S-duality as a beta-deformed Fourier transform

TL;DR

This paper explores the formulation of Gaiotto's S-duality relations through a beta-deformed Fourier transform, connecting conformal blocks, Nekrasov functions, and matrix models, with explicit calculations of corrections.

Contribution

It provides a quantitative formulation of S-duality using matrix models and reveals how beta-deformation affects the Fourier transform structure of Nekrasov functions.

Findings

Corrections vanish at beta=1.

S-duality reduces to Legendre transformation in Seiberg-Witten limit.

Explicit calculation of beta-deformation effects on Fourier transform.

Abstract

An attempt is made to formulate Gaiotto's S-duality relations in an explicit quantitative form. Formally the problem is that of evaluation of the Racah coefficients for the Virasoro algebra, and we approach it with the help of the matrix model representation of the AGT-related conformal blocks and Nekrasov functions. In the Seiberg-Witten limit, this S-duality reduces to the Legendre transformation. In the simplest case, its lifting to the level of Nekrasov functions is just the Fourier transform, while corrections are related to the beta-deformation. We calculate them with the help of the matrix model approach and observe that they vanish for beta=1. Explicit evaluation of the same corrections from the U_q(sl(2)) infinite-dimensional representation formulas due to B.Ponsot and J.Teshner remains an open problem.