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

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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.

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Notes

  1. Various other generalizations of the Frobenius–Schur indicator were studied in the literature. Firstly, we can consider generalized Frobenius–Schur indicators for finite groups, see e.g. [BG04] and [GI18, Sect. 11]. Secondly, we can extend the concept of the Frobenius–Schur indicator to fusion categories, see e.g. [FS01, Sect. 3], which is further extended to fermionic fusion categories in e.g. [BWHV17, Sect. 3.3]. This last generalization should agree with ours when specialized to fermionic fusion categories arising from our superalgebra. It would be interesting to check this.

  2. The following three paragraphs were added in v2. The authors thank J. Murray for the information.

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  1. Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan

    Takumi Ichikawa & Yuji Tachikawa

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The research of Yuji Tachikawa is supported by in part supported by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo, and in part by JSPS KAKENHI Grant-in-Aid (Wakate-A), No. 17H04837 and JSPS KAKENHI Grant-in-Aid (Kiban-S), No. 16H06335. The authors thank Arun Debray for careful reading and instructive comments on an earlier draft of this paper. The authors also thank John Murray for helpful email exchanges after v1 appeared on the arXiv, informing them the relevance of [Gow79] to our work. The authors also thank Luuk Stehouwer for pointing out a few typos in Sect. 3 which persisted until v2.

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Ichikawa, T., Tachikawa, Y. The Super Frobenius–Schur Indicator and Finite Group Gauge Theories on Pin\(^-\) Surfaces. Commun. Math. Phys. 400, 417–428 (2023). https://doi.org/10.1007/s00220-022-04601-9

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