Bethe Ansatz and supersymmetric vacua

Supersymmetric vacua of two dimensional Ν = 4 gauge theories with matter, softly broken by the twisted masses down to Ν = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU(2) XXX spin chain which is mapped to the two dimensional U(N) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactifled on a circle, the XYZ spin chain and eight-vertex model which are related to the four dimensional theorycompactifled on T². A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as cotangent bundles to (partial) flag varieties and the ring of quantum integrals of motion of various spin chains. 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 like Hubbard model. Compactiflcations of four dimensional Ν = 1 theories on a two-sphere lead to the instanton-corrected Bethe equations.