Wilson lines construction of sl₃ toroidal conformal blocks

We study W₃ toroidal conformal blocks for degenerate primary fields in AdS/CFT context. In the large central charge limit W₃ algebra reduces to sl₃ algebra and sl₃ blocks are defined as contributions to W₃ blocks coming from the generators of sl₃ subalgebra. We consider the construction of sl₃ toroidal blocks in terms of Wilson lines operators of 3d Chern-Simons gravity in the thermal AdS₃ space-time. According to the correspondence, degenerate primary fields are associated with Wilson lines operators acting in the corresponding finite-dimensional sl₃ representations. We verify this dual construction for one-point toroidal block using sl₃ tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.