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Symplectic microgeometry. I: Micromorphisms. (English) Zbl 1201.53082
The authors introduce the notion of symplectic microfolds and symplectic micromorphisms. They form a monoidal category, which is a version of the “category” of symplectic manifolds and canonical relations obtained by localizing them around lagrangian submanifolds in the spirit of Milnor’s microbundles.
Symplectic microfolds are equivalence classes \((M,A)\) of pairs consisting of a symplectic manifold \(M\) and a Lagrangian submanifold \(A\subset M\), called the core. The equivalence reflects the fact that these objects really describe the geometry of a neighborhood of \(A\) in \(M\).
This extended category can be used as a sort of heuristic guideline in an attempt to quantize Poisson manifolds in a geometric way, see [A. Weinstein, Noncommutative geometry and geometric quantization. Symplectic Geometry and Mathematical Physics, Symplectic geometry and mathematical physics, Proc. Colloq., Aix-en- Provence/ Fr. 1990, Prog. Math. 99, 446–461 (1991; Zbl 0756.58022)].

53D05 Symplectic manifolds (general theory)
53D50 Geometric quantization
Zbl 0756.58022
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