Multisymplectic Manifolds Research Articles

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe mechanical systems), as well as certain cases of multisymplectic manifolds, while extends the Darboux theorem in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories, i.e. with locally non-invertible Legendre maps. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

We consider higher generalizations of both a (twisted) Poisson structure and the equivariant condition of a momentum map on a symplectic manifold. On a Lie algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we introduce a Lie algebroid differential form called a compatible $E$-$n$-form. This differential form satisfies a compatibility condition, which is consistent with both the Lie algebroid structure and the (pre-)(multi)symplectic structure. There are many interesting examples such as a Poisson structure, a twisted Poisson structure and a twisted $R$-Poisson structure for a pre-$n$-plectic manifold. Moreover, momentum maps and momentum sections on symplectic manifolds, homotopy momentum maps and homotopy momentum sections on multisymplectic manifolds have this structure.

Algebraic and geometric reduction of multisymplectic manifolds

In this talk, we discuss an extension of the Marsden–Weinstein–Meyers symplectic reduction theorem to multisymplectic manifolds, and an adaptation of the Śniatycki–Weinstein, Dirac and Arms–Gotay–Cushman Poisson algebra reduction theorems to L∞ -algebras of multisymplectic observables. This is based on joint work with A Miti and L Ryvkin.

Homotopy momentum sections on multisymplectic manifolds

We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

Multisymplectic actions of compact Lie groups on spheres

We investigate the existence of homotopy comoment maps (comoments) for high-dimensional spheres seen as multisymplectic manifolds. Especially, we solve the existence problem for compact effective group actions on spheres and provide explicit constructions for such comoments in interesting particular cases.

An invitation to multisymplectic geometry

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in “multisymplectic terms”. Furthermore, we investigate basic questions on normal forms of multisymplectic manifolds, notably the questions whether and when Darboux-type theorems hold, and “how many” diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.

CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS

Given a vector field on a manifold $M$, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into $M$. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

Remarks on Multisymplectic Reduction

The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries.

Variational principles and symmetries on fibered multisymplectic manifolds

Abstract The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.

L∞-algebra models and higher Chern–Simons theories

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern–Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of [Formula: see text]-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu–Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie [Formula: see text]-algebra extensions of [Formula: see text]. Finally, we study a number of [Formula: see text]-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.

Existence and unicity of co-moments in multisymplectic geometry

Given a multisymplectic manifold (M,ω) and a Lie algebra g acting on it by infinitesimal symmetries, Fregier–Rogers–Zambon define a homotopy (co-)moment as an L∞-algebra-homomorphism from g to the observable algebra L(M,ω) associated to (M,ω), in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and by deriving from it existence results concerning partial co-moment maps, as e.g. covariant multimomentum maps and multi-moment maps.

Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

It is shown that the geometry of locally homogeneous multisymplectic manifolds(that is, smooth manifolds equipped with a closed nondegenerate form of degree$> 1$, which is locally homogeneous of degree $k$ with respect toa local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend thefamily of classical geometries possessing a similar property: symplectic,volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra ofinfinitesimal automorphisms, and on the study of the local properties ofHamiltonian vector fields on locally multisymplectic manifolds.In particular it is proved that thegroup of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

Constructing a class of solutions for the Hamilton-Jacobi equation in field theory

A new approach leading to the formulation of the Hamilton-Jacobi equation for field theories is investigated within the framework of jet bundles and multisymplectic manifolds. An algorithm associating classes of solutions to given sets of boundary conditions of the field equations is provided. The paper also puts into evidence the intrinsic limits of the Hamilton-Jacobi method as an algorithm to determine families of solutions of the field equations, showing how the choice of the boundary data is often limited by compatibility conditions.

THE POISSON BRACKET FOR POISSON FORMS IN MULTISYMPLECTIC FIELD THEORY

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

A vertical exterior derivative in multisymplectic geometry and a graded poisson bracket for nontrivial geometries

A vertical exterior derivative is constructed which is needed for a graded Poisson structure on multisymplectic manifolds over nontrivial vector bundles. In addition, the properties of the Poisson bracket are proved and first examples are discussed.

On the geometry of multisymplectic manifolds

AbstractA 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 exterior k-forms on 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.

On the multisymplectic formalism for first order field theories

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.

A Darboux theorem for multi-symplectic manifolds

A class of geometric structures defined by i+1-forms that generalize the notion of a symplectic form is introduced. Examples of these structures occur in multi-dimensional variational calculus. An extension of the Darboux-Moser-Weinstein theorem is proved for these structures and a characterization for their pseudogroups is given.