Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory

Details Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory

2010-06-07

arxiv.org/abs/1006.1214v3

Phys. Rev. D 82, 025020 (2010)

Physics

High Energy Physics

High Energy Physics - Theory

17 pages, submitted to PRD; v2, typos corrected, references added; v3, published version

Scientific paper

10.1103/PhysRevD.82.025020

It is recently claimed by Nekrasov and Shatashvili that the $\mathcal {N}=2$ gauge theories in the $\Omega$ background with $\epsilon_1=\hbar, \epsilon_2=0$ are related to the quantization of certain algebraic integrable systems. We study the special case of SU(2) pure gauge theory, the corresponding integrable model is the A$_1$ Toda model, which reduces to the sine-Gordon quantum mechanics problem. The quantum effects can be expressed as the WKB series written analytically in terms of hypergeometric functions. We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space. We also discuss the relation between the electric-magnetic duality of gauge theory and the action-action duality of the integrable system.

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