A Lagrangian subspace 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 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 of a weak symplectic space which imply that the induced canonical relation from to 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.
[Arn67] Characteristic class entering in quantization conditions, Funkts. Anal. Prilozh., Volume 1 (1967) no. 1, pp. 1-14 | Zbl
[CF01] Poisson sigma models and symplectic groupoids, Quantization of Singular Symplectic Quotients (Landsman, Nicolaas P.; J., Pflaum Markus; Schlichenmaier, Martin, eds.) (Progress in Mathematics), Volume 198, Springer, 2001, pp. 61-93 | DOI | MR
[CMR12] Classical and quantum Lagrangian field theories with boundary (2012) (https://arxiv.org/abs/1207.0239)
[Con13] Relational symplectic groupoids and Poisson sigma models with boundary (2013) (Ph. D. Thesis)
[Rud91] Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1991 | Zbl