Representations of Homotopy Lie-Rinehart Algebras

I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie-Rinehart connections.Comment: v2: 29 pages, 3 tables, title changed, examples and references added. v3: 32 pages, examples added. Typos and a (minor) mistake corrected. v4: typos corrected, accepted for publication on Math. Proc. Camb. Phil. Soc., Comments still welcome

On the Strong Homotopy Associative Algebra of a Foliation

An involutive distribution $C$ on a smooth manifold $M$ is a Lie-algebroid acting on sections of the normal bundle $TM/C$. It is known that the Chevalley-Eilenberg complex associated to this representation of $C$ possesses the structure $\mathbb{X}$ of a strong homotopy Lie-Rinehart algebra. It is natural to interpret $\mathbb{X}$ as the (derived) Lie-Rinehart algebra of vector fields on the space $\mathbb{P}$ of integral manifolds of $C$. In this paper, I show that $\mathbb{X}$ is embedded in a strong homotopy associative algebra $\mathbb{D}$ of (normal) differential operators. It is natural to interpret $\mathbb{D}$ as the (derived) associative algebra of differential operators on $\mathbb{P}$. Finally, I speculate about the interpretation of $\mathbb{D}$ as the universal enveloping strong homotopy algebra of $\mathbb{X}$.Comment: 28 pages, comments welcom

Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs

Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e., those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three hour minicourse that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2-5, 2013, Evora, Portugal.Comment: 26 pages, short elementary review submitted for publication on the Proceedings of XXII IFWG

Dirac-Jacobi Bundles

We show that a suitable notion of Dirac-Jacobi structure on a generic line bundle $L$, is provided by Dirac structures in the omni-Lie algebroid of $L$. Dirac-Jacobi structures on line bundles generalize Wade's $\mathcal E^1 (M)$-Dirac structures and unify generic (i.e.~non-necessarily coorientable) precontact distributions, Dirac structures and local Lie algebras with one dimensional fibers in the sense of Kirillov (in particular, Jacobi structures in the sense of Lichnerowicz). We study the main properties of Dirac-Jacobi structures and prove that integrable Dirac-Jacobi structures on line-bundles integrate to (non-necessarily coorientable) precontact groupoids. This puts in a conceptual framework several results already available in literature for $\mathcal E^1 (M)$-Dirac structures.Comment: v6: 55 pages, corrected some minor mistakes, final version, to appear in J. Sympl. Geom, 16 (2018

Tulczyjew Triples in Higher Derivative Field Theory

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrary high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that, the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.Comment: 29 pages, v2: minor revisions. Accepted for publication in J. Geom. Mec

Generalized Contact Bundles

In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators, it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.Comment: Short Note: 8 pages. Minor revisions. Published in C. R. Math. Comments welcome

L-infinity Algebras From Multicontact Geometry

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 manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical $L_{\infty}$-algebra to any (pre-)multicontact structure. Such $L_{\infty}$-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic $L_\infty$-algebra of Rogers and Zambon, the multicontact $L_\infty$-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the $L_{\infty}$-algebra associated to the Cartan distribution on jet spaces.Comment: 19 pages, v2: exposition slightly changed. to appear in Diff. Geom. Appl. Comments still welcome

Vector Bundle Valued Differential Forms on NQ\mathbb{N} Q-manifolds

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known results, and new results about degree one $\mathbb N Q$-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in literature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular presymplectic and multisymplectic structures (not yet present in literature).Comment: 32 pages, v3: minor reisions, final version to appear in Pacific J. Math.; v2: added a section on degree one presymplectic NQ-manifolds and Dirac manifold

The Hamilton-Jacobi Formalism for Higher Order Field Theories

We extend the geometric Hamilton-Jacobi formalism for hamiltonian mechanics to higher order field theories with regular lagrangian density. We also investigate the dependence of the formalism on the lagrangian density in the class of those yelding the same Euler-Lagrange equations.Comment: 25 page

On the Strong Homotopy Lie-Rinehart Algebra of a Foliation

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the space P of integral manifolds (of any dimension) of the characteristic distribution C of F. Similarly, characteristic cohomologies with local coefficients in the normal bundle TM/C of F may be interpreted as vector fields on P. In particular, they possess a (graded) Lie bracket and act on characteristic cohomology H. In this paper, I discuss how both the Lie bracket and the action on H come from a strong homotopy structure at the level of cochains. Finally, I show that such a strong homotopy structure is canonical up to isomorphisms.Comment: 41 pages, v2: almost completely rewritten, title changed; v3: presentation partly changed after numerous suggestions by Jim Stasheff, mathematical content unchanged; v4: minor revisions, references added. v5: (hopefully) final versio