$G/G$--Topological Field Theories by Cosetting $G_k$

TL;DR

This paper constructs and analyzes $G/G$ topological field theories derived from $G_k$ WZW models, revealing their spectrum, amplitudes, and connections to conformal blocks and fusion rules, with similarities to 2D gravity systems.

Contribution

It introduces a formulation of $G/G$ theories via Complex BRST cohomology and explores their spectral and amplitude properties, providing new insights into their structure and relations to other models.

Findings

Finite physical spectrum matches conformal blocks of $G_k$

Amplitudes are expressed through $G_k$ fusion rules

Complex BRST cohomology includes states of arbitrary ghost number

Abstract

$G/G$ topological field theories based on $G_k$ WZW models are constructed and studied. These coset models are formulated as Complex BRST cohomology in $G^c_k$, the complexified level $k$ current algebra. The finite physical spectrum corresponds to the conformal blocks of $G_k$ .The amplitudes for $G/G$ theories are argued to be given in terms of the $G_k$ fusion rules. The $G_k/G_k$ character is the Kac-Weyl numerator of $G_k$ and is interpreted as an index. The Complex BRST cohomology is found to contain states of arbitrary ghost number. Intriguing similarities of $G/G$ to $c\leq 1$ matter systems coupled to two dimensional gravity are pointed out.