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arXiv:2508.04990 · Progress of Theoretical and Experimental Physics

Generalization of Anomaly Formula for Time-Reversal Symmetry in (2+1)D Abelian Bosonic Topological Quantum Field Theories

Ippo Orii

Published Oct 30, 2025• 0 citations
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Abstract

Abstract We study time-reversal symmetry in $(2+1)$D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by $\mathbb {Z}_2 \times \mathbb {Z}_2$ symmetry-protected topological (SPT) phases in $(3+1)$D, and can be diagnosed via partition functions on manifolds such as $\mathbb {RP}^4$ and $\mathbb {CP}^2$. These partition functions are related by the anomaly formula $Z(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}$, where $\theta _\mathcal {M}$ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $\xi _n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb {CP}^2) = \xi _1$, we propose a generalization of the anomaly formula that involves both the higher central charges $\xi _n$ and a new time-reversal invariant $\eta _n$. Introducing a distinguished subset $\mathcal {M}^n \subset \mathcal {A}$ of anyons, we establish the relation $\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},$which generalizes the known anomaly formula. We analyze the algebraic structure of $\mathcal {M}^n$, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.

DOI
10.1093/ptep/ptaf155

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