Deformations of nearly K\"ahler instantons

TL;DR

This paper develops a deformation theory for instantons on nearly Kähler six-manifolds, identifying the deformation space via elliptic operators and demonstrating rigidity of abelian instantons, with applications to canonical connections.

Contribution

It introduces a new framework using spinors and Dirac operators to analyze instanton deformations on nearly Kähler manifolds, revealing rigidity and deformation properties.

Findings

Irreducible instantons with semisimple structure groups have deformation spaces as kernels of elliptic operators.

Abelian instantons are shown to be rigid with no nontrivial deformations.

Canonical connections on certain nearly Kähler manifolds are rigid instantons, but can deform when viewed as tangent bundle instantons.

Abstract

We formulate the deformation theory for instantons on nearly K\"ahler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly K\"ahler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3).