N=2 supersymmetric gauge theories and quantum integrable systems

Understand

We study N=2 supersymmetric gauge theories on the product of a two-sphere and a cylinder.

• We show that the low-energy dynamics of a BPS sector of such a theory is described by a quantum integrable system, with the Planck constant set by the inverse of the radius of the sphere.
• If the sphere is replaced with a hemisphere, then our system reduces to an integrable system of the type studied by Nekrasov and Shatashvili.
• In this case we establish a correspondence between the effective prepotential of the gauge theory and the Yang-Yang function of the integrable system.