Critical points of glueball superpotentials and equilibria of integrable systems

We compare the matrix model and integrable system approaches to calculatingthe exact vacuum structure of general N=1 deformations of either the basic N=2theory or its generalization with a massive adjoint hypermultiplet, the N=2*theory. We show that there is a one-to-one correspondence between arbitrarycritical points of the Dijkgraaf-Vafa glueball superpotential and equilibriumconfigurations of the associated integrable system. The latter being either theperiodic Toda chain, for N=2, or the elliptic Calogero-Moser system, for N=2*.We show in both cases that the glueball superpotential at the crtical pointequals the associated Hamiltonian. Our discussion includes an analysis of thevacuum structure of the N=1* theory with an arbitrary tree-level superpotentialfor one of the adjoint chiral fields.Comment: 15 pages, JHEP3.cls, Hamiltonian and glueball superpotential agree for N=1* as wel