Dirac structures and Dixmier-Douady bundles

TL;DR

This paper establishes a connection between Dirac structures on vector bundles and Dixmier-Douady bundles, providing a functorial construction with applications to Lie groups and twisted Spin-c structures.

Contribution

It introduces a method to associate Dixmier-Douady bundles to Dirac structures, with functorial properties and applications to Lie groups and moment maps.

Findings

The spin Dixmier-Douady bundle over a compact Lie group is multiplicative.

A canonical twisted Spin-c-structure can be constructed on spaces with group-valued moment maps.

The construction has good functorial properties relative to Morita morphisms.

Abstract

A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum of V with its dual. We show how to associate to any Dirac structure a Dixmier-Douady bundle A, that is, a Z/2Z-graded bundle of C*-algebras with typical fiber the compact operators on a Hilbert space. The construction has good functorial properties, relative to Morita morphisms of Dixmier-Douady bundles. As applications, we show that the `spin' Dixmier-Douady bundle over a compact, connected Lie group (as constructed by Atiyah-Segal) is multiplicative, and we obtain a canonical `twisted Spin-c-structure' on spaces with group valued moment maps.