### Metadata

### Abstract

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.

### References

[Arn67] Characteristic class entering in quantization conditions, Funkts. Anal. Prilozh., Volume 1 (1967) no. 1, pp. 1-14 | Zbl 0175.20303

[Cat14] Coisotropic submanifolds and dual pairs, Lett. Math. Phys., Volume 104 (2014) no. 3, pp. 243-270 | Article | MR 3164609 | Zbl 1326.53119

[CC15] Relational symplectic groupoids, Lett. Math. Phys., Volume 105 (2015) no. 5, pp. 723-767 | Article | MR 3339206 | Zbl 1351.53100

[CF00] A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys., Volume 212 (2000) no. 3, pp. 591-611 | Article | MR 1779159 | Zbl 1038.53088

[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 | Article | MR 1938552

[CM14] Wave relations, Commun. Math. Phys., Volume 332 (2014) no. 3, pp. 1083-1111 | Article | MR 3262621 | Zbl 1300.53069

[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)

[EM99] Boundary value problems and symplectic algebra for ordinary and quasi-differential operators, Mathematical Surveys and Monographs, Volume 61, American Mathematical Society, 1999 | MR 1647856 | Zbl 0909.34001

[Got82] On coisotropic imbeddings of presymplectic manifolds, Proc. Am. Math. Soc., Volume 84 (1982), pp. 111-114 | Article | MR 633290 | Zbl 0476.53020

[Kon03] Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216 | Article | MR 2062626 | Zbl 1058.53065

[KS82] A symplectic reflexive Banach space with no Lagrangian subspaces, Trans. Am. Math. Soc., Volume 273 (1982) no. 1, pp. 385-392 | Article | Zbl 0525.58018

[LW15] Decomposition of (co)isotropic relations (2015) (https://arxiv.org/abs/1509.04035) | Zbl 1362.18005

[Rud91] Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1991 | Zbl 0867.46001

[Wei71] Symplectic manifolds and their Lagrangian submanifolds, Adv. Math., Volume 6 (1971), pp. 329-346 | Article | MR 286137 | Zbl 0213.48203

[Wei10] Symplectic Categories, Port. Math. (N.S.), Volume 67 (2010) no. 2, pp. 261-278 | Article | MR 2662868 | Zbl 1193.53173