Prepotential and instanton corrections in Script N = 2 supersymmetric SU(N1) × SU(N2) Yang Mills theories

In this paper we analyse the non-hyperelliptic Seiberg-Witten curves derivedfrom M-theory that encode the low energy solution of N=2 supersymmetrictheories with product gauge groups. We consider the case of a SU(N_1)xSU(N_2)gauge theory with a hypermultiplet in the bifundamental representation togetherwith matter in the fundamental representations of SU(N_1) and SU(N_2). By meansof the Riemann bilinear relations that hold on the Riemann surface defined bythe Seiberg--Witten curve, we compute the logarithmic derivative of theprepotential with respect to the quantum scales of both gauge groups. As anapplication we develop a method to compute recursively the instantoncorrections to the prepotential in a straightforward way. We present explicitformulas for up to third order on both quantum scales. Furthermore, we extendthose results to SU(N) gauge theories with a matter hypermultiplet in thesymmetric and antisymmetric representation. We also present some non-trivialchecks of our results.Comment: 21 pages, 2 figures, minor changes and references adde