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

TL;DR

This paper constructs $rak{sl}_3$ toroidal conformal blocks using Wilson lines in 3D Chern-Simons gravity, verifying the duality with boundary CFT methods and advancing the understanding of AdS/CFT correspondence for higher-rank algebras.

Contribution

It introduces a Wilson line approach to compute $rak{sl}_3$ toroidal blocks in AdS/CFT, connecting bulk gravity and boundary conformal field theory techniques.

Findings

Verified dual construction for one-point toroidal block

Applied $rak{sl}_3$ tensor techniques in bulk theory

Used AGT correspondence algorithm in boundary CFT

Abstract

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 $\mathfrak{sl}_3$ algebra and $\mathfrak{sl}_3$ blocks are defined as contributions to $\mathcal{W}_3$ blocks coming from the generators of $\mathfrak{sl}_3$ subalgebra. We consider the construction of $\mathfrak{sl}_3$ 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 $\mathfrak{sl}_3$ representations. We verify this dual construction for one-point toroidal block using $\mathfrak{sl}_3$ tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.