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

TL;DR

This work extends the Seiberg–Witten framework to 5d and 6d SUSY gauge theories on $R^4\times S^1$ and $R^4\times T^2$, identifying integrable-system correspondences: twisted $XXZ$ chains for 5d and an $XYZ$ chain for 6d. It derives spectral curves and generating differentials from these integrable models, formulates perturbative prepotentials including Chern–Simons terms, and connects the results to M-theory geometrical engineering and toric diagrams via brane pictures. The paper also shows how degenerations (to relativistic Toda and other limits) encode various field-theory limits (pure YM, matter content) and discusses constraints in 6d, notably the anomalous consistency condition $2N_c=N_f$. Overall, the work highlights deep links between integrable systems, string/brane constructions, and low-energy effective actions across dimensions, providing a unified stringy underpinning for these gauge theories.

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.