## 19/10/2009, Monday, 16:30–17:30

I describe four dimensional $N=2$ supersymmetric gauge theory in the Omega-background with the two dimensional $N=2$ super-Poincaré invariance. I explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional $N=2$ theory. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. I present the thermodynamic-Bethe-ansatz like formulae for this Yang-Yang function and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. Particular examples of the many-body systems include the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions. Gauge theory gives a complete characterization of the $L^2$-spectrum for these integrable systems.