Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decomposition of the {\em variational Lie derivative} of the generalized Euler--Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that {\em within} a gauge-natural invariant Lagrangian variational principle, the gauge-natural lift of infinitesimal principal automorphism {\em is not} intrinsically arbitrary. As a consequence the existence of {\em canonical} global superpotentials for gauge-natural Noether conserved currents is proved without resorting to additional structures.Comment: 24 pages, minor changes, misprints corrected, a misprint in the coordinate expression of the Jacobi morphism corrected; final version to appear in Arch. Math. (Brno

Some aspects of the homogeneous formalism in Field Theory and gauge invariance

We propose a suitable formulation of the Hamiltonian formalism for Field Theory in terms of Hamiltonian connections and multisymplectic forms where a composite fibered bundle, involving a line bundle, plays the role of an extended configuration bundle. This new approach can be interpreted as a suitable generalization to Field Theory of the homogeneous formalism for Hamiltonian Mechanics. As an example of application, we obtain the expression of a formal energy for a parametrized version of the Hilbert--Einstein Lagrangian and we show that this quantity is conserved.Comment: 9 pages, slightly revised, to appear in Proc. Winter School "Geometry and Physics", Srni (CZ) 200

A variational perspective on classical Higgs fields in gauge-natural theories

Higgs fields on gauge-natural prolongations of principal bundles are defined by invariant variational problems and related canonical conservation laws along the kernel of a gauge-natural Jacobi morphism.Comment: 10 page

Infinitesimal Algebraic Skeletons for a (2 + 1)-dimensional Toda Type System

A tower for a (2+1)-dimensional Toda type system is constructed in terms of a series expansion of operators which can be interpreted as generalized Bessel coefficients; the result is formulated as an analog of the Baker-Campbell-Hausdorff formula. We tackle the problem of the construction of infinitesimal algebraic skeletons for such a tower and discuss some open problems arising along our approach

Lagrangian reductive structures on gauge-natural bundles

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.Comment: 11 pages, remarks and comments added, this version published in ROM

Covariant gauge-natural conservation laws

When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. {\em Vice versa} all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as {\em canonical} generators of covariant gauge-natural physical charges.Comment: 16 pages; presented at XXXVI Symposium on Math. Phys., Torun 09/06-12/06/04; the last paragraph of Section 3 has been reformulated, in particular a mistake in the equation governing the vertical part of gauge-natural lifts with respect to prolongations of principal connections (appearing e.g. in the vertical superpotential) has been correcte