Higher-power harmonic maps, instantons and Yang-Mills theory

Abstract

Let $(M,g)$ and $(N,h)$ be two pseudo-Riemannian manifolds. We study field theoretic properties of higher-power harmonic maps (also called r-harmonic maps) $\phi :M\to N$, which are a natural generalization of standard harmonic maps first introduced by C. Wood. In particular, we discuss the coupled system of higher-power harmonic maps and the Einstein-Hilbert action and prove a sufficient condition for a map to be r-harmonic, which is highly motivated by classical field equations like the harmonic map equation or the Yang-Mills equation. Furthermore, we derive an instanton theory for r-harmonic maps on 2r-dimensional base manifolds and investigate conformal properties of general higher-power harmonic maps. Finally, since the theory of higher-power harmonic maps bears striking similarities with Yang-Mills theory, we provide a comprehensive comparison between the two theories which explains in more detail surprisingly many analogies.