International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 643605, 36 pages
Research Article

Formal Lagrangian Operad

1Institut für Mathematik, Universität Zürich—Irchel, Winterthurerstraße 190, 8057 Zürich, Switzerland
2Department of Mathematics, Utrecht University, Budapestlaan 6, 3584 CD Utrecht, The Netherlands
3D-MATH, ETH-Zentrum, 8092 Zürich, Switzerland

Received 14 July 2010; Accepted 7 December 2010

Academic Editor: A. Zayed

Copyright © 2010 Alberto S. Cattaneo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given a symplectic manifold 𝑀 , we may define an operad structure on the the spaces 𝑂 𝑘 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 ( 𝑑 ) is a deformation of the trivial symplectic groupoid structure of 𝑇 𝑑 .