More on time-reversal anomaly of 2+1d topological phases

TL;DR

The paper proves an explicit formula for the time-reversal anomaly of 2+1d fermionic TQFTs, showing that $\exp(\frac{2\pi i\nu}{16}) = \frac{1}{D} \sum_p \eta_p d_p e^{-2\pi i h_p}$, where $D$ is the total quantum dimension, $d_p$ the quantum dimension, $h_p$ the topological spin, and $\eta_p$ the $\mathsf{T}^2_p$-related eigenvalue. This is achieved by deriving an explicit crosscap state $|{\mathbb{CC}}\rangle = S\sum_p \eta_p |p\rangle$ and relating $\eta_p$ to the $\mathsf{T}^2$ eigenvalues through the action of symmetry and orientation-reversing operations, thereby extending bosonic results to fermionic theories. The work clarifies how nonorientable-manifold data encodes the boundary anomaly and provides a computable criterion to identify surface theories of 3+1d topological superconductors. The approach integrates modular tensor category data (the $S$-matrix, quantum dimensions, and topological spins) with $\mathsf{T}$-structure and $\mathsf{T}^2$ eigenvalues to produce a concrete, checkable formula for the anomaly.

Abstract

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 crosscap state in terms of the modular S matrix and $\mathsf{T}^2$ eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.