Elliptic Integrable Systems: a Comprehensive Geometric Interpretation
Résumé
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
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