More on time-reversal anomaly of 2+1d topological phases
  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. 
Yuji Tachikawa, Kazuya Yonekura
hep-th cond-mat.str-el