Wilson lines construction of ${\mathfrak{s}\mathfrak{l}}_3$ toroidal conformal blocks

We study ${\mathcal{W}}_3$ toroidal conformal blocks for degenerate primary fields in AdS/CFT context. In the large central charge limit ${\mathcal{W}}_3$ algebra reduces to ${\mathfrak{s}\mathfrak{l}}_3$ algebra and ${\mathfrak{s}\mathfrak{l}}_3$ blocks are defined as contributions to ${\mathcal{W}}_3$ blocks coming from the generators of ${\mathfrak{s}\mathfrak{l}}_3$ subalgebra. We consider the construction of ${\mathfrak{s}\mathfrak{l}}_3$ toroidal blocks in terms of Wilson lines operators of $3d$ Chern-Simons gravity in the thermal AdS${}_3$ space-time. According to the correspondence, degenerate primary fields are associated with Wilson lines operators acting in the corresponding finite-dimensional ${\mathfrak{s}\mathfrak{l}}_3$ representations. We verify this dual construction for one-point toroidal block using ${\mathfrak{s}\mathfrak{l}}_3$ tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.