Applied Categorical Structures

, Volume 19, Issue 1, pp 175–232

The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters

Article

DOI: 10.1007/s10485-009-9189-0

Cite this article as:
Bartlett, B. Appl Categor Struct (2011) 19: 175. doi:10.1007/s10485-009-9189-0

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 Grothendieck category of unitary 2-representations to the category of unitary conjugation equivariant vector bundles over the group.

Keywords

2-Category Unitary 2-representation 2-Character Geometric quantization Adjunction Equivariant gerbe 

Mathematics Subject Classifications (2000)

Primary 57R56 Secondary 18D05 53D50 

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.University of SheffieldSheffieldUK
  2. 2.Stellenbosch UniversityStellenboschSouth Africa