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

TL;DR

This paper demonstrates the equivalence between Konishi anomaly equations and matrix model loop equations in N=1* gauge theories, extending the analysis to classical gauge groups and including unoriented diagrams.

Contribution

It explicitly derives the field theory loop equations from the Konishi anomalies and relates them to matrix model equations for all classical gauge groups, including SO/Sp.

Findings

Konishi anomaly equations reproduce matrix model loop equations.

Field theory resolvents match matrix model resolvents.

In SO/Sp cases, unoriented diagrams are incorporated into the loop equations.

Abstract

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 RP2 resolvent.