Path Integral Formulation of the Conformal Wess-Zumino-Witten to Toda Reductions

TL;DR

This paper develops a conformally invariant path integral formulation for Wess-Zumino-Witten to Toda reductions, utilizing the Batalin-Fradkin-Vilkovisky method to elucidate the role of zero modes and anomalies.

Contribution

It introduces a conformally invariant path integral approach to WZW to Toda reductions, highlighting the significance of zero modes and anomalies in gauge invariance.

Findings

Zero modes produce the Toda potential.

Gradients produce the WZW anomaly.

Anomaly is key for gauge invariance proof.

Abstract

The phase space path integral Wess-Zumino-Witten $\to$ Toda reductions are formulated in a manifestly conformally invariant way. For this purpose, the method of Batalin, Fradkin, and Vilkovisky, adapted to conformal field theories, with chiral constraints, on compact two dimensional Riemannian manifolds, is used. It is shown that the zero modes of the Lagrange multipliers produce the Toda potential and the gradients produce the WZW anomaly. This anomaly is crucial for proving the Fradkin-Vilkovisky theorem concerning gauge invariance.