The super Frobenius-Schur indicator and finite group gauge theories on pin${}^{-}$ surfaces

Abstract

It is well-known that the value of the Frobenius-Schur indicator $|G{|}^{-1}{\sum }_{g\in G}\chi (g^2)=\pm 1$ of a real irreducible representation of a finite group $G$ determines which of the two types of real representations it belongs to, i.e. whether it is strictly real or quaternionic. We study the extension to the case when a homomorphism $\phi :G\to \mathbb{Z}/2\mathbb{Z}$ gives the group algebra $\mathbb{C}[G]$ the structure of a superalgebra. Namely, we construct of a super version of the Frobenius-Schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in [Wall1964] it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin${}^{-}$ surfaces.

arXiv comment11 pages; v2: additional references