Formal Lagrangian Operad

Given a symplectic manifold $M$, we may define an operad structure on the the spaces ${\text{op}}^k$ of the Lagrangian submanifolds of $(\bar{M}{)}^k\times M$ 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 semi-classical part of Kontsevich’s deformation of $C^{\mathrm{\infty }}({\mathbb{R}}^d)$ is a deformation of the trivial symplectic groupoid structure of $T^{\ast }{\mathbb{R}}^d$.