The Liouville Theory and SL(2,R) Symmetry

TL;DR

This paper discusses the role of Liouville theory as an effective action in 2D quantum gravity and explores the residual SL(2,R) symmetry after gauge fixing, highlighting quantum symmetry violations.

Contribution

It analyzes the residual SL(2,R) invariance in gauge-fixed Liouville theory and its implications for 2D quantum gravity.

Findings

Liouville action arises as the effective 2D gravity action.

Quantum effects cause violations of classical symmetries.

Residual SL(2,R) invariance persists in gauge-fixed theory.

Abstract

The Liouville action emerges as the effective action of 2-d gravity in the process of path integral quantization of the bosonic string. It yields a measure of the violation of classical symmetries of the theory at the quantum level. Certain aspects of the residual SL(2,R) invariance of the gauge-fixed Liouville theory are disscussed. ( Lectures presented at the Danube Workshop '93, June 1993, Belgrade, Yugoslavia.)