Gauge reduction in covariant field theory

TL;DR

This paper develops a Lagrangian reduction framework for covariant gauge field theories using generalized principal connections, enabling systematic reduction and reconstruction of equations with gauge symmetries.

Contribution

It introduces a novel reduction method for covariant field theories with gauge symmetries using generalized principal connections, extending classical reduction techniques.

Findings

Derived reduced equations for gauge theories.

Established reconstruction conditions and relation to Noether's theorem.

Applied the theory to electromagnetism and non-Abelian gauge theories.

Abstract

In this work, we develop a Lagrangian reduction theory for covariant field theories with gauge symmetries. These symmetries are modeled by a Lie group fiber bundle acting fiberwisely on a configuration bundle. In order to reduce the variational principle, we utilize generalized principal connections, a type of Ehresmann connections that are equivariant by the fiberwise action. After obtaining the reduced equations, we give the reconstruction condition and we relate the vertical reduced equation with the Noether theorem. Lastly, we illustrate the theory with several examples, including the classical case (Lagrange-Poincar\'e reduction), Electromagnetism, symmetry-breaking and non-Abelian gauge theories.