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 $$\varphi :G\rightarrow \mathbb {Z}/2\mathbb {Z}$$ 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 Wall (in J Reine Angew Math 213:187–199, 1963/64. https://doi.org/10.1515/crll.1964.213.187 ) it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin $$^-$$ surfaces.