On the geometry of multi-Dirac structures and Gerstenhaber algebras

Joris Vankerschaver, Hiroaki Yoshimura, Melvin Leok

    Research output: Contribution to journalArticle

    7 Citations (Scopus)

    Abstract

    In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a graded multiplication and a multi-Courant bracket on the space of sections of a multi-Dirac structure, so that the space of sections has the structure of a Gerstenhaber algebra. We then show that the graph of a k-form on a manifold gives rise to a multi-Dirac structure and also that this multi-Dirac structure is integrable if and only if the corresponding form is closed. Finally, we show that the multi-Courant bracket endows a subset of the ring of differential forms with a graded Poisson bracket, and we relate this bracket to some of the multisymplectic brackets found in the literature.

    Original languageEnglish
    Pages (from-to)1415-1425
    Number of pages11
    JournalJournal of Geometry and Physics
    Volume61
    Issue number8
    DOIs
    Publication statusPublished - 2011 Aug

    Fingerprint

    Gerstenhaber Algebra
    Dirac Structures
    brackets
    algebra
    Brackets
    geometry
    Multi-symplectic
    Classical Field Theory
    multiplication
    Poisson Bracket
    set theory
    Differential Forms
    Multiplication
    If and only if
    rings
    Ring
    Closed
    Subset
    Graph in graph theory

    Keywords

    • Classical field theories
    • Multi-Dirac structures
    • Multisymplectic forms

    ASJC Scopus subject areas

    • Mathematical Physics
    • Physics and Astronomy(all)
    • Geometry and Topology

    Cite this

    On the geometry of multi-Dirac structures and Gerstenhaber algebras. / Vankerschaver, Joris; Yoshimura, Hiroaki; Leok, Melvin.

    In: Journal of Geometry and Physics, Vol. 61, No. 8, 08.2011, p. 1415-1425.

    Research output: Contribution to journalArticle

    Vankerschaver, Joris ; Yoshimura, Hiroaki ; Leok, Melvin. / On the geometry of multi-Dirac structures and Gerstenhaber algebras. In: Journal of Geometry and Physics. 2011 ; Vol. 61, No. 8. pp. 1415-1425.
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