Gauge-natural field theories and Noether Theorems: canonical covariant conserved currents

Recently we found that canonical gauge-natural superpotentials are obtained as global sections of the {\em reduced} $(n-2)$-degree and $(2s-1)$-order quotient sheaf on the fibered manifold \bY_{\zet} \times_{\bX} \mathfrak{K}, where $\mathfrak{K}$ is an appropriate subbundle of the vector bundle of (prolongations of) infinitesimal right-invariant automorphisms $\bar{\Xi}$. In this paper, we provide an alternative proof of the fact that the naturality property \cL_{j_{s}\bar{\Xi}_{H}}\omega (\lambda, \mathfrak{K})=0 holds true for the {\em new} Lagrangian $\omega (\lambda, \mathfrak{K})$ obtained contracting the Euler--Lagrange form of the original Lagrangian with $\bar{\Xi}_{V}\in \mathfrak{K}$. We use as fundamental tools an invariant decomposition formula of vertical morphisms due to Kol\'a\v{r} and the theory of iterated Lie derivatives of sections of fibered bundles. As a consequence, we recover the existence of a canonical generalized energy--momentum conserved tensor density associated with $\omega (\lambda, \mathfrak{K})$.Comment: 16 pages, abstract rewritten, body slightly revised, Proc. Winter School "Geometry and Physics" (Srni,CZ 2005

Constructing towers with skeletons from open Lie algebras and integrability

We provide a given algebraic structure with the structure of an infinitesimal algebraic skeleton. The necessary conditions for integrability of the absolute parallelism of a tower with such a skeleton are dispersive nonlinear models and related conservation laws given in the form of associated linear spectral problems.Comment: misprints corrected, title changed, some remarks adde

Noether identities in Einstein--Dirac theory and the Lie derivative of spinor fields

We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restrictions on the Lie derivative of fields.Comment: 11 pages, completely rewritten, contains an example of application to the coupling of gravity with spinors; in v4 misprints correcte

Variational Lie derivative and cohomology classes

We relate cohomology defined by a system of local Lagrangian with the cohomology class of the system of local variational Lie derivative, which is in turn a local variational problem; we show that the latter cohomology class is zero, since the variational Lie derivative `trivializes' cohomology classes defined by variational forms. As a consequence, conservation laws associated with symmetries ensuring the vanishing of the second variational derivative of a local variational problem are globally defined.Comment: 7 pages, misprints in Corollary 2 and a misleading in the abstract and the introduction corrected, XIX International Fall Workshop on Geometry and Physic

Variational derivatives in locally Lagrangian field theories and Noether--Bessel-Hagen currents

The variational Lie derivative of classes of forms in the Krupka's variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application we determine the condition for a Noether--Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.Comment: 20 page

Variational Sequences, Representation Sequences and Applications in Physics

This paper is a review containing new original results on the finite order variational sequence and its different representations with emphasis on applications in the theory of variational symmetries and conservation laws in physics