Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914262178332672 |
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| author | Orii, Ippo |
| author_facet | Orii, Ippo |
| contents | 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 $θ_\mathcal{M}$ is the Dehn twist phase associated with the crosscap state.
Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $ξ_n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb{CP}^2) = ξ_1$, we propose a generalization of the anomaly formula that involves both the higher central charges $ξ_n$ and a new time-reversal invariant $η_n$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs Orii, Ippo High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics 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 $θ_\mathcal{M}$ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $ξ_n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb{CP}^2) = ξ_1$, we propose a generalization of the anomaly formula that involves both the higher central charges $ξ_n$ and a new time-reversal invariant $η_n$. 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. |
| title | Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2508.04990 |