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.

S-duality and modular transformation as a non-perturbative deformation of the ordinary pq-duality

“…This explains the perturbative result of [60,61] for the properly normalized conformal blocks and straightforwardly provides its non-perturbative generalization, which is in accordance with [27,28].…”

“…In [60] this was shown for the central charge c = 1 − 6(β − 1/β) 2 = 1 (β = 1), and, after a more accurate analysis of normalization factors in [61], this result was extended to an arbitrary β. This claim was reconsidered and confirmed from a slightly different viewpoint in [62,63].…”

“…Making use of this idea and calculational advances in AGT studies, we conjectured recently [60,61] that the S-duality is actually reduced to an ordinary Fourier transform in all orders of perturbation expansion in string coupling constant. In [60] this was shown for the central charge c = 1 − 6(β − 1/β) 2 = 1 (β = 1), and, after a more accurate analysis of normalization factors in [61], this result was extended to an arbitrary β.…”

“…Calculations of [60,61] are quite tedious, what seems strange for such a simple outcome. Clearly some very simple explanation should exist, which does not require long calculations.…”

“…Now, (4.19) implies that L A and L B form a pair of dual operators explicitly realized in pq-variables, much similar to the example of s.2. Hence, the corresponding modular transformation is nothing but the Fourier transform, in accordance with [60,61].…”