Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with $G_2$-structure

TL;DR

This paper investigates Yang-Mills fields and Spin(7)-instantons on cylinders over seven-dimensional coset spaces with special geometric structures, reducing the problem to classical mechanics and analyzing solutions with finite energy.

Contribution

It provides a detailed analysis of Yang-Mills solutions and Spin(7)-instantons on specific coset spaces with G_2 and SU(3) structures, including critical points and finite-energy solutions.

Findings

Identification of critical points of the potential for each coset space

Construction of finite-energy Yang-Mills solutions

Explicit Spin(7)-instanton configurations on the cylinders

Abstract

We study $\mathfrak{g}$-valued Yang-Mills fields on cylinders $Z(G/H)=\mathbb{R} \times G/H$, where G/H is a compact seven-dimensional coset space with $G_2$-structure, $\mathfrak{g}$ is the Lie algebra of G, and Z(G/H) inherits a Spin(7)-structure. After imposing a general G-invariance condition, Yang-Mills theory with torsion on Z(G/H) reduces to Newtonian mechanics of a point particle moving in $\mathbb{R}^n$ under the influence of some quartic potential and possibly additional constraints. The kinematics and dynamics depends on the chosen coset space. We consider in detail three coset spaces with nearly parallel $G_2$-structure and four coset spaces with SU(3)-structure. For each case, we analyze the critical points of the potential and present a range of finite-energy solutions. We also study a higher-dimensional analog of the instanton equation. Its solutions yield G-invariant Spin(7)-instanton configurations on Z(G/H), which are special cases of Yang-Mills configurations with torsion.