Symplectic microgeometry I: micromorphisms

  title={Symplectic microgeometry I: micromorphisms},
  author={Alberto S. Cattaneo and Benoit Richard Umbert Dherin and Alan D. Weinstein},
  journal={Journal of Symplectic Geometry},
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 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
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
Formal Lagrangian Operad
It turns out that the semiclassical part of Kontsevich's deformation of () is aDeformation of the trivial symplectic groupoid structure of .
Symplectic microgeometry, IV: Quantization
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the
On Lie algebroids, L1 algebras, and the homotopy Poisson structure on shifted conormal bundles of coisotropic submanifolds
In this semester research project, we will quickly review Lie algebroids, L1 algebras and related structures. We will also elaborate general aspects of the theory of supergeometry. Finally, we will
Relational symplectic groupoids and Poisson sigma models with boundary
We introduce the notion of relational symplectic groupoid as a way to integrate Poisson manifolds in general, following the construction through the Poisson sigma model (PSM) given by Cattaneo and
Double Groupoids and the Symplectic Category
We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects
Quantizations of Momentum Maps and G-Systems
In this note, we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of quantizations
G-Systems and Deformation of G-Actions On Rd
Given a (smooth) action of a Lie group G on Rd we construct a DGA whose Maurer-Cartan elements are in one to one correspondence with some class of defomations of the (induced) G-action on the ring of
Integration of Exact Courant Algebroids
In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [32] inverts our integration.


Formal Lagrangian Operad
It turns out that the semiclassical part of Kontsevich's deformation of () is aDeformation of the trivial symplectic groupoid structure of .
Functoriality for Lagrangian correspondences in Floer theory
Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that
Quantization of symplectic vector spaces over finite fields
In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result,
Categories for the Working Mathematician
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large
Lectures on the geometry of quantization
Symplectic manifolds and their lagrangian submanifolds
Noncommutative Geometry and Geometric Quantization
Categories for the Working Mathematician”, Second edition, Graduate Texts in Mathematics 5, Springer-Verlag
  • 1998
Noncommutative geometry and geometric quantization”, Symplectic geometry and mathematical physics
  • Progr. Math
  • 1991