The universal gerbe, Dixmier-Douady class, and gauge theory

TL;DR

This paper explores the relationship between the Dixmier-Douady class and the curvature of determinant bundles in infinite-dimensional gauge theories, providing explicit formulas for Dirac operators on 3D manifolds.

Contribution

It clarifies the connection between the Dixmier-Douady class and determinant bundle curvature, offering explicit expressions in the context of Dirac operators on three-dimensional manifolds.

Findings

Explicit formulas for the Dixmier-Douady class and curvature forms.

Connection established between universal B-field and gauge theory.

Simplified expressions for Dirac operators on 3D manifolds.

Abstract

We clarify the relation between the Dixmier-Douady class on the space of self adjoint Fredholm operators (`universal B-field') and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple and explicit expression for both forms.