S-duality as Fourier transform for arbitrary $\epsilon_1,\epsilon_2$

TL;DR

This paper extends the conjecture that S-duality acts as a Fourier transform on conformal blocks from the special case of equal deformation parameters to arbitrary values of _1 and _2, supported by perturbative calculations.

Contribution

It proposes that S-duality can be represented as a Fourier transform for any _1,_2 values, generalizing previous results limited to _1/_2=1.

Findings

S-duality corresponds to Fourier transform for general _1,_2.

Perturbative evidence from string coupling and mass expansions supports the conjecture.

Extension of the Fourier transform representation beyond the _1/_2=1 case.

Abstract

The AGT 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 form of the ordinary Fourier transform when $\beta\equiv-\epsilon_1/\epsilon_2=1$. Here we extend this conjecture to general values of $\epsilon_1,\epsilon_2$. Namely, we 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 $\epsilon_1,\epsilon_2$. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant $g^2\equiv-\epsilon_1\epsilon_2$ and hypermultiplet masses.