Deformed N=2 theories, generalized recursion relations and S-duality

TL;DR

This paper explores non-perturbative aspects of N=2 superconformal field theories in four dimensions, deriving recursion relations and analyzing S-duality properties using localization, instanton calculations, and modular functions.

Contribution

It introduces a modular anomaly recursion relation for deformed N=2 theories and demonstrates how S-duality can be preserved through redefinitions of the prepotential and coupling.

Findings

Derived explicit instanton corrections to the partition function.

Established a modular anomaly recursion relation involving epsilon-deformations.

Showed S-duality relations can be maintained with suitable redefinitions up to third order.

Abstract

We study the non-perturbative properties of N=2 super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic epsilon-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov's approach. These results allow to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly epsilon-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.