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

Domenico Fiorenza, Antonio Michele Miti|SPIRE - Sciences Po Institutional REpository|Feb 16, 2026

Homotopy and Cohomology in Algebraic Topology被引用 0

一句话总结

论文提供了一个通用框架，可从扭曲的高 Courant 代数到非扭曲的 L_{inf}-态射，适用于任意 r，并建立了它们的几何与同伦基础。

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].

研究动机与目标

• Motivate and formalize how closed (r+1)-forms twist higher Courant algebroids and their L_{inf}-algebras.
• Provide a conceptual framework that yields canonical L_{inf}-morphisms from twisted to untwisted higher Courant L_{inf}-algebras for all r.
• Clarify the homotopical and geometrical structures underlying the twisted-to-untwisted correspondence.
• Connect the twisted Courant framework with Rogers’ multisymplectic observables via a coherent homotopy-theoretic picture.

提出的方法

• Develop a Cartan DGLA model encoding Cartan calculus on M to simplify embeddings and twisting.
• Use the g_r = Cartan ⋉ Ω[r] construction with a Maurer–Cartan element σ ∈ Ω^{r+1}_{cl}(M) to form the σ-twisted DGLA g_{r,σ}.
• Show higher Courant algebras are homotopy fibers of inclusion morphisms in the DGLA/L_{inf} setting (Getzler/cone framework).
• Construct a 2-term L_{inf}-morphism Φ between g_r,σ and g_{r+1} and twist to obtain Φ_σ compatible with nonnegative truncations.
• Establish a canonical L_{inf}-morphism from twisted to untwisted Courant algebras via a commutative square of L_{inf}-morphisms and the homotopy fiber universal property.
• Relate Rogers’ multisymplectic observables to twisted Courant algebras using a homotopically commutative square of DGLAs.

实验结果

研究问题

• RQ1 Do canonical L_{inf}-morphisms exist that connect twisted higher Courant algebroids to untwisted ones for arbitrary r?
• RQ2 Can a small, conceptual model clarify the origin of these morphisms and their interaction with truncations to nonnegative degrees?
• RQ3 How do multisymplectic observables relate to twisted Courant L_{inf}-algebras within this homotopical framework?
• RQ4 Can Cartan calculus-based (Cartan DGLA) formulations simplify the construction and interpretation of these morphisms?

主要发现

• A general framework using homotopy fibers in the DGLA/L_{inf} setting yields canonical L_{inf}-morphisms from twisted to untwisted higher Courant L_{inf}-algebras for arbitrary r.
• A smaller, tractable DGLA model (Cartan DGLA) shows the higher Courant algebra as a homotopy fiber of a suitable inclusion, clarifying the construction.
• A 2-term L_{inf}-morphism between g_r,σ and g_{r+1} can be σ-twisted to yield a morphism compatible with nonnegative degree truncations, realizing the conjectured morphism.
• The framework provides a natural morphism from the twisted higher Courant L_{inf}-algebra to the untwisted one via a commutative square and the universal property of homotopy fibers.
• The work connects Rogers’ multisymplectic observables to twisted Courant algebras through a homotopically commutative square of DGLAs.
• The constructions illuminate the geometric and homotopical underpinnings of higher Dirac and Courant structures and their interrelations.