Resolvents and Seiberg-Witten representation for a Gaussian β-ensemble

Abstract

The exact free energy of a matrix model always satisfies the Seiberg-Witten equations on a complex curve defined by singularities of the semiclassical resolvent. The role of the Seiberg-Witten differential is played by the exact one-point resolvent in this case. We show that these properties are preserved in the generalization of matrix models to β-ensembles. But because the integrability and Harer-Zagier topological recursion are still unavailable for β-ensembles, we must rely on the ordinary Alexandrov-Mironov-Morozov/Eynard-Orantin recursion to evaluate the first terms of the genus expansion. We restrict our consideration to the Gaussian model.