The Freed--Quinn line bundle from higher geometry

Abstract

For a finite group $G$, and level $\alpha\in Z^3(BG;{\rm U}(1))$, Freed and Quinn construct a line bundle over the moduli space of $G$-bundles on surfaces. Global sections determine the values of Chern--Simons theory at level $\alpha$ on surfaces. In this paper, we provide an alternate construction using tools from higher geometry: the pair $(G,\alpha)$ determines a 2-group group, and the Freed--Quinn line arises as a categorical truncation of the bicategory of 2-group bundles.