L-infinity algebras from multisymplectic geometry

TL;DR

This paper constructs Lie n-algebras and differential graded Leibniz algebras from n-plectic manifolds, generalizing previous 2-plectic results and exploring their algebraic structures in multisymplectic geometry.

Contribution

It explicitly generalizes the construction of Lie n-algebras from 2-plectic to n-plectic manifolds and compares these with Leibniz algebra structures.

Findings

Explicit construction of Lie n-algebras from n-plectic manifolds

Identification of differential graded Leibniz algebras in this context

Discussion of the relationship between these algebraic structures

Abstract

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, we showed that just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L-infinity algebras: graded vector spaces which are equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras.