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

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 ǫ-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton cor- rections to the partition function and the prepotential using Nekrasov's approach. These results allow us 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 ǫ-dependent term. We then investigate the implications of this recursion relation on the modular prop- erties 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 deforma- tion parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.