Split Canonical Relations

Cattaneo, Alberto; Contreras, Ivan

Annales Henri Lebesgue, Tome 4 (2021), pp. 155-185.

Un sous-espace Lagrangien $L$ d’un espace vectoriel symplectique faible est appelé Lagrangien scindé s’il a un complément isotrope (donc Lagrangien). Lorsque la structure symplectique est forte, il suffit que $L$ ait un complément fermé, qui peut ensuite être déplacé pour devenir isotrope. Le but de cette note est de développer la théorie des compositions et des réductions des relations canoniques scindées pour les espaces vectoriels symplectiques. Nous donnons des conditions sur un sous-espace coisotrope $C$ d’un espace symplectique faible $V$ qui impliquent que la relation canonique induite $L_C$ de $V$ à $C/C^{\omega }$ est scindée, et en déduisons des conditions suffisantes pour que les relations canoniques scindées soient composables. Nous prouvons que les relations canoniques résultant du modèle sigma de Poisson dans l’approche lagrangienne de la théorie des champs sont scindées, donnant une description des groupïdes symplectiques intégrant les variétés de Poisson en termes de relations canoniques scindées.

A Lagrangian subspace $L$ of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for $L$ to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace $C$ of a weak symplectic space $V$ which imply that the induced canonical relation $L_C$ from $V$ to $C/C^{\omega }$ is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.

Classification : 53D22, 17B63, 53D17
Mots clés : Canonical relations, symplectic reduction, Poisson sigma model, symplectic groupoids

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