On the Coulomb Branch of a Marginal Deformation of N=4 SUSY Yang-Mills

We determine the exact vacuum structure of a marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N). The Coulomb branch of the theory consists of several sub-branches which are governed by complex curves of the form Sigma_{n_{1}} U Sigma_{n_{2}} U Sigma_{n_{3}} of genus N=n_{1}+n_{2}+n_{3}. Each sub-branch intersects with a family of Higgs and Confining branches permuted by SL(2,Z) transformations. We determine the curve by solving a related matrix model in the planar limit according to the prescription of Dijkgraaf and Vafa, and also by explicit instanton calculations using a form of localization on the instanton moduli space. We find that Sigma_{n} coincides with the spectral curve of the n-body Ruijsenaars-Schneider system. Our results imply that the theory on each sub-branch is holomorphically equivalent to certain five-dimensional gauge theory with eight supercharges. This equivalence also implies the existence of novel confining branches in five dimensions.