arXiv:hep-th/9304011 v1 Apr 5 1993

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From our experience with conformal field theories in chapt. 1, we may expect that if we insert a “macroscopic loop operator” W C (l) = δ ( ∮ l − dŝ e 1 γφ) 2 (3.86) C in the Liouville path integral and then shrink the circumference to zero, then W C (l) may be replaced by an infinite sum W (l) ∼ ∑ j≥0 l x j σ j , (3.87) where σ j are local operators which can couple to the boundary state created by (3.86). Exercise. Exponents in Loop Expansion Suppose that there is an expansion like (3.87) in which the operators σ j Liouville charge α j , i.e. σ j ∼ P(∂ ∗ φ, ∂ ∗ φ) e α j φ , have where P is a polynomial. Show that the exponents x j of (3.87) can then be found by a variant of the simple scaling argument we have used in (3.49) and earlier. Consider a Liouville path integral with the operator W (l) inserted, and shift φ, remembering to take into account the change in Euler character from shrinking the hole. Show that the path integral scales as e − 1 2 Q δφ+α j δφ+ 1 2 γ x j δφ , from which follows x j = Q/γ − 2α j /γ = 2 γ ( 1 2 Q − α j) . (3.88) Note that x j ≥ 0. It turns out that, because of the geometrical nature of Liouville theory, the expansion (3.87) is only valid under certain circumstances. This may be seen by a semiclassical study of amplitudes with loops [36], analogous to the semiclassical considerations above. The main results of this study are the following: 1) Let −sγ = ∑ α i − 1 2Qχ, where χ = 2 − 2h − B on a surface with h handles and B boundaries. If −sγ > 0, the l → 0 behavior of W (l) is equivalent to a sum of local operators. In particular, this is always the case if there are two or more loops on the surface (including the one that shrinks). 2) As noted in the above exercise, x j ≥ 0 for local operators. Coefficients of negative powers of l as l → 0 arise from small area divergences and are analytic in µ (and other coupling constants, in the context of 2D gravity). Therefore they are interpreted as arising from infinitesimal size surfaces, and such terms are classified as non-universal contributions when comparing with matrix model answers. 49
4. 2D Euclidean Quantum Gravity II: Canonical Approach It will be useful to work with the canonical approach to two dimensional gravity. In this chapter, we are led to introduce in particular some of the details of the algebraic (BRST) point of view (to be pursued further in secs. 14.4, 14.5 ). We hope that providing a common language will help bridge the schism between the algebraicists and the matrix model theorists, who are after all employing two complementary approaches to study the same subject. 4.1. Canonical Quantization of Gravitational Theories For a review of the canonical approach to Einsteinian general relativity, see [58,59]. Since gravitational theories are gauge theories, we are immediately led to study constrained dynamics. The canonical approach applies to spacetimes M which admit a space-time foliation: we assume there is a diffeomorphism Σ × IR → M, where Σ is a D-dimensional spacelike manifold. Choosing a unit normal n µ to the surface Σ, we may project the metric onto components parallel and perpendicular to the surface. The restriction (D) g of the metric g to the spatial surface defines the canonical coordinates, while the time–time and time–space components of the metric are expressed in terms of Lagrange multipliers for constraints of the theory (the “lapse” and “shift”). In Einsteinian general relativity, the constraints associated with the lapse and shift are the time–time and time–space components of the Einstein equations: G 00 = 0, G 0i = 0. In the canonical theory, these are the generators of time and space diffeomorphisms. In the canonical quantization of gravity, wavefunctions are functions of the spatial metric (and other fields in the theory): Ψ = Ψ[ (D) g, matter]. The requirement of gauge invariance states that wavefunctions are required to obey operator versions of the space and time diffeomorphism constraints. The Wheeler–DeWitt equation is the equation expressing the invariance under the generator of time-diffeomorphisms [60–62] and plays a fundamental role in the theory. In non-Einsteinian metric theories of gravity, including for example gravity in one and two spacetime dimensions, a completely analogous formulation may still be obtained by performing constrained quantization of a theory with diffeomorphism invariance. 50
Page 1 and 2: YCTP-P23-92 LA-UR-92-3479 hep-th/93
Page 3 and 4: 7.1. Orthogonal polynomials . . . .
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: 3.11. Surfaces with boundaries The
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 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
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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
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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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