The Universal Generating Function of Analytical Poisson Structures

The notion of generating functions of Poisson structures was first studied in math.SG/0312380.They are special functions which induce, on open subsets of $\R^d$, a Poisson structure together with the local symplectic groupoid integrating it. A universal generating function was provided in terms of a formal power series coming from Kontsevich star product. The present article proves that this universal generating function converges for analytical Poisson structures and compares the induced local symplectic groupoid with the phase space of Karasev--Maslov.Comment: 15 pages, 2 figures, shorter version, introductive part remove

Symplectic Microgeometry I: Micromorphisms

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. 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.Comment: 16 pages, 1 figure, typos correcte

The Universal Generating Function of Analytical Poisson Structures

Generating functions of Poisson structures are special functions which induce, on open subsets of $$\mathbb{R}^d$$ , a Poisson structure together with the local symplectic groupoid integrating it. In a previous paper by A. S. Cattaneo, G. Felder and the author, a universal generating function was provided in terms of a formal power series coming from Kontsevich star product. The present article proves that this universal generating function converges for analytical Poisson structures and shows that the induced local symplectic groupoid coincides with the phase space of Karasev-Maslo

A Graphical Calculus for Classical and Quantum Microformal Morphisms

We develop a graphical calculus for the microformal or thick morphisms introduced by Ted Voronov. This allows us to write the infinite series arising from pullbacks, compositions, and coordinate transformations of thick morphisms as sums over bipartite trees. The methods are inspired by those employed by Cattaneo-Dherin-Felder in their work on formal symplectic groupoids. We also extend this calculus to quantum thick morphisms, which are special types of Fourier integral operators quantizing classical thick morphisms. The relationship between the calculi for classical and quantum thick morphisms resembles the relationship between the semi-classical and full perturbative expansions over Feynman diagrams in quantum field theory.Comment: 23 pages, 3 figures - changes: addition to acknowledgment

Formal Lagrangian Operad

Given a symplectic manifold M, we may define an operad structure on the the spaces Ok of the Lagrangian submanifolds of Mk × M via symplectic reduction. If M is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semiclassical part of Kontsevich’s deformation of C∞d is a deformation of the trivial symplectic groupoid structure of T∗d