Real representation theory of finite categorical groups

Abstract

We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a group $\mathsf{G}$, or more generally a finite categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g, ω) \in \mathsf{G} \times \hat{\mathsf{G}}$, where $\hat{\mathsf{G}}$ is the background Real structure on $\mathsf{G}$. This collection of numbers defines the Real $2$-character of the Real representation. We also define various forms of induction for Real representations of finite categorical groups and compute the result at the level of Real $2$-characters. We interpret results in Real categorical character theory in terms of geometric structures, namely gerbes, vector bundles and functions on iterated unoriented loop groupoids. This perspective naturally leads to connections with the representation theory of unoriented versions of the twisted Drinfeld double of $\mathsf{G}$ and with discrete torsion in $M$-theory with orientifold. We speculate on an interpretation of our results as a generalized Hopkins--Kuhn--Ravenel-type character theory in Real equivariant homotopy theory.