Moduli of coisotropic sections and the BFV-complex

TL;DR

This paper links the deformation theory of coisotropic submanifolds in Poisson manifolds to the BFV-complex, providing a new algebraic description of the moduli space of coisotropic sections.

Contribution

It introduces a differential graded Poisson algebra called the BFV-complex to describe the groupoid of coisotropic sections and establishes an isomorphism with the moduli space of Maurer-Cartan elements.

Findings

The groupoid of coisotropic sections is described via the BFV-complex.

An explicit surjective morphism from the BFV-based groupoid to the coisotropic groupoid is constructed.

The moduli space of coisotropic sections is isomorphic to the moduli space of geometric Maurer-Cartan elements.

Abstract

We consider the local deformation problem of coisotropic submanifolds inside Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident, it is usually a very intricate object. We provide a description of the groupoid of coisotropic sections in terms of a differential graded Poisson algebra, called the BFV-complex. This description is achived by constructing a groupoid from the BFV-complex and a surjective morphism from this groupoid to the groupoid of coisotropic sections. The kernel of this morphism can be easily chracterized. As a corollary we obtain an isomorphism between the moduli space of coisotropic sections and the moduli space of geometric Maurer-Cartan elements of the BFV-complex.