Regge-Liouville action from group theory

TL;DR

This paper derives constraints from group invariance principles on a discretized Liouville action, fixing its form and extending the approach to supersymmetric cases, with implications for quantum gravity models.

Contribution

It provides a group-theoretic derivation of the discretized Liouville action, fixing its form and extending the framework to supersymmetric theories.

Findings

Constraints from SL(2,C) invariance fix the discretized action.

Constraints from modular invariance further determine the action.

The approach can be extended to supersymmetric Liouville theory.

Abstract

We work out the constraints imposed by SL(2C) invariance for sphere topology and modular invariance for torus topology, on the discretized form of Liouville action in Polyakov's non local covariant form. These are sufficient to completely fix the discretized action except for the overall normalization constant and a term which in the continuum limit goes over to a topological invariant. The treatment can be extended to the supersymmetric case.