arXiv subjects

Frédéric Hélein

Publications and source records attributed to Frédéric Hélein.

18 recordsLinked to original sources

arXiv2023

Gauge and Gravity theories on a dynamical principal bundle

In this paper we present original variational formulations of Yang-Mills, Einstein's gravitation and Kaluza-Klein theories, where, in the spirit of General Relativity, the principal bundle structure over the space-time is not fixed a priori but is dynamical. In the Yang-Mills case only a topological fibration is given a priori. In the gravity and the Kaluza-Klein theories no fibration is assumed: any critical point of the action functional defines a foliation of the manifold and the leaves make up the space-time. The latter is naturally equipped with a pseudo-Riemannian metric and, under some hypotheses, this foliation is actually a fibration. In all cases the apparition of a (at least local) principal bundle structure and a connection follows from the dynamics. Moreover the metric and the connection thus constructed are solutions of the Yang-Mills, the Einstein-Cartan or the Yang-Mills-Einstein equations, depending on the model. A crucial point is that we face the difficulty that some Lagrange multiplier fields (which are responsible for the foliation, the principal bundle structure and the connection) create unwanted terms in the equations. This difficulty is overcome by the observation that, if the structure group is compact, these terms miraculously cancel.

math.DG↗

arXiv2022

Dynamical mechanisms for Kaluza-Klein theories

We present variational formulations of gauge theories and Einstein--Yang-Mills equations in the spirit of Kaluza-Klein theories. For gaugetheories, only a topological fibration is assumed. For gravitation coupled with gauge fields, no fibration is assumed: Fields are defined on a 'space-time' $Y$ of dimension $4 + r$ without any structure a priori, where $r$ is the dimension of the structure group. If the latter is compact and simply connected, classical solutions allow to construct a manifold $X$ of dimension 4 to be the physical space-time, in such a way that $Y$ acquires the structure of a principal bundle over $X$ and leads to solutions of the Einstein--Yang-Mills systems. The special case of the Einstein-Maxwell system is also discussed: It suffices that at least one fiber closes in on a circle to deduce that the five-dimensional space-time has afiber bundle structure.

math.DG↗

arXiv2019

A variational principle for Kaluza-Klein type theories

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on $\mathbb{R}^n$ and an $\hbox{Ad}$-invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem on fields defined on an arbitrary oriented $(n+\hbox{dim}\mathfrak{G})$-dimensional manifold $\mathcal{Y}$. We show that, if $\mathfrak{G}$ is compact and simply connected, any global solution of the Euler--Lagrange equations leads, through a spontaneous symmetry breaking, to identify $\mathcal{Y}$ with the total space of a principal bundle over an $n$-dimensional manifold $\mathcal{X}$. Moreover $\mathcal{X}$ is then endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein--Yang--Mills system of equations with a cosmological constant.

math-ph↗

arXiv2018

A remark on compact H-surfaces into R^3

Using the functional associated with the optimal Wente inequality for pairs of functions on 2-dimensional domains, we show the existence of solutions to the H-system on an annulus satisfying Neumann boundary conditions.

math.AP↗

arXiv2018

Isoperimetric inequalities and calibrations

The subject of these Notes is the new proof, proposed in [F. H{é}lein, In{é}galit{é} isop{é}rim{é}trique et calibrations, Annales de l'Institut Fourier 44, 4 (1994), 1211-1218] of the classical isoperimetric inequality in the plane. This proof is far from being the first one, but its interest is that it uses essentially integration by parts and Stoke's formula in a simple manner, like in a calibration. Here we expound again this proof and discuss in which sense our proof works like a calibration, or it can be understood in the framework of the theory of null Lagrangians, which goes back to Weierstrass, Mayer, Hilbert, Weyl, Carath{é}odory, and Lepage.

math.DG↗

arXiv2016

Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker)

The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front set satisfies some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces D\_$Γ$ of distributions having a wave front set included in a given closed cone $Γ$ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on D\_$Γ$ , and the tensor product is not separately continuous. In this paper, a new topology is defined for which the pull-back and push-forward are continuous, the tensor and convolution products and the multiplication of distributions are hypocontinuous.

math.FA↗

arXiv2015

Curved space-times by crystallization of liquid fiber bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of $((α,ω),\varpi)$, where $(α,ω)$ is a 1-form with coefficients in the Lie algebra of the Poincaré group and $\varpi$ is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein--Cartan equations.

