Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

Abstract

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. 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.

The Gauging of Two-Dimensional Bosonic Sigma Models on World-Sheets With Defects

“…Our analysis will lead to a natural completion of the analysis of Ref. [GSW10] that accommodates the full-blown 2-categorial background of a general σ-model characterised in Refs. [FSW08,RS09,Sus11].…”

“…In Section 4, we discuss the behaviour of the gauged amplitudes under large gauge transformations, completing the analysis of global gauge anomalies for topologically trivial gauge fields performed in Ref. [GSW10] for σ-models without defects. In Section 5, we illustrate the preceding discussions on the example of the WZW model with defects.…”

“…[GSW10] in the simpler case of the mono-phase σ-model. Completing the earlier studies of the subject [JJMO90,HS91,FOS94a,FOS94b], that work also covered the case of gauge fields in nontrivial principal bundles of the symmetry group G and provided an exhaustive study of global gauge anomalies that may obstruct invariance under 'large' gauge transformations non-homotopic to the identity.…”

“…The interested reader may consult Ref. [GSW10] for a description of the physical background and for a more detailed account of the history of the topic. The main point of the present work is to extend the analysis of Ref.…”

“…The main point of the present work is to extend the analysis of Ref. [GSW10] to the case of multi-phase σ-models in which the different phases of the two-dimensional theory are separated by one-dimensional domain walls, called defect lines that may intersect at defect junctions. Our analysis will lead to a natural completion of the analysis of Ref.…”