Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs

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 \begin{equation*} Z(\mathbb{RP}^4)\, Z(\mathbb{CP}^2) = θ_{\mathcal{M}}, \end{equation*} 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 \begin{equation*} η_n \cdot ξ_n = \frac{\sum_{a \in \mathcal{M}^n} θ(a)^n}{\left| \sum_{a \in \mathcal{M}^n} θ(a)^n \right|}, \end{equation*} 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.

arXiv comment28 pages, 3 figures, 1 table; v2: affiliation added; v3: minor corrections. Published version

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