Differential cocycles and Dixmier–Douady bundles

Abstract

This paper exhibits equivalences of 2-stacks between certain models of S 1gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to S 1 -bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting of S 1 -gerbes over Lie groupoids.

Groupoid Equivariant Prequantization

“…Indeed, given a differential character (z, y, [x]), choose a cocycle ξ in DC 3 1 (G) that represents it (see Lemma 3.9) and set DD(z, y, [x]) = [pr(ξ)], where pr : DC * 1 (G) → C * (G) denotes the natural projection. This is indeed the DD-class, since isomorphism classes of differential characters are in bijection with H 3 (DC 1 (G)) ∼ = H 3 (G; Z) (where the isomorphism is induced by pr-see [12]).…”

“…In this section, we recall some perspectives on Dixmier-Douady bundles and differential characters. We also briefly review their groupoid equivariant counterparts as in [12]. Dixmier-Douady bundles are geometric models for S 1 -gerbes.…”

“…Analogous to the classification of complex line bundles by their Chern class, Dixmier-Douady bundles are classified by a degree 3 cohomology class called the Dixmier-Douady class [26]. In [12], it was verified that if Remark 3.12. Note that the corresponding notion of a DD-class for G-equivariant differential characters is automatic.…”

“…Following E. Meinrenken's approach in [20] for quasi-Hamiltonian group actions, this paper presents a definition of prequantization (see Definition 4.1) for Hamiltonian actions of quasi-presymplectic groupoids (with no exactness assumption), using the theory of Dixmier-Douady bundles (DD-bundles), which are bundles of C * -algebras with typical fibre the compact operators on an infinite dimensional separable Hilbert space. An important aspect of this definition relates to the notion of equivariance for DD-bundles used in the present context and investigated further in [12], which makes use of the 'higher structure' inherent in DD-bundles (that is also present for other models of S 1 -gerbes, e.g. see [24]).…”

“…As stated earlier, the definition of prequantization given in this paper uses the theory of DD bundles. In Section 3, we review some elements of Dixmier-Douady theory (following [1]) and recall their groupoid-equivariant counterparts from [12], which makes use of the fact that DD bundles from a bicategory. We also recall the equivalent (strict) 2-category of differential characters of degree 3 (as well as its groupoid-equivariant counterpart), which we make use of in the proofs of several of our results.…”