Anomaly constraint on chiral central charge of (2+1)d topological order

In this short paper, we argue that the chiral central charge $c_{-}$ of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with $T^2=(-1{)}^F$ labeled as $\nu \in {\mathbb{Z}}_{16}$, the TQFT must have $c_{-}=1\mathrm{/}4$ mod $1\mathrm{/}2$ for odd $\nu$, while $c_{-}=0$ mod $1\mathrm{/}2$ for even $\nu$. This generalizes the fact that the bosonic TQFT with $T$ anomaly in a particular class must carry $c_{-}=4$ mod $8$ to fermionic cases. We also study such a constraint for fermionic TQFT with $U(1)\times CT$ symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.