Benoit Dherin

Formal Symplectic Groupoid

The multiplicative structure of the trivial symplectic groupoid over ℝd associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich’s deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge–Kutta theory of the numeric integration of ODE’s.

Symplectic microgeometry I: micromorphisms

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a symmetric 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 microgeometry II: generating functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.

Integration of Lie algebroid comorphisms

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.

Symplectic microgeometry III: Monoids

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and Lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.

The Universal Generating Function of Analytical Poisson Structures

AbstractGenerating functions of Poisson structures are special functions which induce, on open subsets of

G-systems and deformation of G-actions on Rd

\mathbb{R}^d

Deformations of momentum maps and G-systems

, 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–Maslov