Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors

Marí Beffa, Gloria

Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2405-2434.

Nous décrivons un repère mobile non local pour les courbes de spineurs purs dans $\mathrm{O}(2m,2m)/P$, et la base correspondante d’invariants différentiels. Nous montrons que l’espace des invariants différentiels de type Schwarzien définit une sous-variété de crochets de Poisson géométriques de spineurs purs. La restriction résultante est donnée par un systéme découplé de crochets de Poisson de KdV . Nous définissons une généralisation de l’évolution de Schwarz-KdV pour les courbes de spineurs purs et nous montrons que, en restriction à un niveau fixé, cela induit un système d’équations de KdV découplé pour les invariants de type projectif. Nous décrivons par ailleurs la transformation correspondante de Miura et le système non commutatif modifié de KdV.

In this paper we describe a non-local moving frame along a curve of pure spinors in $\mathrm{O}(2m,2m)/P$, and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generalization of the Schwarzian-KdV evolution for pure spinor curves and we prove that it induces a decoupled system of KdV equations on the invariants of projective type, when restricted to a certain level set. We also describe its associated Miura transformation and non-commutative modified KdV system.

Classification : 37K, 53D55
Keywords: Moving frame, spinor evolutions, geometric Poisson brackets, KdV equations, differential invariants, Miura transformation, non-commutative modified KdV system
Mot clés : repères mobiles, evolution de spineurs, crochet de Poisson géométriques, équations de KdV, invariants différentiels, transformation de Miura, système non commutatif modifié de KdV

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