Deformations of Symplectic Foliations

TL;DR

This paper develops a deformation theory for symplectic foliations, establishing an $L_$-algebra framework that links small deformations to Maurer-Cartan elements, revealing potential obstructions.

Contribution

It introduces an $L_$-algebra controlling symplectic foliation deformations and connects these to Poisson structures, highlighting obstructions in infinitesimal deformations.

Findings

Each symplectic foliation has an associated controlling $L_$-algebra.

Infinitesimal deformations can be obstructed.

Obstructed deformations of symplectic foliations may become unobstructed in Poisson structures.

Abstract

We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its deformation problem. Indeed, viewing symplectic foliations as regular Poisson structures, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated $L_\infty$-algebra. Using this, we show that infinitesimal deformations of symplectic foliations can be obstructed. Further, we relate symplectic foliations with foliations on one side and with (arbitrary) Poisson structures on the other, showing that obstructed infinitesimal deformations of the former may give rise to unobstructed infinitesimal deformations of the latter.