Abstract
A Lagrangian subspace $L$ of a weak symplectic vector space is called \emph{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.
Highlights
The functorial description of classical and quantum field theories requires enhancement of the usual symplectic category, allowing infinite dimensional objects, and allowing general canonical relations, not just symplectomorphisms, as morphisms
The last part of the paper (Section 6) is devoted to proving that the evolution relations arising from the Poisson sigma model (PSM), a two dimensional topological theory, are split canonical relations obeying a neat intersection condition which allows compositions and reductions
Example 2.13. — Each Lagrangian subspace L ⊆ V may be identified with the canonical relation {0} × L ⊆ {0} ⊕ V from the zero dimensional vector space {0} to V
Summary
The functorial description of classical and quantum field theories requires enhancement of the usual symplectic category, allowing infinite dimensional objects (weak symplectic manifolds), and allowing general canonical relations, not just symplectomorphisms, as morphisms. The objective of this paper is to analyze the special case in which we allow the objects to be (infinite dimensional) weak symplectic vector spaces and the morphisms to be split canonical relations, which are isotropic subspaces with isotropic complements. The last part of the paper (Section 6) is devoted to proving that the evolution relations arising from the Poisson sigma model (PSM), a two dimensional topological theory, are split canonical relations obeying a neat intersection condition which allows compositions and reductions These considerations help us to provide an alternative and shorter proof that such evolution relations are Lagrangian for the ANNALES HENRI LEBESGUE. The study of split Lagrangian spaces can be naturally extended to the framework of Banach manifolds, for which the existence of split Lagrangian submanifolds implies the existence of complementary isotropic smooth distributions, which play an important role in the symplectic formulation of field theories with boundary. We intend to extend the construction of split coisotropic subspaces to Banach manifolds, and adapt the existence of c-triples (Definition 3.2) to the existence and local uniqueness of coisotropic embeddings of finite dimensional presymplectic manifolds, due to Gotay [Got82]
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