On elliptic algebras and large-\(n\) supersymmetric gauge theories (Q2836120)

In this paper the authors continue their study of the known duality between \(5d\), \(N=1^*\), \(\mathrm{U}(n)\) Yang-Mills theory and the elliptic Ruijsenaars-Schneider integrable \(n\)-body system. They also study the resulting duality in the large \(n\) limit which leads to the understanding of the so-called non-Abelian finite difference intermediate long wave hydrodynamics model, a generalization of the large \(n\) (or hydrodynamical) limit of the Ruijsenaars-Schneider \(n\)-body system, in terms of the large \(n\) (or large color) limit of the original gauge theory.NEWLINENEWLINEMore precisely, the authors consider a certain \(\mathrm{U}(n)\) gauge theory on \({\mathbb R}^4\times S^1\) of \(\hat{A}_0\)-type with eight supercharges whose known dual theory is the elliptic Ruijsenaars-Schneider integrable \(n\)-body system. The aforementioned \(5d\) gauge theory has already a known description in the \(n\rightarrow +\infty\) limit in terms of a \(3d\) quiver gauge theory with four supercharges. The understanding of the original \(5d\) gauge theory in a novel slightly different \(n\rightarrow +\infty\) limit and making use of the known gauge theory/integrable system duality dictionary allows the authors to obtain a novel \(n\rightarrow +\infty\) limit of the elliptic Ruijsenaars-Schneider integrable \(n\)-body system. This so-called \textit{non-Abelian finite difference intermediate long wave hydrodynamical limit} is a non-Abelian and finite difference (i.e., effective) generalization of the previously known hydrodynamical limit of the Ruijsenaars-Schneider integrable system.