More on Wilson toroidal networks and torus blocks

Abstract

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, ℝ) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of sl(2, ℝ) representations: (1) 3mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl(2, ℝ) representation.