A closed form for unitons

TL;DR

This paper derives a simple closed-form formula for unitons, harmonic spheres in unitary groups, using monad representations, and explores their properties and classifications.

Contribution

It introduces a monad-based formula for unitons, showing real triviality is automatic and linking uniton number to jumping lines, with classifications for certain cases.

Findings

Derived a closed-form formula for unitons using monads.

Proved real triviality of uniton bundles is automatic.

Connected uniton bundles to based maps into Grassmannians.

Abstract

Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson's representation of instanton bundles to obtain a simple formula for the unitons. Using the monads, we show that real triviality for uniton bundles is automatic. We interpret the uniton number as the `length' of a jumping line of the bundle, and identify the uniton bundles which correspond to based maps into Grassmannians. We also show that energy-$3$ unitons are $1$-unitons, and give some examples.