More on Wilson toroidal networks and torus blocks

TL;DR

This paper explores Wilson line networks in 3D Chern-Simons gravity with toroidal boundaries, providing explicit formulas for torus conformal blocks using $sl(2,R)$ representations and novel computational methods.

Contribution

It explicitly derives one- and two-point torus conformal blocks via Wilson network operators, extending previous results with new computational approaches.

Findings

Explicit formulas for one-point and two-point torus blocks.

Two alternative computational methods using $3mj$ symbols and symmetric tensor products.

Extension of known results in Wilson network calculations for toroidal boundaries.

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.