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

Abstract

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 antiunitary global symmetry. For example, if a (2+1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T^{2}=(−1)^{F} labeled as ν∈Z_{16}, the TQFT must have c_{−}=1/4 mod 1/2 for odd ν, while c_{−}=0 mod 1/2 for even ν. 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)×CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.