Abstract. We study four dimensional N = 2 supersymmetric gauge theory in the Ω-background with the two dimensional N = 2 super-Poincare invariance. We 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. The ε-parameter of the Ω-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. 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. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L 2 -spectrum. We very briefly discuss the quantization of Hitchin system.

This note is a short announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg SU (2) XXX spin chain is mapped to the two dimensional U (N ) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model are related to the four dimensional theory compactified on T 2 .The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N = 2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories.a On leave of absence from ITEP, Moscow, Russia for discussions. The results of this note, as well as those in [14], were presented at various conferences and workshops 3 and we thank the organizers for the opportunity to present our results. We thank various agencies and institutions 4 for supporting this research. The gauge theoryHere we give a brief review of the relevant gauge theories. 3 The IHES seminars and the theoretical physics conference dedicated to the 50th anniversary of IHES (Bures-sur-Yvette,

We develop some useful techinques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkahler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce these volumes for a large class of hyperkahler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on IR 4 and ALE manifolds, Hitchin spaces, and moduli spaces of (parabolic)Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schrödinger system. We discuss some applications of our results.