Yang-Mills flows on nearly Kaehler manifolds and G_2-instantons

TL;DR

This paper studies G-invariant Yang-Mills connections on special manifolds with G_2 and Spin(7) structures, reducing the problem to particle mechanics and finding explicit instanton solutions.

Contribution

It introduces a G-invariant ansatz for Yang-Mills theory with torsion on these manifolds, deriving explicit solutions and connecting G_2- and Spin(7)-instantons.

Findings

Explicit particle trajectories for specific torsion values.

Identification of G_2-instantons as solutions to the equations.

Unified description of G_2- and Spin(7)-instantons from a single equation.

Abstract

We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, RxG/H and R^2xG/H, where G/H is a compact nearly Kaehler six-dimensional homogeneous space, and the manifolds RxG/H and R^2xG/H carry G_2- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on RxG/H is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G_2-structures on RxG/H. It is shown that both G_2-instanton equations can be obtained from a single Spin(7)-instanton equation on R^2xG/H.