Harmonic maps of finite uniton number into $G_2$

TL;DR

This paper provides explicit formulas and algebraic characterizations for harmonic maps of finite uniton number into the exceptional Lie group G_2, advancing understanding of their structure and solutions.

Contribution

It introduces explicit canonical factorizations and describes the Frenet frame data for harmonic maps into G_2, linking harmonic spheres to algebraic systems.

Findings

Explicit formulas for factorizations of harmonic maps into G_2.

Description of Frenet frame data for these harmonic maps.

Harmonic spheres into G_2 correspond to solutions of algebraic quadratic and cubic systems.

Abstract

We establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional Lie group $G_2$ in terms of the Grassmannian model for the group of based algebraic loops in $G_2$. A description of the ``Frenet frame data" for such harmonic maps is given. In particular, we show that harmonic spheres into $G_2$ correspond to solutions of certain algebraic systems of quadratic and cubic equations.