On Instantons on Nearly Kahler 6-manifolds

Abstract

Abstract. We study ω-instantons on nearly Kähler 6-manifolds. These are defined as connections A whose curvatures F satisfy * F = −ω ∧ F . First, we show these connections enjoy nice properties: they are Yang-Mills and variational. Second, we discuss their relation with instantons over the G 2 cones. Third, we derive a Weitzenböck formula for the infinitesimal deformation and derive some rigidity results. Fourth, we construct some SO(4)-invariant examples over open sets of S 6 .

Instantons and Killing spinors

“…The special case of this proposition where M is nearly Kähler was previously obtained using a different method by Xu [18]. We will give some alternative proofs of this proposition in the following section.…”

“…Thus F satisfies the ordinary Yang-Mills equation, confirming proposition 2.1. This argument is due to Xu [18].…”

“…So on a nearly parallel G 2 -manifold, the instanton equation on the cylinder is equivalent to the gradient flow for W . In all other cases the instanton equation on the cylinder is equivalent to the gradient flow for W , together with a number of constraints (see [14,18] for discussions of the nearly Kähler case).…”

“…The instanton equations make sense on any manifold with a Gstructure, and it is hoped that their study will result in new invariants for such manifolds, just as the original instanton equations were the main ingredient in Donaldon's 4-manifold invariants [6,7,8]. Thus the search for solutions to the instanton equations is well-motivated, and many examples of instantons have appeared in the literature [9,10,11,12,13,14,15,16,17,18,19,20].…”

Deformations of Nearly Kähler Instantons

“…Instantons on nearly Kähler six-manifolds are Yang-Mills [63,Proposition 2.10], and the tangent bundle over any nearly Kähler six-manifold admits an instanton [32], which is known as the canonical connection and characterised by having skew-symmetric torsion and holonomy contained in SU (3).…”