Three-dimensional extensions of the Alday–Gaiotto–Tachikawa relation

An extension of the two-dimensional (2d) Alday–Gaiotto–Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two $S$-dual supersymmetric Yang–Mills (SYM) models to the 3d Chern–Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov–Shatashvili limit of the 5d SYM theory.