arXiv:hep-th/9304011 v1 Apr 5 1993

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(consistent with the required overall conformal invariance). It remains to determine the coefficient ˜c in (2.11). We have since rescaled ϕ, so we write instead e γϕ and determine γ by the requirement that the physical metric be g = ĝ e γϕ . Geometrically, this means that the area of the surface is represented by ∫ d 2 ξ √ ĝ e γϕ . γ is thereby determined by the requirement that e γϕ behave as a (1,1) conformal field (so that the combination d 2 ξ e γϕ is conformally invariant). For the energy-momentum tensor mentioned after (2.16), the conformal weight 4 of e γϕ is ∆(e γϕ ) = ∆(e γϕ ) = − 1 2γ(γ − Q) . (2.17) Requiring that ∆(e γϕ ) = ∆(e γϕ ) = 1 determines that Q = 2/γ + γ . (2.18) Using (2.16) and solving for γ then gives 5 γ = √ 1 (√ √ ) Q 25 − D − 1 − D = 12 2 − 1 √ Q 2 2 − 8 . (2.19) For spacetime embedding dimension d ≤ 1, we find from (2.16) and (2.19) that Q and γ are both real (with γ ≤ Q/2). The D ≤ 1 domain is thus where the Liouville theory is well-defined and most easily interpreted. For D ≥ 25, on the other hand, both γ and Q are imaginary. To define a real physical metric g = e γϕ ĝ, we need to Wick rotate ϕ → −iϕ. (This changes the sign of the kinetic term for ϕ. Precisely at D = 25 we can interpret X 0 = −iϕ as a free time coordinate. In other words, for a string naively embedded in 25 flat euclidean dimensions, the Liouville mode turns out to provide automatically a single timelike dimension, dynamically realizing a string embedded in 26 dimensional Minkowski spacetime. In general for D ≥ 25, we must have c Liouville < 1 and the kinetic term of the Liouville field changes sign. The conformal mode of the metric in Euclidean space is a “wrong sign” scalar field analogous to the conformal mode in four-dimensional Euclidean 4 Recall that ∆ is given by the leading term in the operator product expansion T (z) e γϕ(w) ∼ ∆ e γϕ /(z − w) 2 +. . . . Recall also that for a conventional energy-momentum tensor T = − 1 2 ∂ϕ∂ϕ, the conformal weight of e ipϕ is ∆ = ∆ = p 2 /2. 5 One method for choosing the root for γ is to make contact with the classical limit of the Liouville action. Note that the effective coupling in (2.14) goes as (25 − D) −1 so the classical limit is given by D → −∞. In this limit the above choice of root has the classical γ → 0 behavior. We shall discuss this issue further in the next chapter. 19
quantum gravity [31], so it would be useful to make sense of this case (if possible). Finally, in the regime 1 < D < 25, γ is complex, and Q is imaginary. As we shall see later in lurid detail, this problem is equivalent to the cosmological constant becoming a macroscopic state operator. Sadly, it is not yet known how to make sense of the Liouville approach for the regime of most physical interest. A useful critical exponent that can be calculated in this formalism is the string susceptibility Γ str . We write the partition function for fixed area A as ∫ Z(A) = Dϕ DX e −S ∫ δ( d 2 ξ √ ) ĝ e γϕ − A , (2.20) where for convenience we now group the ghost determinant and integration over moduli into DX. We define a string susceptibility Γ str by and determine Γ str by a simple scaling argument. Z(A) ∼ A (Γ str−2)χ/2−1 , A → ∞ , (2.21) (Note that for genus zero, we have Z(A) ∼ A Γ str−3 .) Under the shift ϕ → ϕ + ρ/γ for ρ constant, the measure in (2.20) does not change. The change in the action (2.15) comes from the term ∫ Q d 2 ξ √ ĝ 8π ˆR ϕ → Q ∫ d 2 ξ √ ĝ 8π ˆR ϕ + Q ∫ ρ d 2 ξ √ ĝ 8π γ ˆR . ∫ Substituting in (2.20) and using the Gauss-Bonnet formula 1 4π d 2 ξ √ ĝ ˆR = χ together with the identity δ(λx) = δ(x)/|λ| gives Z(A) = e −Qρχ/2γ−ρ Z(e −ρ A). We may now choose e ρ = A, which results in Z(A) = A −Qχ/2γ−1 Z(1) = A (Γ str−2)χ/2−1 Z(1) , and we confirm from (2.16) and (2.19) that Γ str = 2 − Q γ = 1 12( D − 1 − √ (D − 25)(D − 1) ) . (2.22) In the nomenclature of [21], so-called “minimal conformal field theories” (those with a finite number of primary fields) are specified by a pair of relatively prime integers (p, q) and have central charge D = c p,q = 1 − 6(p − q) 2 /pq. The unitary discrete series, for example, is the subset specified by (p, q) = (m + 1, m). After coupling to gravity, the general (p, q) model has critical exponent Γ str = −2/(p + q − 1). Notice that Γ str = −1/m for the values D = 1 − 6/m(m + 1) ) of central charge in the unitary discrete series. (In general, the m th 20
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: diffeomorphism invariance, (2.11) s
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 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
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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
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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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