S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$

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.

17 pages

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