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arXiv · 2007.10494

More on Wilson toroidal networks and torus blocks

K. B. Alkalaev·V. A. Belavin
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Abstract

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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K. B. Alkalaev, V. A. Belavin. 2020-07-20. More on Wilson toroidal networks and torus blocks. https://doi.org/10.1007/jhep11(2020)121

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