On the Moduli Space of the Octonionic Nahm's Equations

TL;DR

This paper investigates the structure of the moduli space of solutions to octonionic Nahm's equations, establishing its geometric properties, constructing solutions from Lie group data, and analyzing symmetries and decoupled equations.

Contribution

It characterizes the moduli space as a star-shaped smooth manifold with a complete metric and introduces new solutions and symmetry analyses for the octonionic Nahm's equations.

Findings

The moduli space is a star-shaped smooth manifold with a complete metric.

Solutions can be constructed from commuting triples in the cotangent bundle of complex Lie groups.

A Kempf-Ness theorem is established for meromorphic solutions of the decoupled equations.

Abstract

In this paper, we study some basic properties of the octonionic Nahm's equations over $[0,1]$. We prove that the moduli space of the smooth solutions to the octonionic Nahm's equations over $[0,1]$ is a star-shaped smooth manifold with a complete metric. In addition, for any commuting triples of the cotangent bundle of a complex Lie group, we construct solutions to the octonionic Nahm's equations. Moreover, we introduce extra symmetry and study a decoupled version of the octonionic Nahm's equations over $[0,1]$. We prove a Kempf-Ness theorem for the meromorphic solutions to the decoupled octonionic Nahm's equations.