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

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.

Complex Chern-Simons from M5-branes on the squashed three-sphere

“…The net result is then a simple functional determinant. 32) where in the above ad Xa denotes the operator defined by X a acting on fields in the adjoint representation.…”

Semiclassical analysis of the3d/3drelation

“…2 This relation should arise from the dimensional reduction of the 6d (2, 0) theory on S 3 Â M 3 , and is a 3d=3d counterpart of the 4d=2d correspondence [Alday-Gaiotto-Tachikawa (AGT) conjecture] [4]. See [5] for a recent discussion, and [6][7][8] for related proposals.…”

q-Virasoro Modular Double and 3d Partition Functions

“…In particular, S-duality domain wall in 4d N = 2 theories of class S [155] are realized by 3d N = 2 U(N ) YM theories on S 3 b [156][157][158][159][160], and their partition functions are modular kernels of Liouville [161][162][163] or Toda theories [160]. Moreover, our construction of the W q,t modular double should easily lift to the elliptic case [164,165].…”