### 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 language | English |
---|---|

Pages (from-to) | 1415-1425 |

Number of pages | 11 |

Journal | Journal of Geometry and Physics |

Volume | 61 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2011 Aug |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Geometry and Physics*,

*61*(8), 1415-1425. https://doi.org/10.1016/j.geomphys.2011.03.005

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

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 61, no. 8, pp. 1415-1425. https://doi.org/10.1016/j.geomphys.2011.03.005

}

TY - JOUR

T1 - On the geometry of multi-Dirac structures and Gerstenhaber algebras

AU - Vankerschaver, Joris

AU - Yoshimura, Hiroaki

AU - Leok, Melvin

PY - 2011/8

Y1 - 2011/8

N2 - 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.

AB - 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.

KW - Classical field theories

KW - Multi-Dirac structures

KW - Multisymplectic forms

UR - http://www.scopus.com/inward/record.url?scp=79953239767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953239767&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2011.03.005

DO - 10.1016/j.geomphys.2011.03.005

M3 - Article

AN - SCOPUS:79953239767

VL - 61

SP - 1415

EP - 1425

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 8

ER -