Global gauge anomalies in two-dimensional bosonic sigma models

TL;DR

This paper analyzes global gauge anomalies in two-dimensional bosonic sigma models with Wess-Zumino terms, classifies these anomalies, and introduces equivariant gerbes to achieve anomaly-free gauge couplings, with applications to conformal field theory models.

Contribution

It provides a full classification of global gauge anomalies in these models and introduces equivariant gerbes to enable consistent gauge couplings.

Findings

Global anomalies can occur despite classical gauge invariance.

Equivariant gerbes can cancel anomalies and allow consistent gauge coupling.

Coset models are shown to be inconsistent with global anomalies.

Abstract

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. The obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.