/9204028) IASSNS-HEP-91-89; “Non-relativistic fermions, coadjoint orbits of w ∞ and string field theory at c = 1” (hep-th/9207011), TIFR-TH-92-40. [130] D. Minic, J. Polchinski, and Z. Yang, “Translation-invariant backgrounds in 1+1 dimensional string theory,” Nucl. Phys. B362 (1991) 125. [131] D. Kutasov, E. Martinec, and N. Seiberg, “Ground Rings and Their Modules in 2D Gravity with c ≤ 1 Matter” (hep-th/9111048), Phys. Lett. B276 (1992) 437. [132] G. Moore, R. Plesser, and S. Ramgoolam, “Exact S-Matrix for 2D String Theory” (hep-th/9111035) Nucl. Phys. B377(1992)143. [133] B. Lee in “Methods in Field Theory,” Les Houches, 1975, section 6.3. [134] C. Callan, S. Giddings, J. Harvey, and A. Strominger, “Evanescent Black Holes” (hepth/9111056), Phys. Rev. D45 (1992) 1005. [135] Al. B. Zamolodchikov and A.B. Zamolodchikov, “Factorized S matrices in twodimensions as the exact solutions of certain relativistic quantum field models,” Ann. Phys. 120 (1979) 253. [136] N. Andrei and J. Loewenstein, Phys. Lett. 91B (1980) 401; V. Korepin, Theor. and Math. Physics 41 (1979) 953. [137] A.B. Zamolodchikov and Al.B. Zamolodchikov, “Massless factorized scattering and sigma models with topological terms,” Nucl. Phys. B379 (1992) 602. [138] Moore and Plesser, “Classical Scattering in 1 + 1 Dimensional String Theory” (hepth/9203060), to appear in Phys. Rev. D. [139] G. Moore, “Gravitational Phase Transitions and the Sine-Gordon Model,” Yale preprint YCTP-P1-92, hep-th/9203061. [140] N. Seiberg and S. Shenker, “A note on background independence” (hep-th/9201017), Phys. Rev. D45 (1992) 4581. [141] R. Dijkgraaf, G. Moore, and R. Plesser, “The Partition Function of 2D String Theory,” hep-th/9208031, submitted to Nucl. Phys. B [142] D.J. Gross and I. Klebanov, Nucl. Phys. B344 (1990) 475; Nucl. Phys. B354 (1991) 459. [143] I. Klebanov and D. Lowe, Nucl. Phys. B363 (1991) 543. [144] S. Cecotti and C. Vafa, Nucl. Phys. B367 (1991) 359. 197
[145] E. Martinec and S. Shatashvili, “Black hole physics and liouville theory,” Nucl. Phys. B368 (1992) 338. [146] M. Bershadsky and D. Kutasov, “Comment on gauged WZW theory,” Phys. Lett. B266 (1991) 345. [147] S. Das, “Matrix models and black holes” (hep-th/9210107), Mod. Phys. Lett. A8 (1993) 69; A. Dhar, G. Mandal, S. Wadia, “Stringy quantum effects in two-dimensional black hole” (hep-th/9210120), Mod. Phys. Lett. A7 (1992) 3703; T. Yoneya, “Matrix models and 2-d critical string theory: 2-D black hole by c = 1 matrix model” (hep-th/9211079), UT-KOMABA-92-13. [148] Jorge G. Russo, “Black hole formation in c = 1 string field theory” (hep-th/9211057), Phys. Lett. B300 (1993) 336. [149] Edward Witten, “Two-dimensional string theory and black holes” (hep-th/9206069), Lecture given at Conf. on Topics in Quantum Gravity, Cincinnati, OH, Apr 3–4, 1992. [150] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “String propagation in a black hole geometry,” Nucl. Phys. B371 (1992) 269. [151] S. Mukhi and C. Vafa, “Two dimensional black hole as a topological coset model of c = 1 string theory” (hep-th/9301083). [152] Edward Witten, “The n matrix model and gauged WZW models,” Nucl. Phys. B371 (1992) 191. [153] J. Distler and C. Vafa, in proceedings of Cargèse 1990 workshop; “A critical matrix model at c = 1,” Mod. Phys. Lett. A6 (1991) 259. [154] J. Minahan and A. Polychronakos, “Equivalence of Two Dimensional QCD and the c = 1 Matrix Model” (hep-th/9303153); M. Douglas, “Conformal Field Theory Techniques for Large N Group Theory” (hepth/9303159). [155] I. Klebanov and A. Pasquinucci, “Correlation functions from two-dimensional string Ward identities” (hep-th/9204052) PUPT-1313. [156] N. Sakai and Y. Tanii, “Operator product expansion and topological states in c = 1 Matter Coupled to 2D Gravity” (hep-th/9111049), Prog. Theor. Phys. Supp. 110 (1992) 117; “Factorization and topological states in c = 1 matter coupled to 2D gravity” (hep-th/9108027), Phys. Lett. B276 (1992) 41. [157] G. Minic and Z. Yang, “Is S = 1 for c = 1?” Phys. Lett. B274 (1992) 27. [158] D. Lowe, “Unitarity Relations in c = 1 Liouville Theory” (hep-th/9204084), Mod. Phys. Lett. A7 (1992) 2647. [159] M. Bershadsky and D. Kutasov, Phys. Lett. B274 (1992) 331-337; Nucl. Phys. B382 (1992) 213. [160] I.R. Klebanov and A.M. Polyakov, “Interactions of Discrete States in Two-Dimensional String Theory,” Mod. Phys. Lett. A6 (1991) 3273. 198
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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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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
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anches can very well evolve into on
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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 −
Page 165 and 166: Property (i) follows from the integ
Page 167 and 168: The key observation is that the com
Page 169 and 170: (13.44) is given by F = F 0 + 1 µ
Page 171 and 172: and in (13.51) we only keep terms t
Page 173 and 174: e) The c = 1 model is equivalent to
Page 175 and 176: 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
Page 183 and 184: 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
Page 189 and 190: We thank especially N. Seiberg for
Page 191 and 192: References [1] M. B. Green and J. S
Page 193 and 194: [36] G. Moore, N. Seiberg, and M. S
Page 195 and 196: [76] S.R. Das and A. Jevicki, “St
Page 197: Phys. B368 (1992) 671; S. Jain and
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