The Matrix Model Curve Near the Singularities

In N=1 supersymmetric SO(N)/USp(2N) gauge theories with the tree-level superpotential W(\Phi) that is an arbitrary polynomial of the adjoint matter \Phi, the massless fluctuations about each quantum vacuum are described by U(1)^n gauge theory. By turning on the parameters of W(\Phi) to the special values, the singular vacua where the additional fields become massless can be reached. Using the matrix model prescription, we study the intersections of n=0 and n=1 branches. The general formula for the matrix model curve at the singularity which is valid for arbitrary N is obtained and this generalizes the previous results for small values of N from strong-coupling approach. Applying the analysis to the degenerated case, we also obtain a general matrix model curve which is not only valid at a special point but also on the whole branch.