Equivariant Cohomology and Gauged Bosonic sigma-Models

TL;DR

This paper uses equivariant cohomology to analyze obstructions in gauging Wess-Zumino terms of bosonic sigma-models, providing new vanishing theorems and homological insights, especially for low-dimensional cases and specific target manifolds.

Contribution

It introduces a homological framework based on equivariant cohomology for gauging sigma-models, generalizes previous results, and constructs equivariant minimal models for complex target spaces.

Findings

Vanishing theorems guarantee no obstructions for low-dimensional models with certain target spaces.

Gauging of compact semisimple Lie groups in three-dimensional sigma-models is always possible under specified conditions.

Obstructions can be interpreted as classes in BRST cohomology, relating to anomalies.

Abstract

We re-examine the problem of gauging the Wess-Zumino term of a d-dimensional bosonic sigma-model. We phrase this problem in terms of the equivariant cohomology of the target space and this allows for the homological analysis of the obstruction. As a check, we recover the obstructions of Hull and Spence and also a generalization of the topological terms found by Hull, Rocek and de Wit. When the symmetry group is compact, we use topological tools to derive vanishing theorems which guarantee the absence of obstructions for low dimension (d<=4) but for a variety of target manifolds. For example, any compact semisimple Lie group can be gauged in a three-dimensional sigma-model with simply connected target space. When the symmetry group is semisimple but not necessarily compact, we argue in favor of the persistence of these vanishing theorems by making use of (conjectural) equivariant minimal models (in the sense of Sullivan). By way of persuasion, we construct by hand a few such equivariant minimal models, which may be of independent interest. We illustrate our results with two examples: d=1 with a symplectic target space, and d=2 with target space a Lie group admitting a bi-invariant metric. An alternative homological interpretation of the obstruction is obtained by a closer study of the Noether method. This method displays the obstruction as a class in BRST cohomology at ghost number 1. We comment on the relationship with consistent anomalies.