[Paper Review] On higher-order Courant brackets

Yunhe Sheng|arXiv (Cornell University)|Mar 9, 2010

Homotopy and Cohomology in Algebraic Topology7 citations

TL;DR

This paper introduces higher-order Courant brackets on the direct sum bundle $TM \oplus \wedge^nT^*M$ for an $m$-dimensional manifold, establishing that the graph of an $(n+1)$-vector field $\pi$ is closed under the bracket if and only if $\pi$ is a Nambu-Poisson structure, inducing a Leibniz algebroid on $\wedge^nT^*\!M$. Similarly, the graph of an $(n+1)$-form $\omega$ is closed iff $\omega$ is a premultisymplectic structure of order $n$, yielding a Lie algebroid on an admissible subbundle $A \subset \wedge^nT^*\!M$, with a 2-plectic structure inducing the Lie 2-algebra from Baez et al.

ABSTRACT

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector field $\pi$ is closed under the higher-order Dorfman bracket iff $\pi$ is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on $\wedge^nT^*M$. The graph of an $(n+1)$-form $\omega$ is closed under the higher-order Dorfman bracket iff $\omega$ is a premultisymplectic structure of order $n$, i.e. $\dM\omega=0$. Furthermore, there is a Lie algebroid structure on the admissible bundle $A\subset\wedge^{n}T^*M$. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in \cite{baez:classicalstring}.

Motivation & Objective

• To generalize Courant algebroid structures to higher-order analogues on $TM \oplus \wedge^nT^*M$ for $m$-dimensional manifolds.
• To clarify the algebraic conditions under which higher Dorfman brackets close on graphs of $(n+1)$-tensor fields.
• To establish the correspondence between closed graphs of $(n+1)$-vector fields and Nambu-Poisson structures.
• To extend the framework to $(n+1)$-forms, identifying premultisymplectic structures as the closure condition.
• To recover known structures such as Lie 2-algebras from 2-plectic forms via induced Lie algebroid structures.

Proposed method

• Defining a higher-order Dorfman bracket on the bundle $TM \oplus \wedge^nT^*M$ using the Schouten-Nijenhuis bracket and contraction operations.
• Analyzing the closure condition of the graph of an $(n+1)$-vector field $\pi$ under the higher Dorfman bracket.
• Proving that closure holds if and only if $\pi$ satisfies the Nambu-Poisson condition, i.e., the Schouten-Nijenhuis bracket $[\pi, \pi]_{SN} = 0$.
• Introducing the concept of a premultisymplectic structure of order $n$ via the condition $\mathrm{d}_M\omega = 0$ for an $(n+1)$-form $\omega$, where $\mathrm{d}_M$ is the de Rham differential.
• Constructing an admissible subbundle $A \subset \wedge^nT^*M$ from the kernel of the contraction with $\omega$, which inherits a Lie algebroid structure.
• Demonstrating that for $n=2$, the resulting structure on $A$ recovers the Lie 2-algebra structure from Baez et al. in the context of classical string theory.

Experimental results

Research questions

• RQ1 Under what conditions is the graph of an $(n+1)$-vector field $\pi$ closed under the higher-order Dorfman bracket on $TM \oplus \wedge^nT^*M$?
• RQ2 How does the closure of the graph of an $(n+1)$-form $\omega$ relate to geometric structures such as multisymplectic or premultisymplectic forms?
• RQ3 What algebraic structures are induced on $\wedge^nT^*M$ when the graph of an $(n+1)$-tensor field is closed under the higher Dorfman bracket?
• RQ4 Can the higher-order Courant bracket framework recover known structures like Lie 2-algebras in the case of 2-plectic forms?
• RQ5 What is the precise relationship between Nambu-Poisson structures and Leibniz algebroid structures on $\wedge^nT^*M$?

Key findings

• The graph of an $(n+1)$-vector field $\pi$ is closed under the higher-order Dorfman bracket if and only if $\pi$ is a Nambu-Poisson structure.
• This closure condition induces a Leibniz algebroid structure on the bundle $\wedge^nT^*M$.
• The graph of an $(n+1)$-form $\omega$ is closed under the higher Dorfman bracket if and only if $\omega$ is a premultisymplectic structure of order $n$, i.e., $\mathrm{d}_M\omega = 0$.
• For such a premultisymplectic $\omega$, there exists an admissible subbundle $A \subset \wedge^nT^*M$ that carries a Lie algebroid structure.
• In the case $n=2$, the induced Lie algebroid structure on $A$ recovers the Lie 2-algebra structure previously constructed by Baez et al. in the context of classical string theory.
• The framework generalizes classical Courant algebroid theory to higher-order tensor fields and forms, unifying Nambu-Poisson and multisymplectic geometry under a single bracket formalism.