More on Wilson toroidal networks and torus blocks

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.

arXiv comment36 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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Cited by in corpus (3)

• Wilson lines construction of $\mathfrak{osp}(1|2)$ conformal blocks
• Wilson lines construction of ${\mathfrak{sl}}_3$ toroidal conformal blocks
• Rényi entropy of single-character CFTs on the torus