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.
Highlights
Conformal blocks are basic ingredients of conformal field theory correlation functions, they play crucial role in the conformal bootstrap program [1, 2]
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
There is an intriguing relation between the space of quantum states in the three-dimensional Chern-Simons theory in the presence of the Wilson lines and the space of conformal blocks in two-dimensional conformal field theory noticed a long ago [58,59,60]
Summary
Conformal blocks are basic ingredients of conformal field theory correlation functions, they play crucial role in the conformal bootstrap program [1, 2]. We formulate and calculate one-point and two-point Wilson network functionals which are dual to one-point and two-point torus conformal blocks for degenerate quasi-primary operators. — in section 3 we define toroidal Wilson network operators with one and two boundary attachments They are the basis for explicit calculations of one-point blocks and two-point blocks in two OPE channels . — in section 4 we consider torus conformal blocks for degenerate quasi-primary operators which are dual to the Wilson networks carrying finite-dimensional representations of the gauge algebra. — section 5 contains explicit calculation of the one-point toroidal Wilson network operator in two different representations, using 3j Wigner symbols and symmetric tensor product representation.
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