Harmonic maps of finite uniton number and their canonical elements

TL;DR

This paper classifies harmonic maps with finite uniton number from Riemann surfaces into compact simple Lie groups using Bruhat decomposition, providing estimates for minimal uniton numbers and explicit descriptions in low-dimensional spin groups.

Contribution

It introduces a classification framework for harmonic maps of finite uniton number into compact Lie groups using algebraic loop groups and canonical elements, linking previous approaches.

Findings

Classification of harmonic maps via Bruhat decomposition

Estimation of minimal uniton numbers for various representations

Explicit descriptions of harmonic spheres in low-dimensional spin groups

Abstract

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group $\Omega_\mathrm{alg}{G}$ of algebraic loops in $G$ and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low dimensional spin groups making use of spinor representations.