On 3d extensions of AGT relation

TL;DR

This paper explores extending the AGT relation into three dimensions, connecting 3d Chern-Simons theory, knot invariants, and 2d conformal theories, revealing similarities and differences in their integral representations and proposing new interpretations.

Contribution

It proposes a novel 3d extension of the AGT relation, linking 3d Chern-Simons theory, knot invariants, and 2d conformal theories, and discusses their mathematical similarities and differences.

Findings

Trace of modular kernels resembles knot invariants as integrals of quantum dilogarithms

Differences in conservation laws distinguish monodromy traces from knot invariants

Knot invariants may be interpreted as solutions to Baxter equations in the relativistic Toda system

Abstract

An extension of the AGT relation from two to three dimensions begins from connecting the theory on domain wall between some two S-dual SYM models with the 3d Chern-Simons theory. The simplest kind of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the both quantities are very similar, especially if represented as integrals of the products of quantum dilogarithm functions. However, there are also various differences, especially in the "conservation laws" for integration variables, which hold for the monodromy traces, but not for the knot invariants. We also discuss another possibility: interpretation of knot invariants as solutions to the Baxter equations for the relativistic Toda system. This implies another AGT like relation: between 3d Chern-Simons theory and the Nekrasov-Shatashvili limit of the 5d SYM.