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Five-Dimensional Gauge Theories and Quantum Mechanical Matrix Models

We show how the Dijkgraaf-Vafa matrix model proposal can be extended to describe five-dimensional gauge theories compactified on a circle to four dimensions. This involves solving a certain quantum mechanical matrix model. We do this for the lift of the N=1* theory to five dimensions. We show that the resulting expression for the superpotential in the confining vacuum is identical with the elliptic superpotential approach based on Nekrasov's five-dimensional generalization of Seiberg-Witten theory involving the relativistic elliptic Calogero-Moser, or Ruijsenaars-Schneider, integrable system.