We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in ($2+1$)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular $\mathbf{S}$ matrix and ${\mathsf{T}}^2$ eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

中文翻译：

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（2 + 1）维拓扑相位的时间反转异常的推导

我们证明了Wang和Levin最近针对（$\mathrm{2个}+\mathrm{1个}$维费米离子拓扑量子场论。关键步骤是根据模块化确定跨帽状态$\mathbf{小号}$ 矩阵和 ${\mathsf{Ť}}^{\mathrm{2个}}$特征值，概括了Barkeshli等人的最新分析。在玻色的情况下。