S-duality as Fourier transform for arbitrary ϵ1, ϵ2

Abstract

The Alday–Gaiotto–Tachikawa relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that, for the four-point conformal block, the modular transform up to the non-perturbative contributions can be written in the form of the ordinary Fourier transform when β ≡ −ϵ1/ϵ2 = 1. Here I extend this conjecture to general values of ϵ1, ϵ2. Namely, I argue that, for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters ϵ1, ϵ2. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 ≡ −ϵ1ϵ2 and hypermultiplet masses.