Quantum Riemann Surfaces, 2d Gravity and the Geometrical Origin of Minimal Models

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere Σ0,m+n with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on Σ0,m+n implies a relation between conformal weights and ramification indices. This formulation works for arbitrary d and admits a standard representation only for d ≤ 1. Furthermore, it turns out that the integerness of the ramification number constrains d = 1− 24/(n − 1) that for n = 2m+ 1 coincides with the unitary minimal series of CFT. Partly supported by the European Community Research Programme ‘Gauge Theories, applied supersymmetry and quantum gravity’, contract SC1-CT92-0789 e-mail: [email protected], mvxpd5::matone 1. Recently in [1] it has been developed an approach to quantum Liouville theory based on the original proposal by Polyakov [2]. The basic object in this theory is the ‘partition function of Σ0,n’ with Σ0,n the Riemann sphere punctured at z1, . . . , zn−1 and zn = ∞ 〈Σ0,n〉 = ∫ C(Σ0,n) Dφe 1 2πhS(0,n)(φ), (1) where the measure is defined with respect to the scalar product ||δφ||2 = Σ0,n eφ|δφ|2, and the integration is performed on the φ’s such that e be a smooth metric on Σ0,n with asymptotic behaviour at the punctures given by the Poincaré metric ecl (see (8)). The functional S denotes the Liouville action S(φ) = lim r→0 S r (φ) = lim r→0 [∫ Σr ( ∂zφ∂z̄φ+ e φ ) + 2π(nlogr + 2(n− 2)log|logr|) ] , (2) where Σr = Σ0,n\ (⋃n−1 i=1 {z||z − zi| < r} ∪ {z||z| > r−1} ) and z is the global coordinate on Σ0,n. An important remark in [1] is that by SL(2,C)-symmetry one gets the exact result 〈Σ0,3〉 = c |z1 − z2|1/h , Σ0,3 = C\{z1, z2}, c = 〈C\{0, 1}〉, (3) which can be interpreted as correlation function of puncture operators e of conformal weight ∆ = ∆ = 1/2h. In [1], after fixing the standard normalization zn−2 = 0, zn−1 = 1, zn = ∞, it is assumed that the theory defined by (1) satisfies the conformal Ward identity 〈T (z)Σ0,n〉 = [ n−1 ∑ i=1 ∆ (z − zi) + n−3 ∑ i=1 ( 1 z − zi + zi − 1 z − zi z − 1 ) ∂ ∂zi ] 〈Σ0,n〉, (4) where T = (φzz − 12φz)/h is the Liouville stress tensor. Eq.(4) is verified at the tree level where ∆cl = 1/2h = ∆ implying that ∆loops = 0. (5) Remarkably, in considering the tree level of (4) one uses [1–3] the well-known relation between the accessory parameters and the classical Liouville action [4] ci = − 1 2π ∂S (0,n) cl ∂zi . Expanding around the Poincaré metric ecl, we obtain the semiclassical approximation [1] log〈Σ0,n〉 = − 1 2πh S (0,n) cl − 1 2 log det(2∆ + 1) +O(h), (6) Note that in getting the second term in (6) one identifies { φcl + ∑ k akψk ∣ak ∈ R } , where the ψk’s are the eigenfunctions of ∆, with the space C (Σ0,n).