来自融合环的非可逆SPT相：通过Tannaka对偶

中文摘要 AI 辅助

在具有非可逆对称性的(1+1)维量子场论中，拓扑线的融合规则通常是最易获取的数据，但它们并不能完全确定对称性是否无反常，即是否允许平凡能隙相。这类相由相应融合范畴的纤维函子刻画。基于Tannaka对偶，我们仅从融合环出发，通过同时求解幺半结构映射和张量自同构来提取纤维函子。这使得我们无需直接求解五边形方程即可找到允许纤维函子的范畴化。我们将此方法应用于已知的$\mathbb{Z}_2\times\mathbb{Z}_2$的Tambara-Yamagami环。我们的主要结果涉及阶为27的两个超特殊群的表示范畴所共有的秩11融合环，该环具有多重度3的通道，且与自三重性相关。在这两种情况下，我们发现了不止一个纤维函子，即多个非可逆SPT相。我们利用这些纤维函子的显式数据，以依赖于非退化交错双特征和可逆对象$\epsilon$的紧凑形式推导出这两个表示范畴的$F$-符号。利用重构的数据，我们计算了两个不同SPT相之间界面上的反常代数。

英文摘要

In a (1+1)-dimensional QFT with non-invertible symmetry, the fusion rules of topological lines are often the most accessible data, yet they do not fully determine whether the symmetry is anomaly-free, i.e. whether it admits trivially gapped phases. Such phases are characterized by fiber functors of the corresponding fusion category. Building on Tannaka duality, we extract fiber functors from the fusion ring alone by solving simultaneously for monoidal structure maps and tensor automorphisms. This allows us to find categorifications admitting a fiber functor without solving the pentagon equations directly. We test this procedure on the known Tambara--Yamagami ring of $\mathbb{Z}_2\times\mathbb{Z}_2$. Our main result concerns the rank-11 fusion ring shared by the representation categories of the two extraspecial groups of order 27, which has a multiplicity 3 channel and is relevant to self-triality. In both cases we find more than one fiber functor, i.e. several non-invertible SPT phases. We use the explicit data of these fiber functors to derive the $F$-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object $ε$. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.

发表机构

• Dipartimento di Fisica, Università di Trieste（的里雅斯特大学物理系）
• INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）

机构由 AI 辅助整理，请以论文原文为准。

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69 pages + appendices, 5 figures, Mathematica notebooks included as ancillary files