The Super Frobenius–Schur Indicator and Finite Group Gauge Theories on Pin$$^-$$ Surfaces

It is well-known that the value of Frobenius–Schur indicator $$|G|^{-1} \sum _{g\in G} \chi (g^2)=\pm 1$$ a real irreducible representation finite group G determines which two types representations it belongs to, i.e. whether strictly or quaternionic. We study extension to case when homomorphism $$\varphi :G\rightarrow \mathbb {Z}/2\mathbb {Z}$$ gives algebra $$\mathbb {C}[G]$$ structure superalgebra. Namely, we construct super version whose for an eighth root unity, distinguishing eight described in Wall (in J Reine Angew Math 213:187–199, 1963/64. https://doi.org/10.1515/crll.1964.213.187 ) to. also discuss its significance context two-dimensional finite-group gauge theories on pin $$^-$$ surfaces.