math-ph↗

arXiv2014

Multisymplectic formulation of Yang--Mills equations and Ehresmann connections

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 assume the equivariance condition a priori, since this condition is a consequence of the dynamical equations.

math-ph↗

arXiv2014

A smooth introduction to the wavefront set

The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.

math-ph↗

arXiv2013

First integrals for nonlinear dispersive equations

Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series are constructed by means of generating functions. All this theory is based on a new suitable formulation of the dynamics of solutions of dispersive equations.

math.AP↗

arXiv2011

Multisymplectic formalism and the covariant phase

The formulation of a relativistic dynamical problem as a system of Hamilton equations by respecting the principles of Relativity is a delicate task, because in their classical form the Hamilton equations require the use of a time coordinate, which of course contradicts the Relativity. Two interesting solutions have been proposed during the last century: the covariant phase space and the multisymplectic formalism. These two approaches were inspired at the beginning by different points of view. However, as shown by works by Kijowski-Szczyrba, Forger-Romero and Vitagliano, a synthetic vision of the two theories leads probably to the most satisfactory answer to the basic question of understanding the Hamiltonian structure of relativistic fields theory.

math-ph↗

arXiv2010

A note on some overdetermined elliptic problem

We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis. We deduce a classification of all such surfaces assuming some further natural hypotheses and prove a Bernstein type theorem.

math-ph↗

arXiv2009

Manifolds obtained by soldering together points, lines, etc

This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. We explore the notion of solder (or soldering form) in differential geometry and propose an alternative interpretation of it, motivated by the search of an accurate mathematical description of the General Relativity. This new interpretation leads naturally to imagine a rich family of new geometries which has not yet a satisfactory definition in general. We try however to communicate to the reader an intuition of such geometries through some examples and review quickly some possible applications in physics. The basic objects in this geometry are not points (i.e. 0-dimensional), but (p-1)-dimensional.

math-ph↗

arXiv2008

Some examples of `second order elliptic integrable systems associated to a 4-symmetric space'

In this text we expound recent results by Idrisse Khemar on the construction of various geometric completely integrable systems generalizing the structure of Hamiltonian stationary Lagrangian surfaces (HSLS) discovered by F. Hélein and P. Romon. We first explain that, for surfaces in a 4-dimensional Euclidean space, the HSLS are a particular case of a more general theory of surfaces with harmonic left Gauss map and that this theory includes also the classical constant mean curvature surfaces in 3-space. Then we expound a generalization of this theory for surfaces in an 8-dimensional Euclidean space, using octonions. Lastly we discuss supersymmetric harmonic and primitive maps, a theory which also covers the HSLS cas.

math-ph↗

arXiv2006

A representation formula for maps on supermanifolds

In this paper we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e. following Berezin, Leites, Manin, etc.). We establish a representation formula for all morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of R^{p|q}. We then derive two consequences of this result. The first one is that we can integrate the data associated with a morphism in order to get a (non unique) map defined on an ordinary space (and uniqueness can achieved by restriction to a scheme). The second one is a simple and intuitive recipe to compute pull-back images of a function on a manifold by a map defined on a superspace.

math-ph↗

arXiv2005

Hamiltonian stationary tori in the complex projective plane

We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.

math.DG↗

arXiv2002

Covariant Hamiltonian formalism for the calculus of variations with several variables

The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. For instance the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory, while ignoring singularities whenever the Lagrangian is degenerate. Moreover we show that there exist two different definitions of the observable (n-1)-forms which allows one to construct observable functionals by integration (which correspond to two different points of view: generalizing the law {p,q} = 1 or the law dF\dt = {H,F}), oddly enough we prove that these two definitions coincides only in the LP situation. Finally other contributions concerning this subject and examples are also given.

math-ph↗

arXiv2000

Finite dimesional Hamiltonian formalism for gauge and field theories

We discuss in this paper the canonical structure of classical field theory in finite dimensions within the {\it{pataplectic}} Hamiltonian formulation, where we put forward the role of Legendre correspondance. We define the generalized Poisson $\mathfrak{p}$-brackets which are the analogues of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of $\mathfrak{p}$-brackets. As illustration of our formalism we present three examples: the interacting scalar fields, conformal string theory and the electromagnetic field.

math-ph↗