Conservation Laws and Variational Sequences in Gauge-Natural Theories

Abstract

In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant gauge--natural bundle this density is the divergence of a skew--symmetric (tensor) density, which is called a {\em superpotential} for the conserved currents. We describe gauge--natural superpotentials in the framework of finite order variational sequences according to Krupka. We refer to previous results of ours on {\em variational Lie derivatives} concerning abstract versions of Noether's theorems, which are here interpreted in terms of ``horizontal'' and ``vertical'' conserved currents. The gauge--natural lift of principal automorphisms implies suitable linearity properties of the Lie derivative operator. Thus abstract results due to Kolář, concerning the integration by parts procedure, can be applied to prove the {\em existence} and {\em globality} of superpotentials in a very general setting.