Abstract
Given a symplectic manifoldM, we may define an operad structure on the the spacesOkof the Lagrangian submanifolds of(M¯)k×Mvia symplectic reduction. IfMis 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 ofC∞(ℝd) is a deformation of the trivial symplectic groupoid structure ofT∗ℝd.
Highlights
Symplectic groupoids, in the extended symplectic category, may be thought as the analog of associative algebras in the category of vector spaces
We provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras
Rephrased appropriately, most constructions of the deformation theory of algebras can be extended to symplectic groupoids, at least for the trivial one over Êd
Summary
Symplectic groupoids, in the extended symplectic category, may be thought as the analog of associative algebras in the category of vector spaces. The combinatorics of bicolored Runge-Kutta trees was borrowed from the numerical analysis of ODE see 5 We used it first in 2 to expand the structure equation called the “SGA equation” for symplectic groupoid generating functions in formal power series. This combinatorics happens to control the compositions in the formal lagrangian operad over T∗Êd. There is no formal version of the microsymplectic category to date, and the combinatorics presented here to deal with the compositions in the formal lagrangian operad over T∗Êd have no equivalent in terms of a ”formal microsymplectic category”; this is, at the time of writing, still a work in progress
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