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arXiv 2007.10494hep-th
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关于Wilson环形网络和环面块的更多信息

More on Wilson toroidal networks and torus blocks

K. B. Alkalaev, V. A. Belavin

备注：
36 pages, v2: 43 pages, more explanations and clarifying comments, more references, two new Sections on 2-point torus blocks in terms of 3j Wigner symbols, a journal version
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摘要：

本文考虑了Chern-Simons $3d$引力理论的Wilson线网络，该网络具有环面边界条件，计算了环面$2d$ CFT中退化准素性算子的整体共形块。在总结和进一步推广文献中已有结果的基础上，通过$sl(2,\mathbb{R})$代数的不可约有限维表示中环形Wilson网络算子的特殊矩阵元，明确地得到了单点环面块和两点环面块.用不同的方法处理$sl(2,\mathbb{R})$表示的多重张量积，得到了两种不同形式的表达式：（1）高价的$3mj$ Wigner符号和缠结，（2）基本$sl(2,\mathbb{R})$表示的全对称张量积.

英文摘要：

We consider the Wilson line networks of the Chern-Simons $3d$ gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus $2d$ CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of $sl(2,\mathbb{R})$ algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of $sl(2,\mathbb{R})$ representations: (1) $3mj$ Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental $sl(2,\mathbb{R})$ representation.

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