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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)].

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