The geometry of unitary 2-representations of finite groups and their 2-characters

TL;DR

This paper explores the geometric structure of unitary 2-representations of finite groups and their 2-characters, establishing a categorified geometric quantization framework linking 2-representations to equivariant gerbes.

Contribution

It introduces a geometric perspective on 2-representations, demonstrating their correspondence to equivariant gerbes and establishing the functoriality of 2-characters.

Findings

2-representations correspond to equivariant gerbes

2-characters are functorial with respect to morphisms

The 2-character functor is unitarily fully faithful

Abstract

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are `categorified' in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerbes. We also show how the 2-character of a 2-representation can be made functorial with respect to morphisms of 2-representations. Under the geometric correspondence, the 2-character of a 2-representation corresponds to the geometric character of its associated equivariant gerbe. This enables us to show that the complexified 2-character is a unitarily fully faithful functor from the complexified homotopy category of unitary 2-representations to the category of unitary equivariant vector bundles over the group.