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.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 171, No. 1, pp. 96–115, April, 2012.
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Mironov, A.D., Morozov, A.Y., Popolitov, A.V. et al. Resolvents and Seiberg-Witten representation for a Gaussian β-ensemble. Theor Math Phys 171, 505–522 (2012). https://doi.org/10.1007/s11232-012-0049-y
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DOI: https://doi.org/10.1007/s11232-012-0049-y