Quantum integrability and supersymmetric vacua

TL;DR

This paper establishes a correspondence between supersymmetric vacua in 2D N=2 gauge theories and eigenstates of integrable spin chain Hamiltonians, revealing deep links between quantum field theory and integrable systems.

Contribution

It demonstrates a novel isomorphism between the equivariant quantum cohomology of cotangent bundles and quantum integrals of motion of spin chains, connecting supersymmetric vacua with integrable models.

Findings

Supersymmetric vacua correspond to eigenstates of integrable spin chains.

Isomorphism between quantum cohomology and spin chain integrals established.

Detailed review of the Heisenberg spin chain and 2D U(N) theory correspondence.

Abstract

This is an announcement of some of the 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 correspondence between the Heisenberg spin chain and the two dimensional U(N) theory with fundamental hypermultiplets is reviewed in detail. We demonstrate the isomorphism of the equivariant quantum cohomology of the cotangent bundle to the Grassmanian manifold Gr(N,L) and the ring of quantum integrals of motion of the length L SU(2) XXX spin chain, in the N-particle sector. This paper accompanies arXiv:0901.4744