离散 $C^*$-Frobenius 代数与代数量子场论中的无限指标扩张
Discrete $C^*$-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
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中文总结 AI 辅助
本文引入离散 $C^*$-Frobenius 代数作为范畴数据，构造 Möbius 协变网的无限指标扩张，并给出简单电流扩张和离散 Longo-Rehren 构造两类例子。
中文摘要 AI 辅助
我们引入离散 $C^*$-Frobenius 代数作为构造 Möbius 协变网的无限指标扩张的范畴数据。其底层表示是可数直和的简单对偶模，而直和本身不一定是可对偶的。因此，我们不要求单一的有界乘法态射，而是处理相容的有界左乘和右乘态射族。利用 Bin Gui 的范畴扩张理论，我们将每个离散 $C^*$-Frobenius 代数关联到两个（通常非局部的）Möbius 协变扩张，分别由左荷场和右荷场生成。若离散 $C^*$-Frobenius 代数是交换的，则这两个扩张重合，并产生一个局部 Möbius 协变扩张。作为应用，对于每个由具有平凡自辫的可逆对象生成的 $C^*$-张量范畴，且其融合积规则由 $\mathbb{Z}$ 给出，我们构造两类例子：由 $\mathbb{Z}$ 生成的简单电流扩张和离散 Longo-Rehren 构造。
英文摘要
We introduce discrete $C^*$-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete $C^*$-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete $C^*$-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every $C^*$-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by $\mathbb{Z}$, we construct two classes of examples: the simple current extension by $\mathbb{Z}$ and the discrete Longo-Rehren construction.
