Split Canonical Relations
Annales Henri Lebesgue, Volume 4 (2021), pp. 155-185.


Keywords Canonical relations, symplectic reduction, Poisson sigma model, symplectic groupoids


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 ω 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] Arnol’d, Vladimir I. Characteristic class entering in quantization conditions, Funkts. Anal. Prilozh., Volume 1 (1967) no. 1, pp. 1-14 | Zbl

[Cat14] Cattaneo, Alberto S. Coisotropic submanifolds and dual pairs, Lett. Math. Phys., Volume 104 (2014) no. 3, pp. 243-270 | DOI | MR | Zbl

[CC15] Cattaneo, Alberto S.; Contreras, Ivan Relational symplectic groupoids, Lett. Math. Phys., Volume 105 (2015) no. 5, pp. 723-767 | DOI | MR | Zbl

[CF00] Cattaneo, Alberto S.; Felder, Giovanni A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys., Volume 212 (2000) no. 3, pp. 591-611 | DOI | MR | Zbl

[CF01] Cattaneo, Alberto S.; Felder, Giovanni 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

[CM14] Cattaneo, Alberto S.; Mnëv, Pavel Wave relations, Commun. Math. Phys., Volume 332 (2014) no. 3, pp. 1083-1111 | DOI | MR | Zbl

[CMR12] Cattaneo, Alberto S.; Mnëv, Pavel; Reshetikhin, Nikolai Classical and quantum Lagrangian field theories with boundary (2012) (https://arxiv.org/abs/1207.0239)

[Con13] Contreras, Ivan Relational symplectic groupoids and Poisson sigma models with boundary, Ph. D. Thesis, Zürich University, Switzerland (2013)

[EM99] Everitt, William N.; Markus, Lawrence Boundary value problems and symplectic algebra for ordinary and quasi-differential operators, Mathematical Surveys and Monographs, 61, American Mathematical Society, 1999 | MR | Zbl

[Got82] Gotay, Mark J. On coisotropic imbeddings of presymplectic manifolds, Proc. Am. Math. Soc., Volume 84 (1982), pp. 111-114 | DOI | MR | Zbl

[Kon03] Kontsevich, Maxim Lvovich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216 | DOI | MR | Zbl

[KS82] Kalton, Nigel J.; Swanson, R. Charles A symplectic reflexive Banach space with no Lagrangian subspaces, Trans. Am. Math. Soc., Volume 273 (1982) no. 1, pp. 385-392 | DOI | Zbl

[LW15] Lorand, Jonathan; Weinstein, Alan D. Decomposition of (co)isotropic relations (2015) (https://arxiv.org/abs/1509.04035) | Zbl

[Rud91] Rudin, Walter Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1991 | Zbl

[Wei71] Weinstein, Alan D. Symplectic manifolds and their Lagrangian submanifolds, Adv. Math., Volume 6 (1971), pp. 329-346 | DOI | MR | Zbl

[Wei10] Weinstein, Alan D. Symplectic Categories, Port. Math. (N.S.), Volume 67 (2010) no. 2, pp. 261-278 | DOI | MR | Zbl