Quantum integrability and supersymmetric vacua

Abstract

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 eig enstates of integrable spin chain Hamiltonians. Examples include: t he 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 dimensiona l super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model which are related to the fo ur dimensional theory compactified on T 2 . A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as T ? Gr( N,L ) and the ring of quantum integrals of motion of various spin chains. The co rrespondence extends to any spin group, representations, boundary conditions, and inhomog eneity, it includes Sinh-Gordon and non-linear Schr ?odinger models as well as the dynamical spi n chains like Hubbard model. We give the gauge-theoretic interpretation of Drinfeld polynomia ls and Baxter operators. The two-sphere compactifications of the four dimensional N = 2 theories lead to the instanton corrected Bethe equations. We suggest the Yangian, quantum affine, and ellipt ic algebras are a completely novel kind of symmetry of the (collections of the) interacting qua ntum field theories