New constructions of twistor lifts for harmonic maps

TL;DR

This paper establishes conditions for the existence of twistor lifts of harmonic maps from Riemann surfaces to symmetric spaces, providing explicit algebraic formulas for those with finite uniton number.

Contribution

It introduces new constructions for twistor lifts of harmonic maps, characterizing their existence via nilconformality and providing explicit formulas for finite uniton number cases.

Findings

Twistor lifts exist if and only if the harmonic map is nilconformal.

Explicit algebraic formulas for twistor lifts of finite uniton number harmonic maps.

Provides a comprehensive method for constructing twistor lifts in symmetric spaces.

Abstract

We show that given a harmonic map $\varphi$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $\varphi$ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.