arXiv research

Spin(7) geometry linked 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 que…

Paper constructs observables using multisymplectic geometry and algebraic methods.

Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual f…

Novel multisymplectic framework for pseudo-Fueter curves in Hamiltonian field theory.

In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in particular, we discuss existence and make a number of identifications of the spa…

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 to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplec…

Introduces a new bracket for multicontact geometry and applies it to field theories.

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constr…

A notion of orthogonality in multisymplectic geometry has been developed by Cantrijn, Ibort and de León and used by many authors. In this paper, we review this concept and propose a new type of orthogonality in multisymplectic geometry; we prove a number of results regarding this orthogonality and its associated subspa…

Given a multisymplectic manifold $(M,ω)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,ω)$ associated to $(M,ω)$, in analogy with and generalizing the notio…

We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher'' versions of Poisson structures by identifying the infinitesimal counterparts of mul…

This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.

This lecture is devoted to review some of the main properties of multisymplectic geometry. In particular, after reminding the standard definition of multisymplectic manifold, we introduce its characteristic submanifolds, the canonical models, and other relevant kinds of multisymplectic manifolds, such as those where th…

This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce th…

In this paper the notion of Tulczyjew's triples in classical mechanics is extended to classical field theories, using the so-called multisymplectic formalism, and a convenient notion of lagrangian submanifold in multisymplectic geometry. Accordingly, the dynamical equations are interpreted as the local equations defini…

This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generaliz…

The Darboux theorem in symplectic geometry implies that any two points in a connected symplectic manifold have neighbourhoods symplectomorphic to each other. The impossibility of such a theorem in the more general multisymplectic framework appears to be, at least, folkloristic, but no explicit counterexample seems to e…

We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential top…

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In parti…

On a 6-dimensional real vector space $V$ there are three types of multisymplectic 3-forms. We present in this paper a unified treatment of these three types. Forms of each type represent a subset of $Λ^3 V^*$. In two cases they are open subsets, in the third one it is a submanifold of codimension 1. We study the geomet…

We compare existence and equivariance phenomena for weak moment maps and homotopy moment maps in multisymplectic geometry.

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$+$1, typically taken to be spacetime. Given a conn…

This work generalizes Hamiltonian mechanics using closed differential forms.

Embeds pre-multisymplectic manifolds into coisotropic ones.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

Reduces multisymplectic Lie systems through symmetry analysis.

We describe a new algebraic multisymplectic formulation of the classical BRST symmetry. The analogue of Marsden-Weinstein reduction for multisymplectic manifolds is described. We then give a homological description of Multisymplectic Marsden-Weinstein reduction.

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg …

Develops a new geometric framework for non-conservative field theories.

The study of multisymplectic structures using Spencer cohomology.

Characterizes density-valued symplectic forms on multisymplectic manifolds.

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian condition (which agrees with it only when $k=1$). Higher Dirac stru…

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum obs…

Introduces homotopy momentum sections on multisymplectic manifolds.

A geometric multisymplectic formulation of the classical BRST symmetry of constrained first-order classical field theories is described. To effect this we introduce graded analogues of the bundles and manifolds of the multisymplectic formulation of first-order field theories. The Lagrange-d'Alembert formalism is also d…

We introduce the concepts of a multisymplectic structure and a polysymplectic structure on a general fiber bundle over a general base manifold, define the concept of the symbol of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, and prove Darboux theorems for the exist…

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…

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well…

The paper introduces a new form on Lie algebroids over multisymplectic manifolds.

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise i…

We present a multisymplectic formulation of the Yang--Mills equations. The connections are represented by normalized equivariant 1-forms on the total space of a principal bundle, with values in a Lie algebra. Within the multisymplectic framework we realize that, under reasonable hypotheses, it is not necessary to assum…

We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation…

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.

Classifies solutions in multisymplectic field theories using geometric gauge freedom.

Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate exist…

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…

In this note the long standing problem of the definition of a Poisson bracket in the framework of a multisymplectic formulation of classical field theory is solved. The new bracket operation can be applied to forms of arbitary degree. Relevant examples are discussed and important properties are stated with proofs sketc…