Mathematical Proceedings of the Cambridge Philosophical Society

Conservation laws and variational sequences in gauge-natural theories 1

a1 Department of Mathematics, University of Turin, Via C. Alberto 10, 10123 Turin, Italy. e-mail:,,


In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable vector density is known to generate the so-called 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 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 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 Kolár, concerning the integration by parts procedure, can be applied to prove the existence and globality of superpotentials in a very general setting.

(Received June 28 1999)
(Revised December 2 1999)


1 Work partially supported by GNFM of INDAM, MURST, University of Turin.