BFV-complex and higher homotopy structures

TL;DR

This paper establishes a deep connection between the BFV-complex and strong homotopy Lie algebroids for coisotropic submanifolds, showing they are $L_{ extinfty}$ quasi-isomorphic and control the same deformation problems, with some distinctions in non-formal information.

Contribution

It demonstrates that the BFV-complex can be homotopically transferred to produce the strong homotopy Lie algebroid, revealing their quasi-isomorphism and differences in non-formal data.

Findings

BFV-complex and strong homotopy Lie algebroid are $L_{ extinfty}$ quasi-isomorphic.

A one-to-one correspondence exists between coisotropic submanifolds and Maurer-Cartan elements of the BFV-complex.

The correspondence does not hold when using the strong homotopy Lie algebroid instead.

Abstract

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are $L_{\infty}$ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.