S-duality and supersymmetry on curved manifolds

TL;DR

This paper investigates the nature of S-duality in ${ m N}=2$ supersymmetric abelian theories on curved manifolds, proposing a Fourier transform interpretation over the traditional Legendre transform, and introduces a coholomological prepotential concept.

Contribution

It clarifies the proper mathematical interpretation of S-duality in these theories and introduces the notion of a coholomological prepotential for abelian theories.

Findings

Localization and S-duality as Legendre transform are incompatible.

S-duality should be interpreted as Fourier transform.

Proposes a coholomological prepotential matching non-abelian theories.

Abstract

We perform a systematic study of S-duality for ${\cal N}=2$ supersymmetric non-linear abelian theories on a curved manifold. Localization can be used to compute certain supersymmetric observables in these theories. We point out that localization and S-duality acting as a Legendre transform are not compatible. For these theories S-duality should be interpreted as Fourier transform and we provide some evidence for this. We also suggest the notion of a coholomological prepotential for an abelian theory that gives the same partition function as a given non-abelian supersymmetric theory.