Formal Lagrangian Operad

Abstract

Given a symplectic manifold M, we may define an operad structure on the the spaces Ok of the Lagrangian submanifolds of (M¯)k×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∞(ℝd) is a deformation of the trivial symplectic groupoid structure of T∗ℝd.