On the Coulomb branch of a marginal deformation of Script N = 4 SUSY Yang-Mills

We determine the exact vacuum structure of a marginal deformation of N=4 SUSYYang-Mills with gauge group U(N). The Coulomb branch of the theory consists ofseveral sub-branches which are governed by complex curves of the formSigma_{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 branchespermuted by SL(2,Z) transformations. We determine the curve by solving arelated matrix model in the planar limit according to the prescription ofDijkgraaf and Vafa, and also by explicit instanton calculations using a form oflocalization on the instanton moduli space. We find that Sigma_{n} coincideswith the spectral curve of the n-body Ruijsenaars-Schneider system. Our resultsimply that the theory on each sub-branch is holomorphically equivalent tocertain five-dimensional gauge theory with eight supercharges. This equivalencealso implies the existence of novel confining branches in five dimensions.Comment: LaTeX file. 48 page