S. Shatashvili, «SUSY gauge theories and quantization of integrable systems»

I describe four dimensional N=2 supersymmetric gauge theory in the Ω-background with two dimensional N=2 super-Poincare 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 a 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 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 L2-spectrum for these integrable systems.