Differential cocycles and Dixmier–Douady bundles

This paper exhibits equivalences of 2-stacks between certain models of S¹-gerbes 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 S¹-bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting of S¹-gerbes over Lie groupoids, and can be applied to the setting of S¹-gerbes over orbifolds.