Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

Abstract

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k^{\prime }$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k^{\prime }}$ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the primitive case ($m<m_{k^{\prime }}$), the determined case ($m_{k^{\prime }}\le m\le k^{\prime }-1$) and the underdetermined case ($m\ge k^{\prime }$). 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.