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.