S-duality as Fourier transform for arbitrary $ε_1,ε_2$

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 $β\equiv-ε_1/ε_2=1$. Here we extend this conjecture to general values of $ε_1,ε_2$. 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 $ε_1,ε_2$. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant $g^2\equiv-ε_1ε_2$ and hypermultiplet masses.

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