Gerbes, (twisted) K-theory, and the supersymmetric WZW model

TL;DR

This paper explores the role of gerbes and twisted K-theory in quantum field theory, especially in symmetry breaking and the supersymmetric WZW model, highlighting their mathematical structures and physical implications.

Contribution

It introduces a comprehensive framework connecting gerbes, Dixmier-Douady classes, and twisted K-theory to symmetry breaking in quantum field theories and the supersymmetric WZW model.

Findings

Gerbes can be described via cohomology, local line bundles, or prolongation problems.

Central extensions of symmetry groups are related to gerbe structures.

Twisted K-theory classes are constructed using families of supercharges in the WZW model.

Abstract

The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, Dixmier-Douady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinite-dimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (broken) symmetry groups. We also discuss the construction of twisted K-theory classes by families of supercharges for the supersymmetric Wess-Zumino-Witten model.