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

TL;DR

This paper generalizes the anomaly formula for time-reversal symmetry in (2+1)D abelian bosonic topological phases, linking higher central charges and new invariants to better understand anomalies and gapped boundaries.

Contribution

It introduces a generalized anomaly formula involving higher central charges and a new invariant, expanding the understanding of time-reversal anomalies in topological phases.

Findings

Derived the relation between higher central charges and new invariants.

Analyzed the algebraic structure of the distinguished anyon subset.

Connected the generalized formula to the original anomaly relation.

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.