Cattaneo, A S; Dherin, B; Felder, G (2010). Formal Lagrangian Operad. International Journal of Mathematics and Mathematical Sciences:1-36.

## Abstract

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

## Abstract

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

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