We study the relation between the $\mathcal{N}=2$ gauge theory in the $\Omega$ background and the quantized integral system recently proposed by Nekrasov and Shatashvili. We focus on the simplest case, the pure Yang-Mills theory with the $SU(2)$ gauge group and the corresponding sine-Gordon integral model on the complex plane. We analyze the periodic wave function and the corresponding energy spectrum of the sine-Gordon quantum mechanics, and find this model contains information of the low energy effective theory of the gauge theory.

• Received 8 January 2010

DOI:https://doi.org/10.1103/PhysRevD.81.105017