Bundle Gerbes Applied to Quantum Field Theory

TL;DR

This paper explores the application of bundle gerbes in quantum field theory, providing explicit computations and examples related to index theory, gauge anomalies, and string structures, thereby advancing geometric methods in quantum physics.

Contribution

It introduces new applications of bundle gerbes in quantum field theory, including explicit calculations of anomalies and the Dixmier-Douady class, and connects these to physical models like the WZW model.

Findings

Explicit computation of the Dixmier-Douady class for bundle gerbes.

Demonstration of how bundle gerbes relate to gauge anomalies and Schwinger terms.

Extension of the framework to fermions in external fields and string structures.

Abstract

This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson-Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier-Douady class of the associated bundle gerbe. The method works also in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the `existence of string structures' question. We conclude by showing how global Hamiltonian anomalies fit within this framework.