Conservation Laws and Variational Sequences in Gauge-Natural Theories

TL;DR

This paper explores the formulation of conservation laws and superpotentials in gauge-natural field theories using variational sequences, providing a general framework for their existence and properties.

Contribution

It introduces a variational sequence approach to gauge-natural superpotentials, extending previous results on Noether theorems and Lie derivatives.

Findings

Superpotentials can be globally defined in a broad setting.

The variational sequence framework clarifies the structure of conserved currents.

Existence of superpotentials is established using Kolár's integration by parts results.

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 Kol\'a\v{r}, concerning the integration by parts procedure, can be applied to prove the {\em existence} and {\em globality} of superpotentials in a very general setting.