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

Domenico Fiorenza, Antonio Michele Miti

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].

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

Abstract

In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" [arXiv:1003.1004], Marco Zambon constructs an -algebra associated with any higher standard or twisted Courant algebroid (also known as a Vinogradov algebroid), and exhibits an explicit -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 -morphisms exist in higher degrees -- namely, for any standard higher Courant algebroid twisted by a closed -form. We adfirmatively answer this question, presenting a general framework that naturally yields such canonical -morphisms for arbitrary , 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 -algebra of a pre--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].
Paper Structure (17 sections, 16 theorems, 134 equations)

This paper contains 17 sections, 16 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Higher Courant algebroids
  4. Higher Courant Lie infinity-algebras
  5. Multisymplectic Observables L-infinity-algebra
  6. A morphism between the twisted and untwisted higher Courant L-infinity-algebras
  7. The Cartan DGLA and a smaller model for the higher Courant L-infinity-algebra
  8. The canonical morphism between higher Courant L-infinity-algebras
  9. The canonical morphism between twisted and untwisted higher Courant L-infinity-algebras
  10. A morphism between the multisymplectic L-infinity-algebra and the twisted higher Courant L-infinity-algebra
  11. Homotopy comomentum maps and the pentagonal diagram
  12. Outlook
  13. A primer on L-infinity-algebras
  14. L-infinity-structures as Maurer-Cartan elements
  15. L-infinity-morphisms between DGLAs as Maurer-Cartan elements
  16. ...and 2 more sections

Key Result

proposition 1

Let $M$ be a smooth manifold, $r\geq 1$ an integer and $\sigma \in \Omega^{r+1}_{\mathrm{cl}}(M)$ a closed differential form. The $\sigma$-twisted higher Courant $L_\infty$-algebra of degree $r{-}1$ is a model for the homotopy fiber of the canonical inclusion of DGLAs $\mathfrak{G}_{r,\sigma}^{\ge 0 where $\mathfrak{G}_{r,\sigma}^{\ge 0}$ is the non-negatively graded truncation of the twisted cano

Theorems & Definitions (59)

  • definition 1: Higher Courant Algebroid
  • remark 1
  • remark 2: Naming and generalization
  • definition 2: The canonical DGLA associated with the standard higher Courant algebroid
  • definition 3: Canonical DGLA associated with the twisted higher Courant algebroid
  • definition 4: Higher Courant $L_\infty$-algebra Zambon2012
  • proposition 1: Higher Courant $L_\infty$-algebra as homotopy fiber
  • lemma 1
  • proof
  • definition 5: Hamiltonian pairs
  • ...and 49 more