任意子与固有复杂的F符号
Anyons and Inherently Complex F-symbols
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中文总结 AI 辅助
研究二维加一维空间中任意子的F符号，通过展示特定编织融合范畴的F符号不能取实数值，表明其“固有复杂”，分析了最小秩的此类范畴及对应任意子模型，还将例子与任意子分类的最新结果相联系。
中文摘要 AI 辅助
二维加一维空间中的任意子不仅具有奇异的编织统计特性，还具有复杂的融合属性。两个任意子可融合到多个拓扑电荷扇区，三个任意子融合产生固定电荷扇区的结合性由F符号控制。虽然编织不变量（如模数据）通常是复数值，但对一般任意子模型的完整描述还需要理解其融合结合性数据的算术性质。许多最常见的二维加一维拓扑序的F符号，包括所有阿贝尔任意子模型以及斐波那契和伊辛任意子，都可以取实数值。我们通过展示其F符号不能取实数值的编织融合范畴，表明这种现象并不普遍。我们将这样的F符号称为“固有复杂的”。我们研究的例子缺乏电荷共轭对称性，因此我们的结果与在相关工作中证明的一个陈述的逆命题一致，该陈述将编织融合范畴中的实F符号与合适的电荷共轭对称性的存在联系起来。我们分析了已知的具有固有复杂F符号的最小秩编织融合范畴：${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$和${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$。因此，相应的$\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$和$\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$任意子模型也具有固有复杂的F符号。我们的展示将这些例子与关于超越模数据对任意子进行分类的最新结果联系起来。
英文摘要
Anyons in $2+1$ dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by $F$-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The $F$-symbols for many of the most common $2+1$d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose $F$-symbols cannot be made real. We call such $F$-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real $F$-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex $F$-symbols: ${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$ and ${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$. Consequently, the corresponding $\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$ and $\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$ anyon models also have inherently complex $F$-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.
