Deformations of pre-symplectic structures and the Koszul $L_{\infty }$-algebra

Abstract

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_{\infty }$-algebra, which we call Koszul $L_{\infty }$-algebra. This $L_{\infty }$-algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul $L_{\infty }$-algebra is isomorphic to the $L_{\infty }$-algebra which controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed.

30 pages, final version to be published. The geometric interpretation in terms of Dirac geometry has been removed from this paper and put, in an improved form, in ArXiv 1807.10148

References in corpus (2)

• Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach
• Gauge equivalences for foliations and pre-symplectic structures

Cited by in corpus (3)

• Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach
• Split Courant algebroids as $L_{\infty }$-structures
• Gauge equivalences for foliations and pre-symplectic structures