论文标题
超级Frobenius-schur指标和有限的集体量规理论$^ - $表面
The super Frobenius-Schur indicator and finite group gauge theories on pin$^-$ surfaces
论文作者
论文摘要
众所周知，Frobenius-Schur指示器的值$ | g |^{ - 1} \ sum_ {g \ in G}χ（g^2）= \ pm1 = \ pm1 $ pm1 $的实际不可理解的代表$ g $确定其两种实际表述中的两种实际表示，即严格或quartion。我们研究了同构$φ：g \ to \ mathbb {z}/2 \ mathbb {z} $的情况下的情况。也就是说，我们构建了Frobenius-Schur指示器的超级版本，其对真实不可约的超级表示的价值是统一的第八个根源，区分了它属于[Wall1964]中所述的八种不可还原的真实超级表示中的哪种。我们还讨论了它在PIN $^ - $表面上的二维有限组理论的背景下进行讨论。
It is well-known that the value of the Frobenius-Schur indicator $|G|^{-1} \sum_{g\in G} χ(g^2)=\pm1$ 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 $φ:G\to \mathbb{Z}/2\mathbb{Z}$ 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.
