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We study four dimensional N=2 supersymmetric gauge theory in theOmega-background with the two dimensional N=2 super-Poincare invariance. Weexplain how this gauge theory provides the quantization of the classicalintegrable system underlying the moduli space of vacua of the ordinary fourdimensional N=2 theory.  The epsilon-parameter of the Omega-background is identified with the Planckconstant, the twisted chiral ring maps to quantum Hamiltonians, thesupersymmetric vacua are identified with Bethe states of quantum integrablesystems. This four dimensional gauge theory in its low energy description hastwo dimensional twisted superpotential which becomes the Yang-Yang function ofthe integrable system. We present the thermodynamic-Bethe-ansatz like formulaefor these functions and for the spectra of commuting Hamiltonians following thedirect computation in gauge theory. The general construction is illustrated atthe examples of the many-body systems, such as the periodic Toda chain, theelliptic Calogero-Moser system, and their relativistic versions, for which wepresent a complete characterization of the L^2-spectrum. We very brieflydiscuss the quantization of Hitchin system.

Quantization of Integrable Systems and Four Dimensional Gauge Theories