General Solution of 7D Octonionic Top Equation

TL;DR

This paper derives the general solution for a 7D octonionic top equation, linking it to integrable systems, Riemann surfaces, and self-dual membrane instantons, extending classical Euler top results to higher dimensions.

Contribution

It provides the first explicit general solution for a 7D octonionic top equation using Riemann surface integration, and explores its reductions to lower dimensions.

Findings

Solution expressed via Riemann surface of genus 9

Connections to self-dual membrane instantons

Reductions yield lower-dimensional Euler tops

Abstract

The general solution of a 7D analogue of the 3D Euler top equation is shown to be given by an integration over a Riemann surface with genus 9. The 7D model is derived from the 8D $Spin(7)$ invariant self-dual Yang-Mills equation depending only upon one variable and is regarded as a model describing self-dual membrane instantons. Several integrable reductions of the 7D top to lower target space dimensions are discussed and one of them gives 6, 5, 4D descendants and the 3D Euler top associated with Riemann surfaces with genus 6, 5, 2 and 1, respectively.