Deformations of coisotropic submanifolds and strong homotopy Lie algebroids

Abstract

In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^{\infty }$ deformations of coisotropic submanifolds and define the corresponding $C^{\infty }$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong homotopy Lie algebroid} (or {\it $L_{\infty }$-algebroid}) and associate a canonical isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold $(Y,\omega )$ and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the {\it Maurer-Cartan equation} (or the {\it master equation}) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic $A$-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation.

arXiv comment60 pages, This is the final version that will appear in Invent. Math

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• On the Strong Homotopy Lie-Rinehart Algebra of a Foliation
• Shifted derived Poisson manifolds associated with Lie pairs
• Deformations of coisotropic submanifolds for fibrewise entire Poisson structures
• Deformation spaces and normal forms around transversals
• Strongly homotopy Lie bialgebras and Lie quasi-bialgebras
• Jacobi bundles and the BFV-complex
• Rigidity of integral coisotropic submanifolds of contact manifolds
• On higher rank coisotropic A-branes
• A Lie-Rinehart algebra in general relativity
• Kirillov structures up to homotopy
• Double BFV quantisation of 3d Gravity
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• $A_{\infty }$-Algebras from Lie Pairs