Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups

TL;DR

This paper develops a geometric framework for Euler-Poincaré equations on automorphism groups of principal bundles, revealing new dual pair structures, geodesic flows, and introducing chromomorphism groups for incompressible flows.

Contribution

It formulates Euler-Poincaré equations on automorphism groups, identifies dual pair structures with momentum solutions, and introduces chromomorphism groups for volume-preserving automorphisms.

Findings

Identified geodesic flows on infinite-dimensional semidirect-product Lie groups.

Extended previous geodesic flow results to automorphism groups of principal bundles.

Defined chromomorphism groups as extensions generalizing quantomorphism groups.

Abstract

We formulate Euler-Poincar\'e equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing \delta-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group of volume-preserving automorphisms of a principal bundle. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.