Deformations of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds

TL;DR

This paper investigates the deformation space of $ ext{G}_2$-instantons on nearly $ ext{G}_2$ manifolds, establishing a link with Dirac operators and proving rigidity of abelian instantons, with applications to specific homogeneous manifolds.

Contribution

It introduces a spinor and Dirac operator framework for deformation theory of $ ext{G}_2$-instantons on nearly $ ext{G}_2$ manifolds and proves abelian instanton rigidity.

Findings

Deformation space characterized by kernel of elliptic operator

Abelian instantons are rigid

Deformation space described for canonical connection on specific manifolds

Abstract

We study the deformation theory of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds. There is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$ structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly $\mathrm{G}_2$ manifolds.