On a singular Fredholm-type integral equation arising in N=2 super-Yang-Mills theories

In this work we study the Nekrasov-Shatashvili limit of the Nekrasov instanton partition function of Yang-Mills field theories with N=2 supersymmetry and gauge group SU(Nc). The theories are coupled with Nf flavors of fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved when Nf=2Nc. The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. The instanton energy and thus the instanton contribution to the prepotential of the gauge theory is computed in closed form.