Double Elliptic Dynamical Systems From Generalized Mukai - Sklyanin Algebras

TL;DR

This paper introduces a double-elliptic integrable system based on generalized Mukai-Sklyanin algebras, revealing its geometric structure and potential links to supersymmetric gauge theories.

Contribution

It develops a new double-elliptic dynamical system framework using Mukai-Sklyanin algebras and explores its geometric and physical implications.

Findings

Identified the associated elliptically fibered rational surface.

Analyzed the two-body double-elliptic system in detail.

Suggested generalizations for N-body systems and applications to SUSY gauge theories.

Abstract

We consider the double-elliptic generalisation of dynamical systems of Calogero-Toda-Ruijsenaars type using finite-dimensional Mukai-Sklyanin algebras. The two-body system, which involves an elliptic dependence both on coordinates and momenta, is investigated in detail and the relation with Nambu dynamics is mentioned. We identify the 2D complex manifold associated with the double elliptic system as an elliptically fibered rational ("1/2K3 ") surface. Some generalisations are suggested which provide the ground for a description of the N-body systems. Possible applications to SUSY gauge theories with adjoint matter in $d=6$ with two compact dimensions are discussed.