The Freed--Quinn line bundle from higher geometry

Abstract

For a finite group $G$, and level $\hat{I}\pm \in Z^3(BG;\mathrm{U}(1))$, 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 $\hat{I}\pm$ on surfaces. In this paper, we provide an alternate construction using tools from higher geometry: the pair $(G,\hat{I}\pm )$ determines a 2-group group, and the Freed--Quinn line arises as a categorical truncation of the bicategory of 2-group bundles.