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

TL;DR

This paper investigates the large N limit of beta-ensembles related to Liouville conformal blocks and their connection to deformed Seiberg-Witten relations, revealing how quantized differentials emerge from classical forms.

Contribution

It demonstrates that the free energy in the large N limit satisfies Seiberg-Witten relations and introduces a method to derive quantized differentials from classical ones using differential operators.

Findings

The free energy obeys Seiberg-Witten relations in the large N limit.

Quantized Seiberg-Witten differentials can be obtained from classical forms via differential operators.

The approach connects beta-ensembles with deformed Seiberg-Witten theory through perturbative analysis.

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.