arXiv:hep-th/9304011 v1 Apr 5 1993

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In matrix notation, we write this as λΠ = QΠ, where the matrix Q has components Q nm = √ r m δ m,n+1 + √ r n δ m+1,n . (7.22) Due to the orthonormality property (7.21), we see that ∫ e −V λΠ n Π m = Q nm = Q mn , and Q is a symmetric matrix. In the continuum limit, Q will therefore become a hermitian operator. To see how this works explicitly [34,104], we substitute the scaling ansatz r(ξ) = r c + a 2/m u(z) for the m th multicritical model into (7.22), Q → (r c + a 2/m u(z)) 1/2 e ε ∂ ∂ξ + e −ε ∂ ∂ξ (rc + a 2/m u(z)) 1/2 . With the substitution ε ∂ ∂ξ → −ga1/m ∂ ∂z , we find the leading terms Q = 2r 1/2 c + a2/m √ rc (u + r c κ 2 ∂ 2 z ) , (7.23) of which the first is a non-universal constant and the second is a hermitian 2 nd order differential operator. The other matrix that naturally arises is defined by differentiation, ∂ ∂λ Π n = A nm Π m , (7.24) and automatically satisfies [A, Q] = 1. The matrix A does not have any particular symmetry or antisymmetry properties so it is convenient to correct it to a matrix P that satisfies the same commutator as A. From our definitions, it follows that ∫ ∂ ( 0 = Πn Π m e ) −V ⇒ A + A T = V ′ (Q) , ∂λ where we have differentiated term by term and used ∫ e −V λ l Π n Π m = (Q l ) nm . The matrix P ≡ A − 1 2 V ′ (Q) = 1 2 (A − AT ) is therefore anti-symmetric and satisfies [ P, Q ] = 1 . (7.25) To determine the order of the differential operator Q in the continuum limit, let us assume for example that the potential V is of order 2l, i.e. V = ∑ l k=0 a k λ 2k . For m > n, the integral A mn = ∫ e −V Π n ∂ ∂λ Π m = ∫ e −V V ′ Π n Π m may be nonvanishing for m − n ≤ 2l − 1. That means that P mn ≠ 0 for |m − n| ≤ 2l − 1, and thus has enough 91
parameters to result in a (2l−1) st order differential operator in the continuum. The single condition W ′ = 0 results in P tuned to a 3 rd order operator, and the l − 1 conditions W ′ = . . . = W (l−1) = 0 allow P to be realized as a (2l−1) st order differential operator. In (7.23), we see that the universal part of Q after suitable rescaling takes the form Q = d 2 −u. For the simple critical point W ′ = 0, the continuum limit of P is the antihermitian operator P = d 3 − 3 4 {u, d}, and the commutator 1 = [P, Q] = 4R ′ 2 = ( 3 4 u2 − 1 4 u′′ ) ′ (7.26) is easily integrated with respect to z to give an equation equivalent to (7.17), the string equation for pure gravity. In (7.26), the notation R 2 is conventional for the first member of the ordinary KdV hierarchy. The emergence of the KdV hierarchy in this context is due to the natural occurrence of the fundamental commutator relation (7.25), which also occurs in the Lax representation of the KdV equations. (The topological gravity approach has as well been shown at length to be equivalent to KdV, for a review see [105,106].) In general the differential equations [P, Q] = 1 (7.27) that follow from (7.25) may be determined directly in the continuum. Given an operator Q, the differential operator P that can satisfy this commutator is constructed as a “fractional power” of the operator Q. This method of formulating the continuum theory has a beautiful generalization to a larger class of theories, which is defined in the following two sections. 7.5. Ising Model The first extension of the method of orthogonal polynomials occurs in the solution of the Ising model. The partition function of the Ising model on a random surface can be formulated using the two-matrix model: e Z ∫ = dU dV e −tr( U 2 + V 2 − 2c UV + g N (eH U 4 + e −H V 4 ) ) , (7.28) where U and V are hermitian N × N matrices and H is a constant. In the diagrammatic expansion of the right hand side, we now have two different quartic vertices of the type depicted in fig. 13b, corresponding to insertions of U 4 and V 4 . The propagator is determined by the inverse of the quadratic term, ( ) −1 1 −c = 1 ( ) 1 c −c 1 1 − c 2 c 1 92 .
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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
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
Page 87 and 88: with r n a scalar coefficient indep
Page 89 and 90: gravity. It is natural that pure gr
Page 91: with α = 1 10 . The solution to (7
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
Page 119 and 120: This suggests that instead of the l
Page 121 and 122: Fig. 20: Two loops on a continuum s
Page 123 and 124: Using Feynman diagrams to obtain th
Page 125 and 126: continuous and particle states are
Page 127 and 128: Here λ 1,2 are the two turning poi
Page 129 and 130: As we have seen from the tree-level
Page 131 and 132: Laplace transform the eigenvalue de
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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anches can very well evolve into on
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It follows that if we define a nonl
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12.5. The w ∞ Symmetry of the Har
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The w ∞ symmetry of the inverted
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coadjoint orbit quantization for a
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egarded as space, the time coordina
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The equivalence of the collective f
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Kondo effect, the Callan-Rubakov ef
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13.4. Tree-Level Collective Field T
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presence of a wall. The function R
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ω q R(µ + ω ; V ) R q q > 0 −
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Property (i) follows from the integ
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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
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e) The c = 1 model is equivalent to
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The expansion is only convergent fo
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Example. Let us consider the most g
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free-field techniques. But this cal
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Quite generally, the operator produ
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x p−1 ∼ y q−1 ∼ 1 (where C(
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spin j even at the self-dual radius
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15.2. Disappointments From the quan
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We thank especially N. Seiberg for
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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
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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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