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 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.

arXiv comment16 pages, 2 figures, references added, accepted in Physical Review Research

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• On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds
• The Smith Fiber Sequence and Invertible Field Theories
• Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs