On canonical quantization of the gauged WZW model with permutation branes

TL;DR

This paper performs canonical quantization of gauged WZW models with permutation branes, revealing their phase space structure as equivalent to certain Chern-Simons theories, thus linking boundary conditions to topological field theories.

Contribution

It establishes a symplectomorphism between the phase space of gauged WZW models with permutation branes and double Chern-Simons theories, extending understanding of boundary conditions in topological field theories.

Findings

Phase space of gauged WZW with permutation branes matches double Chern-Simons theory.

For G/G coset, phase space corresponds to Chern-Simons on a genus N-1 surface.

Results connect boundary conditions in WZW models to topological invariants.

Abstract

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the $N$-fold product of the gauged WZW model $G/H$ on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern-Simons theory on a sphere with $N$ holes times the time-line with $G$ and $H$ gauge fields both coupled to two Wilson lines. For the special case of the topological coset $G/G$ we arrive at the conclusion that the phase space of the $N$-fold product of the topological coset $G/G$ on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern-Simons theory on a Riemann surface of the genus $N-1$ times the time-line with four Wilson lines.