Building Vertex Algebras from Parts
Building Vertex Algebras from Parts
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从零件构建顶点代数
DOI：
10.1007/s00220-019-03607-0
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发表时间：
2020
影响因子：
2.4
通讯作者：
Scott Carnahan
中科院分区：
文献类型：
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作者：
Toshiro Kuwabara;Hyohe Miyachi;Kentaro Wada;Yuichiro Tanaka;Yusuke Nakajima;Yuichiro Tanaka;Toshiro Kuwabara;田中雄一郎;Yusuke Nakajima;Scott Carnahan
给定由阿贝尔群参数化的顶点代数模的集合，以及可组合交织算子的一维空间，我们分配 Eilenberg-Mac Lane 空间的上同调的规范元素。该元素描述了对局部性的阻碍，因为该元素的消失等效于由我们的交织算子给出的乘法顶点代数结构的存在，并且给定存在，该结构在同构上是唯一的。同调阻碍简化为“偶数”问题，对于 2 可整群自然消失，因此组织成奇数阶阿贝尔群的简单流总是产生顶点代数。此外，在与共形场论最相关的情况下（即，当我们有良好的对梯度和张量积时），我们自然地获得了交织算子的空间，并且均匀性障碍简化为特定阶二次电流上的对梯度双线性形式是对称还是斜对称的问题。我们证明，如果给定一个简单的正则 VOA 和由融合环中的一组偶数单元参数化的积分权重模块，则直和承认简单正则 VOA 的结构，称为简单电流扩展，并且该结构在同构上是唯一的。
Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg–Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an “evenness” problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order two currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.