F. Cantrijn, A. Ibort and M. De León, "On the geometry of multisymplectic manifolds". DOI: 10.1017/S1446788700036636.

Abstract:

A multisymplectic structure on a manifold is defined by a closed differential form with zero characteristic distribution. Starting from the linear case, some of the basic properties of multisymplectic structures are described. Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a bundle of exteriorfc-formson a manifold. For a class of multisymplectic manifolds admitting a 'Lagrangian' fibration, a general structure theorem is given which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.

J. F. Cariñena, M. Crampin and L. A. Ibort, "On the multisymplectic formalism for first order field theories". DOI: 10.1016/0926-2245(91)90013-Y.

Abstract:

The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noether's theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent.

F. Cantrijn, A. Ibort and M. De León, "Hamiltonian structures on multisymplectic manifolds". URL: https://seminariomatematico.polito.it/rendiconti/cartaceo/54-3/225.pdf.

Abstract:

After a brief review of some basic notions of multisymplectic geometry and a discussion of some examples, special attention is paid to the concepts of Hamiltonian multivector field and Hamiltonian form on a multisymplectic manifold. In particular,it is shown that the space of equivalence classes of Hamiltonian forms, modulo closed forms, can be equipped with a graded Lie algebra structure. Next, it is demonstrated that the tangent bundle of a multisymplectic manifold is also multisymplectic, and that a locally Hamiltonian vector field is determined by a Lagrangian section of this "tangent multisymplectic structure".