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) = \theta_{\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*} \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|}, \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.