Equivariant Kaehler Geometry and Localization in the G/G Model

TL;DR

This paper explores the equivariant supersymmetry of the $G/G$ model, demonstrating localization onto reducible connections and deriving the Verlinde formula through a cohomological framework related to Bismut's theory.

Contribution

It provides a novel cohomological interpretation of the $G/G$ model's supersymmetry and establishes a new connection to equivariant Bott-Chern currents on Kähler manifolds.

Findings

Localization onto reducible connections enables derivation of the Verlinde formula.

Supersymmetry relates to a q-deformation of the moment map in symplectic geometry.

Large k limit recovers the ordinary moment map of 2D gauge theories.

Abstract

We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents on K\"ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. We also show that the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group $G$ is replaced by a representation of its group algebra with commutator $[g,h] = gh-hg$. In the large $k$ limit it reduces to the ordinary moment map of two-dimensional gauge theories.