Prequantization of differential characters of Lie groupoids

Abstract

In this paper, we describe a category ${\mathrm{DC}}_{\mathrm{ex},3-1}^3\left(G\right)$ of degree-3 differential characters of a Lie groupoid $G$ together with a prequantization functor Preq from it to the category $Ger{b}_{\nabla }\left(G\right)$ of $S^1$-central extensions with pseudo-connections over $G$, and show that Preq is an equivalence of categories and the isomorphism classes of $S^1$-central extensions with pseudo-connections over $G$ are classified by the cohomology group $H^3\left(D{C}_{\mathrm{ex},3-1}^{⁎}\right(G\left)\right)$ of degree-3 differential characters. As an application, we characterize closed integral 3-forms with prequantization $S^1$-central extensions and pseudo-connections for all Lie groupoids. This generalizes Behrend–Xu's prequantization result of degree 3-context for Lie groupoids satisfying $H^2\left({\Omega }^0\right(G_{•}),\partial )=0$. Moreover we identify the group of flat $S^1$-central extensions over a Lie groupoid $G$ with the cohomology group $H^2\left(C_{\mathrm{ex}}^{⁎}\right(G,R/Z\left)\right)$ of a modification of the complex of singular cochains with coefficient in $R/Z$. We also extend these results to differentiable stacks.