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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 $(g, \omega) \in \mathsf{G}
\times \hat{\mathsf{G}}$, 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.