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

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.

arXiv comment21 pages, 1 figure. Minor corrections, published version

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• Argyres-Douglas theories and Liouville Irregular States
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
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• Notes on Mayer Expansions and Matrix Models
• Seiberg-Witten equations and non-commutative spectral curves in Liouville theory
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Quantum integrability of $\mathcal{N}=2$ 4d gauge theories
• Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model
• Seiberg-Witten period relations in Omega background
• Large-$N$ Free Energy of Chiral $\mathcal{N}=2$ Chern-Simons-Matter Theories