SUSY field theories, integrable systems and their stringy/brane origin -- II

TL;DR

This paper explores the connections between five and six-dimensional supersymmetric gauge theories with integrable spin chains, using string theory and brane frameworks to relate gauge theories to algebraic and geometric structures.

Contribution

It establishes new correspondences between 5d/6d SUSY gauge theories with matter and specific integrable spin chains, extending the gauge/brane/geometry relations.

Findings

5d theories with fundamental matter relate to twisted XXZ spin chains

Group product 5d theories correspond to higher rank spin chains

6d theories with fundamental matter relate to XYZ spin chain via Sklyanin algebra

Abstract

Five and six dimensional SUSY gauge theories, with one or two compactified directions, are discussed. The 5d theories with the matter hypermultiplets in the fundamental representation are associated with the twisted $XXZ$ spin chain, while the group product case with the bi-fundamental matter corresponds to the higher rank spin chains. The Riemann surfaces for $6d$ theories with fundamental matter and two compact directions are proposed to correspond to the $XYZ$ spin chain based on the Sklyanin algebra. We also discuss the obtained results within the brane and geometrical engeneering frameworks and explain the relation to the toric diagrams.