AGT on the S-duality Wall

TL;DR

The work connects 3D theories on S-duality walls in 4D $N=4$ SYM to the S-duality kernels of Liouville/Toda CFT within the AGT framework. By showing that the mass-deformed wall theory $Z_{3D}^{N=2^*}$ for $T[SU(2)]$ reproduces the Liouville kernel under S-duality (up to universal factors), it establishes a concrete 3D-2D duality link and generalizes to $SU(N)$ with Toda kernels on a one-punctured torus. The results consolidate a picture where domain-wall operators encode duality transformations, with a brane picture based on M5-branes on $S^3\times M_3$ offering geometric intuition. They also outline directions for future work, including squashed $S^3$ and non-canonical R-charge analyses.

Abstract

Three-dimensional gauge theory T[G] arises on a domain wall between four-dimensional N=4 SYM theories with the gauge groups G and its S-dual G^L. We argue that the N=2^* mass deformation of the bulk theory induces a mass-deformation of the theory T[G] on the wall. The partition functions of the theory T[SU(2)] and its mass-deformation on the three-sphere are shown to coincide with the transformation coefficient of Liouville one-point conformal block on torus under the S-duality.