Supersymmetric vacua and Bethe ansatz

TL;DR

This paper establishes a correspondence between supersymmetric vacua in various dimensions and eigenstates of integrable spin chain models, revealing deep connections between quantum field theories and integrable systems.

Contribution

It introduces a novel framework linking supersymmetric gauge theories with integrable spin chains, extending to various models and algebraic structures.

Findings

Supersymmetric vacua correspond to eigenstates of integrable spin chains.

Mappings include Heisenberg, XXZ, XYZ spin chains to different dimensional theories.

Proposes new symmetry actions of Yangian and quantum affine algebras on quantum field theories.

Abstract

An announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg 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 XYZ spin chain and eight-vertex model are related to the four dimensional theory compactified on a torus. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schroedinger 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. This paper accompanies arXiv:0901.4748