Large N limit of beta-ensembles and deformed Seiberg-Witten relations
arXiv:1206.1696 · doi:10.1007/JHEP08(2012)046
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 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.
21 pages, 1 figure. Minor corrections, published version
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