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This article is cited in 27 scientific papers (total in 27 papers)
Resolvents and Seiberg–Witten representation for a Gaussian $\beta$-ensemble
A. D. Mironovab, A. Yu. Morozovb, A. V. Popolitovb, Sh. R. Shakirovbc a Lebedev Physical Institute, RAS, Moscow, Russia
b Institute for Theoretical and Experimental Physics, Moscow,
Russia
c Department of Mathematics,
University of California, Berkeley, CA, USA
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 $\beta$-ensembles. But because the integrability and Harer–Zagier topological recursion are still unavailable for $\beta$-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.
Keywords:
matrix model, $\beta$-ensemble, integrability, Seiberg–Witten theory.
Received: 17.05.2011
Citation:
A. D. Mironov, A. Yu. Morozov, A. V. Popolitov, Sh. R. Shakirov, “Resolvents and Seiberg–Witten representation for a Gaussian $\beta$-ensemble”, TMF, 171:1 (2012), 96–115; Theoret. and Math. Phys., 171:1 (2012), 505–522
Linking options:
https://www.mathnet.ru/eng/tmf6915https://doi.org/10.4213/tmf6915 https://www.mathnet.ru/eng/tmf/v171/i1/p96
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