极值非全纯共形场论的分类
On classification of extremal non-holomorphic conformal field theories
摘要：
有理手征共形场论按照其亏格进行组织，亏格由一个模张量范畴$\mathcal{C}$和一个中心荷$c$组成.一个长期的目标是分类酉有理共形场论的基础上的分类酉模张量类别。我们猜想，对于任意的幺正模张量范畴$\mathcal{C}$，存在幺正手征共形场论$V$，使得它的模张量范畴$\mathcal{C}_V$是$\mathcal{C}$ .在本文中，我们提出了一个数学方案，并围绕这个猜想。定义了一类极值顶点算子代数，其中心荷的最小共形维数尽可能大，并给出了非平凡表示理论。我们证明了至多具有三个不同不可约模的极顶点算子代数V有许多不同的特征标。此外，我们还列出了这类顶点算子代数的所有可能的特征标，这些顶点算子代数的$c$不超过48。
英文摘要：
Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras V possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with $c$ at most 48.
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