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 Σg∈ G (g2)=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 :G Z/2Z gives the group algebra 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.