Harmonic maps into principal bundles and generalized magnetic maps

TL;DR

This work extends Wong’s equations to higher-dimensional initial manifolds by studying harmonic maps into KK principal bundles with $G$-invariant (Kaluza-Klein) metrics. It develops a comprehensive framework: KK metrics are parameterized by a connection and base/adjoint metrics, KK harmonic maps satisfy coupled horizontal/vertical Euler–Lagrange equations, and generalized magnetic maps arise as base projections modulo gauge equivalence, with a gauge-variation/harmonic gauge-fixing structure and existence results. The theory is implemented through local expressions, fiber geometry via O’Neill tensors, and explicit Hopf-fibration examples, including $α$-twisted spherical and Clifford-type immersions, revealing that the unique uncharged instances include the standard Clifford torus and the standard spherical harmonic immersion $S^{3}\times S^{3}\to S^{7}$. Overall, the paper provides a global geometric framework for modeling magnetic-type interactions of extended objects under arbitrary gauge fields, and links generalized magnetic maps to higher-dimensional harmonic map theory with concrete canonical examples.

Abstract

We study harmonic mappings from a Riemannian manifold $N$ into a principal $G$-bundle $P$ endowed with a $G$-invariant Riemannian metric (i.e. a Kaluza-Klein metric). These morphisms are called Kaluza-Klein harmonic maps and naturally lead to the notion of generalized magnetic maps for an arbitrary gauge group $G$, which are just their projections onto the base manifold of $P$ and might provide a geometric formulation for the magnetic interaction of extended objects modelled by $N$ under the action of a generalized Lorentz force. We provide a characterization of Kaluza-Klein harmonic maps and show that the space of generalized magnetic maps is a quotient of the space of Kaluza-Klein harmonic maps under an equivalence relation generated by an appropriate gauge group. We establish a necessary condition that they must satisfy, the gauge variation formula and the harmonic gauge fixing equation, also providing a main existence theorem for them. After analyzing how they are influenced by the geometry of the fibers of the principal bundle, we construct several instances of generalized magnetic maps, including two non-trivial one-parameter families of examples based on $α$-twisted spherical harmonic immersions with values in the complex $S^{3}\longrightarrow S^{2}$ and quaternionic $S^{7}\longrightarrow S^{4}$ Hopf fibrations, proving that among them the unique uncharged ones are the standard Clifford torus and the standard spherical harmonic immersion of $ S^3\times S^3$ into $S^7$.

Theorems & Definitions (93)

• Theorem
• Definition 1
• Lemma 1: Covariance of the energy mapping under automorphisms
• Definition 2
• Proposition 1: Covariance of the Dirichlet energy under diffeomorphisms
• Definition 3
• Proposition 2
• Theorem 1
• Theorem 2
• Example : Geodesic curves