The Quantum Group Dual of the First-Row Subcategory for the Generic Virasoro VOA
The Quantum Group Dual of the First-Row Subcategory for the Generic Virasoro VOA
复制标题
通用 Virasoro VOA 第一行子类别的量子组对偶
DOI：
10.1007/s00220-021-04266-w
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发表时间：
2021
影响因子：
2.4
通讯作者：
Kytola Kalle
中科院分区：
文献类型：
--
作者：
Koshida Shinji;Kytola Kalle
在几个例子中，已经观察到顶点算子代数（VOA）的模范畴等价于某个量子群的表示范畴。本文关注的是发展这样一个二元性的情况下，Virasoro VOA在通用的中央收费，可以说是最基本的所有VOA，但结构复杂。我们不解决通用Virasoro VOA的所有模块的类别，但我们考虑Kac表第一行的无限多个模块。基于库仑气体积分的显式量子群方法，给出了融合规则的新证明，证明了纠缠算子的合成是解析的，并证明了共形块完全由量子群方法确定.最重要的是，我们证明了第一行模之间的交织算子的结合性，并发现这种结合性是由量子群的6 j-符号决定的。我们的结果构成了VOA和量子群之间的具体对偶，它们将作为关键工具来建立通用Virasoro VOA的第一行子范畴的模块和（类型-1）有限维表示的范畴的等价性。
In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of.