arXiv:hep-th/9304011 v1 Apr 5 1993

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particle − # hole number, i.e., nonzero soliton sectors, then nonperturbative unitarity will be restored. A target space string interpretation of the solitons would be quite interesting. 2) By making small perturbations of the matrix model potential fig. 21, we can produce infinitely many nonperturbatively unitary completions of the string S-matrix [132]. In other words, the requirement of nonperturbative unitarity is a very weak constraint on nonperturbative formulations of string theory. Strangely, the situation is opposite to that of unitary c < 1 models coupled to gravity, where no satisfactory nonperturbative definitions exist. In either case, we see that matrix models have been somewhat disappointing as a source of nonperturbative physics. 13.8. Generating functional for S-matrix elements The key formula (13.41) leads to a concise generating functional for all S-matrix elements [141]. A very intriguing aspect of this formula is that it involves the asymptotic conformal field theory in spacetime in a natural way. We have mentioned above that the collective field theory, or equivalently the spacetime tachyon theory T (φ, t), is asymptotically a conformal field theory. In fact there are two asymptotic conformal field theories corresponding to the two different null infinities I ± in the past and the future. According to (13.41), the entire content of broken conformal invariance in the interior is summarized by the potential scattering of fermions: a(E) out = R(E)a(E) in = S −1 a(E) in S (∫ ∞ S ≡ exp dE log ( R(E) )( a † (E) a(E) ) ) in −∞ . (13.42) As we have noted, unitarity of the S-matrix is equivalent to the identity R(E)R(E) ∗ = 1 on the reflection factors. We may use (13.42) to summarize the entire S-matrix as follows. Define vertex operators with normalization Ṽ ± ω = Γ(−iω) Γ(iω) µ1+iω/2 V ± ω (13.43) relative to the normalization of (5.22), and define the generating functional µ 2 F [ t(ω), ¯t(ω) ] ≡ 〈〈 e ∫ ∞ 0 dω t(ω)Ṽ + ω e ∫ ∞ 0 dω ¯t(ω)Ṽ − ω 〉〉 c , (13.44) where 〈〈. . .〉〉 indicates a sum over genus and integral over moduli space, ∑ h≥0 κ−χ ∫ M h,n (as in (11.61)), and the subscript c indicates the connected part. The genus expansion of 167
(13.44) is given by F = F 0 + 1 µ 2 F 1 + · · ·, and thus by KPZ scaling, combining (13.41) and (13.42) we have the formula [141]: µ 2 F [ t(ω), ¯t(ω) ] ∫ = 〈0|e µ ∞ 0 ∫ dω t(ω)α(−ω) · S · e µ ∞ dω ¯t(ω)α(ω) 0 |0〉 c . (13.45) The expression (13.45) has a simple compactified Euclidean space analog. If we take the Euclidean coordinate X to have finite radius β, then from [142,143], we see that the only modification in (11.38) is that bosonic momenta instead lie on a lattice q ∈ 1 β Z and the fermions, now interpreted as being at finite temperature 1 β , have Matsubara frequencies 1 β ( Z + 1 2 ). τ = ∞ τ = 0 . (matrix model wall) Fig. 33: The Euclidean spacetime of the matrix model in natural coordinates. Note that the asymptotic conformal field theory on spacetime is concentrated in the “ultraviolet” region at the center of the disk. The analytic continuation of the asymptotically conformal collective field is given by the standard c = 1 scalar field ∂φ in/out (z) = ∑ n α in/out n z −n−1 , (13.46) where z = e −τ+iX , so that the Euclidean spacetime in the z-plane looks as in fig. 33. In particular, the bosonization becomes the standard one with Weyl fermions in the Neveu-Schwarz sector: ψ(z) = ∑ ψ m+ 1 z −m−1 ψ(z) = ∑ ψ 2 m+ 1 z−m−1 2 m∈ Z m∈ Z {ψ r , ψ s } = δ r+s,0 ∂φ = ψ(z) ¯ψ(z) . (13.47) The Euclidean analog of (13.42) is ψ in −(m+ 1 2 ) = R(µ + ip m) ψ out −(m+ 1 2 ) ¯ψ in −(m+ 1 2 ) = R(µ − ip m) ∗ ¯ψout −(m+ 1 2 ) . (13.48) 168
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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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Page 123 and 124: Using Feynman diagrams to obtain th
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Page 133 and 134: The perturbative expansion of the H
Page 135 and 136: 11.5. Wavefunctions and Wheeler-DeW
Page 137 and 138: x Fig. 24: The function x 2 K 0 (x)
Page 139 and 140: Remarks: 1) The definition (11.59)
Page 141 and 142: p λ Fig. 25: A generic initial con
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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
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Page 157 and 158: Kondo effect, the Callan-Rubakov ef
Page 159 and 160: 13.4. Tree-Level Collective Field T
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Page 165 and 166: Property (i) follows from the integ
Page 167: The key observation is that the com
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 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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