More on Wilson toroidal networks and torus blocks

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.