Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the [Formula: see text]-order gauge-natural bundle of [Formula: see text]-principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

Noether's Second Theorem on natural bundles

Following work by Anderson, in this paper we present a formulation of Noether's Second Theorem that is valid on any natural bundle.

Conservation laws and variational sequences in gauge-natural theories

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 Kolář, concerning the integration by parts procedure, can be applied to prove the existence and globality of superpotentials in a very general setting.