Lagrangian reductive structures on gauge-natural bundles

TL;DR

This paper explores how reductive structures linked to Lagrangian conserved quantities on gauge-natural bundles lead to a natural self-adjointness property, resulting in a reductive split structure on principal bundles.

Contribution

It introduces a novel connection between reductive structures, gauge-natural lifts, and the self-adjointness of the generalized Jacobi morphism in the context of gauge theories.

Findings

Reductive structures are associated with conserved quantities on gauge-natural bundles.

The generalized Jacobi morphism is shown to be naturally self-adjoint.

A reductive split structure on principal bundles is derived from the kernel of the Jacobi morphism.

Abstract

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.