Harmonic maps into the exceptional symmetric space $G_2/SO(4)$

TL;DR

This paper characterizes harmonic maps into the space G_2/SO(4) that admit twistor lifts, relates them to almost complex maps into the 6-sphere, and constructs examples with various properties, including infinite uniton number.

Contribution

It establishes a characterization of harmonic maps with twistor lifts as nilconformal, links them to maps into the 6-sphere, and provides explicit constructions for maps with different uniton properties.

Findings

Harmonic maps with twistor lifts are exactly the nilconformal ones.

Constructed examples of nilconformal harmonic maps not of finite uniton number.

Explicit construction of lifts for harmonic maps of finite uniton number.

Abstract

We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal; for such maps, we show that our lifts can be constructed explicitly from extended solutions.