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arXiv · 1206.1696

Large N limit of beta-ensembles and deformed Seiberg-Witten relations

Jean-Emile Bourgine
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Abstract

We study the beta-ensemble that represents conformal blocks of Liouville theory on the sphere. This quantity is related through AGT conjecture to the Nekrasov instanton partition function of 4d $\mathcal{N}=2$ SU(2) gauge theory with four flavors. We focus on the large N limit, equivalent to the Nekrasov-Shatashvili limit where one of the Omega-background deformation parameters is vanishing. A quantized Seiberg-Witten differential form is defined perturbatively in h-bar as the singular part of the beta-ensemble resolvent. Using the Dyson collective field action, we show that the free energy obeys the Seiberg-Witten relations. As suggested by Mironov and Morozov, the quantized differential form can be obtained from the classical one by the action of a differential operator in the hypermultiplet masses and the Coulomb branch modulus.

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Jean-Emile Bourgine. 2012-08-20. Large N limit of beta-ensembles and deformed Seiberg-Witten relations. https://doi.org/10.1007/jhep08(2012)046

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