Dirac Structures and Dixmier–Douady Bundles

A Dirac structure on a vector bundle V is a maximal isotropic sub-bundle E of the direct sum V ⊕V*. We show how to associate to any Dirac structure a Dixmier–Douady bundle $\mathcal A_{E}$, that is, a $\mathbb Z_2$-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 Dixmier–Douady bundle $\mathcal A^{\mathrm Spin}_{G}\rightarrow G$ over a compact, connected Lie group (as constructed by Atiyah–Segal) is multiplicative, and we obtain a canonical “twisted Spinc-structure” on spaces with group valued moment maps.