Chiral rings , anomalies and loop equations in N = 1 ∗ gauge theories

We examine the equivalence between the Konishi anomaly equations and the matrix model loop equations in N = 1 gauge theories, the mass deformation of N = 4 supersymmetric Yang-Mills. We perform the superfunctional integral of two adjoint chiral superfields to obtain an effective N = 1 theory of the third adjoint chiral superfield. By choosing an appropriate holomorphic variation, the Konishi anomaly equations correctly reproduce the loop equations in the corresponding three-matrix model. We write down the field theory loop equations explicitly by using a noncommutative product of resolvents peculiar to N = 1 theories. The field theory resolvents are identified with those in the matrix model in the same manner as for the generic N = 1 gauge theories. We cover all the classical gauge groups. In SO/Sp cases, both the one-loop holomorphic potential and the Konishi anomaly term involve twisting of index loops to change a one-loop oriented diagram to an unoriented diagram. The field theory loop equations for these cases show certain inhomogeneous terms suggesting the matrix model loop equations for the RP 2 resolvent. Email: [email protected]