Instantons and Chern-Simons flows in 6, 7 and 8 dimensions

TL;DR

This paper explores the mathematical structures of instantons and Chern-Simons flows in higher dimensions, analyzing their equations, solutions, and geometric conditions on manifolds with special holonomy.

Contribution

It introduces new insights into the behavior of instantons and Chern-Simons flows on 7-manifolds with G_2-structures, including analytic and numerical solutions.

Findings

Existence of instantons on nearly Kähler 6-manifolds

Reduction of Yang-Mills equations to quartic dynamics

Analytic kink solutions and numerical interpolations

Abstract

The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant connections, the torsionful Yang-Mills equation reduces to a particular quartic dynamics for a Newtonian particle on C. When the torsion corresponds to one of the G_2-structures, this dynamics follows from a gradient or hamiltonian flow equation, respectively. We present the analytic (kink-type) solutions and plot numerical non-BPS solutions for general torsion values interpolating between the instantonic ones.