顶点算子代数的张量分解、仿费米子、层秩对偶和互反律
Tensor Decomposition, Parafermions, Level-Rank Duality, and Reciprocity Law for Vertex Operator Algebras
摘要：
对于半单李代数$ \frak{sl}_n$，仿射李代数$\widehat{\frak{sl}_{n}}$的基本表示$L_{\widehat{\frak{sl}_{n}}}(1,0)$是格顶点算子代数。本文的第一个主要结果是证明了$ L_{\widehat{\frak{sl}_{n}}}(l,0)$在$l$ -fold张量积$ L_{\widehat{\frak{sl}_{n}}}(1,0)^{\otimes l}$中的交换子顶点算子代数$ L_{\widehat{\frak{sl}_{n}}}(l,0)$同构于仿费米子顶点算子代数$K(\frak{sl}_{l},n)$，后者是海森堡顶点算子代数$L_{\widehat{\frak{h}}}(n,0) $在$L_{\widehat{\frak{sl}_l}}(n,0)$中的交换子.结果提供了一个版本的水平秩对偶。本文的第二个主要结果是证明了第一个结果的更一般的版本：$ L_{\widehat{\frak{sl}_{n}}}(l_1+\cdots +l_s, 0)$在$L_{\widehat{\frak{sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat{\frak{sl}_{n}}}(l_s, 0)$中的交换子同构于由$\frak{sl}_{l_1+\cdots+l_s}$的Levi李子代数生成的顶点算子代数的交换子，该顶点算子代数对应于有理顶点算子代数$ L_{\widehat{\frak{sl}}_{l_1+\cdots +l_s}}(n,0)$中的复合$(l_1, \cdots, l_s)$.这个一般的版本也类似于一个版本的互惠法律讨论豪在约化李群。在主要结果的证明过程中，某些Howe对偶对也出现在顶点算子代数的上下文中。
英文摘要：
For the semisimple Lie algebra $ \frak{sl}_n$, the basic representation $L_{\widehat{\frak{sl}_{n}}}(1,0)$ of the affine Lie algebra $\widehat{\frak{sl}_{n}}$ is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of $ L_{\widehat{\frak{sl}_{n}}}(l,0)$ in the $l$-fold tensor product $ L_{\widehat{\frak{sl}_{n}}}(1,0)^{\otimes l}$ is isomorphic to the parafermion vertex operator algebra $K(\frak{sl}_{l},n)$, which is the commutant of the Heisenberg vertex operator algebra $L_{\widehat{\frak{h}}}(n,0) $ in $L_{\widehat{\frak{sl}_l}}(n,0)$. The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of $ L_{\widehat{\frak{sl}_{n}}}(l_1+\cdots +l_s, 0)$ in $L_{\widehat{\frak{sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat{\frak{sl}_{n}}}(l_s, 0)$ is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of $\frak{sl}_{l_1+\cdots+l_s}$ corresponding to the composition $(l_1, \cdots, l_s)$ in the rational vertex operator algebra $ L_{\widehat{\frak{sl}}_{l_1+\cdots +l_s}}(n,0)$. This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.
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