Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models

TL;DR

This paper introduces a formulation of 2D quantum gravity using Liouville action on Riemann spheres with various points, revealing connections between conformal weights, ramification, and minimal models in conformal field theory.

Contribution

It proposes a novel approach to 2D quantum gravity incorporating parabolic and elliptic points, linking classical limits to Fuchsian projective connections and minimal model series.

Findings

Relation between conformal weights and ramification indices.

Constraints on the parameter d matching minimal series.

Formulation applicable for arbitrary d with standard representation for d≤1.

Abstract

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 $\Sigma_{0,m+n}$ 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 $d\le 1$. 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.