More on Wilson toroidal networks and torus blocks
Konstantin Alkalaev, A. A. Belavin
Abstract
A bstract We consider the Wilson line networks of the Chern-Simons 3 d gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus 2 d 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) 3 mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl (2 , ℝ) representation.