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

TL;DR

This paper demonstrates that the chiral central charge of (2+1)d topological orders is constrained by 't Hooft anomalies of anti-unitary symmetries, linking anomalies to specific values of the central charge.

Contribution

It establishes new anomaly-based constraints on the chiral central charge in fermionic topological quantum field theories with certain symmetries.

Findings

Fermionic TQFT with time reversal anomaly $ u$ must have $c_-=1/4$ mod $1/2$ for odd $ u$.

Fermionic TQFT with even $ u$ has $c_-=0$ mod $1/2$.

Constraints extend to fermionic TQFT with $U(1) imes CT$ symmetry.

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 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 $c_-=1/4$ mod $1/2$ for odd $\nu$, while $c_-=0$ mod $1/2$ 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.