Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

TL;DR

This paper provides a comprehensive geometric interpretation of elliptic integrable systems associated with symmetric spaces, linking them to sigma models with Wess-Zumino terms and twistors, and classifies cases based on an integer parameter.

Contribution

It introduces a unified geometric framework for all elliptic integrable systems related to symmetric spaces, including new interpretations via sigma models and twistors.

Findings

Classification of elliptic integrable systems into three cases

Interpretation in terms of sigma models with Wess-Zumino terms

Geometric interpretation using twistors

Abstract

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the primitive case ($m < m_{k'}$), the determined case ($m_{k'}\leq m \leq k'-1$) and the underdetermined case ($m \geq k'$). We prove that we have an interpretation in terms of a sigma model with a Wess-Zumino term. Moreover we prove that we have a geometric interpretation in terms of twistors. See the abstract in the paper for more precisions.