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

International audienceWe study the non-perturbative properties of N=2 super conformal fieldtheories in four dimensions using localization techniques. In particular weconsider SU(2) gauge theories, deformed by a generic epsilon-background, withfour fundamental flavors or with one adjoint hypermultiplet. In both cases weexplicitly compute the first few instanton corrections to the partitionfunction and the prepotential using Nekrasov's approach. These results allow toreconstruct exact expressions involving quasi-modular functions of the baregauge coupling constant and to show that the prepotential terms satisfy amodular anomaly equation that takes the form of a recursion relation with anexplicitly epsilon-dependent term. We then investigate the implications of thisrecursion relation on the modular properties of the effective theory and findthat with a suitable redefinition of the prepotential and of the effectivecoupling it is possible, at least up to the third order in the deformationparameters, to cast the S-duality relations in the same form as they appear inthe Seiberg-Witten solution of the undeformed theory