$L_\infty$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r{+}1)$-Lie algebroids

TL;DR

This paper constructs canonical $L_$-morphisms between twisted and untwisted higher Courant algebroids, generalizing previous results and clarifying the underlying geometric and homotopical structures.

Contribution

It provides a general framework for canonical $L_$-morphisms in higher degrees, extending prior work on twisted Courant algebroids and their associated $L_$-algebras.

Findings

Established existence of $L_$-morphisms for arbitrary $r$

Clarified geometric and homotopical structures underlying these morphisms

Connected the framework to observable $L_$-algebras of pre-$r$-plectic manifolds

Abstract

In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" [arXiv:1003.1004], Marco Zambon constructs an $L_\infty$-algebra associated with any higher standard or twisted Courant algebroid (also known as a Vinogradov algebroid), and exhibits an explicit $L_\infty$-morphism from the Lie algebra associated with a standard Lie algebroid twisted by a closed 2-form to the Lie-2 algebra of the standard Courant algebroid. He poses the question of whether analogous canonical $L_\infty$-morphisms exist in higher degrees -- namely, for any standard higher Courant algebroid twisted by a closed $(r+1)$-form. We adfirmatively answer this question, presenting a general framework that naturally yields such canonical $L_\infty$-morphisms for arbitrary $r$, while at the same time clarifying the geometrical and homotopical structures underlying the construction. We also show how this framework accommodates the canonical morphism between Roger's observable $L_\infty$-algebra of a pre-$r$-plectic manifold and the higher Courant algebra described by Zambon and one of the authors in "Observables on multisymplectic manifolds and higher Courant algebroids" [arXiv:2209.05836].