Path Integral Formulation of the Conformal Wess-Zumino-Witten to Liouville Reduction

TL;DR

This paper develops a path integral approach to connect the Wess-Zumino-Witten model with Liouville theory, highlighting the role of zero modes and resolving gauge dependence issues in the process.

Contribution

It introduces a phase space path integral formulation for the WZW to Liouville reduction, emphasizing zero modes' significance and solving gauge dependence problems.

Findings

Zero modes are crucial for Liouville potential and anomaly production.

The gauge dependence of the Virasoro center is resolved.

A consistent path integral framework for the reduction is established.

Abstract

The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved.