Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

Idrisse Khemar (1, 2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IECN - Institut Élie Cartan de Nancy

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.

Mots clés: Integrable Systems; Geometric PDE; Differential Geometry; Homogeneous space; Homogeneous fibre bundle; Skew-symmetric torsion; Symmetric spaces; Harmonic maps; Vertically harmonic maps; Twistors; J-holomorphic curves; F-structures; k-symmetric spaces; Loop Groups; Kac-Moody algebras; Supersymmetry; sigma model; Wess-Zumino; Connection with torsion; Variational calculus

Domaines: Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Idrisse Khemar. Elliptic Integrable Systems: a Comprehensive Geometric Interpretation. Memoirs of the American Mathematical Society, 2012, 219 (1031), ⟨10.1090/S0065-9266-2012-00651-4⟩. ⟨hal-00374546v7⟩