QUANTUM RIEMANN SURFACES, 2-D GRAVITY AND THE GEOMETRICAL ORIGIN OF MINIMAL MODELS

Based on a recent paper by Takhtajan, we propose a formulation of 2-D quantum gravity whose basic object is the Liouville action on the Riemann sphere Sigma(0,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 Sigma(0,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 less than or equal to 1. Furthermore, it turns out that the integerness of the ramification number constrains d = 1 - 24/(n(2) - 1) that for n = 2m + 1 coincides with the unitary minimal series of CFT.