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∞ deformations of coisotropic submanifolds and define the corresponding C∞-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 closed one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of strong homotopy Lie algebroid (or L∞-algebroid) and associate a canonical isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold (Y,ω) 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 Maurer-Cartan equation (or the 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.