Defects in G/H coset, G/G topological field theory and discrete Fourier-Mukai transform

TL;DR

This paper constructs and analyzes defects in coset G/H theories, establishing their relation to Chern-Simons theory and proposing a categorical description involving a discrete Fourier-Mukai transform.

Contribution

It provides a canonical quantization of gauged WZW models with defects and links these to Chern-Simons theory, introducing a novel categorical perspective on defects in topological field theories.

Findings

Defects in G/H theories are related to Chern-Simons theory with Wilson lines.

G/G theory with defects corresponds to Chern-Simons on a torus with Wilson lines.

Proposes a categorical description of defects via a discrete Fourier-Mukai transform.

Abstract

In this paper we construct defects in coset $G/H$ theory. Canonical quantization of the gauged WZW model $G/H$ with $N$ defects on a cylinder and a strip is performed and the symplectomorphisms between the corresponding phase spaces and those of double Chern-Simons theory on an annulus and a disc with Wilson lines are established. Special attention to topological coset $G/G$ has been paid. We prove that a $G/G$ theory on a cylinder with $N$ defects coincides with Chern-Simons theory on a torus times the time-line $R$ with 2N Wilson lines. We have shown also that a $G/G$ theory on a strip with $N$ defects coincides with Chern-Simons theory on a sphere times the time-line $R$ with $2N+4$ Wilson lines. This particular example of topological field theory enables us to penetrate into a general picture of defects in semisimple 2D topological field theory. We conjecture that defects in this case described by a 2-category of matrices of vector spaces and that the action of defects on boundary states is given by the discrete Fourier-Mukai transform.