Localization of strings on group manifolds

TL;DR

This paper develops a novel method using supersymmetric localization to compute the partition function of WZW models on group manifolds, providing explicit formulas and verifying results for specific groups like SU(2).

Contribution

It introduces a new approach to compute WZW partition functions via supersymmetric localization, applicable to various group manifolds including non-compact cases.

Findings

Derived a formula for the WZW partition function as a sum over classical solutions.

Verified the formula matches the Weyl-Kac character formula for SU(2).

Extended the method to SL(2,R) and hyperbolic target spaces.

Abstract

We compute the partition function of the WZW model with target a compact Lie group $G$ by adapting a method used by Choi and Takhtajan to compute the heat kernel of the group manifold. The basic idea is to compute the partition function of a supersymmetric version of the WZW model using a form of supersymmetric localization and then use the fact that, since the fermions of the supersymmetric WZW model are actually decoupled from the bosons, this also determines the partition function of the purely bosonic WZW model. The result is a formula for the partition function as a sum over contributions from abelian classical solutions. We verify for $G=SU(2)$ that this formula agrees with the result for the same partition function that comes from the Weyl-Kac character formula. We extend the method of supersymmetric localization to certain related models such as the $SL(2,\mathbb{R})$ WZW model and a Wick-rotated version of this model in which the target space is hyperbolic three-space $H_3^+$.