Deformations of Asymptotically Conical $G_2$-Instantons

TL;DR

This paper develops a deformation theory for $G_2$-instantons on asymptotically conical manifolds, relating deformations to Dirac operators, and uses it to classify and prove uniqueness of certain instantons.

Contribution

It introduces a spinorial deformation framework for $G_2$-instantons on asymptotically conical manifolds and applies it to classify unobstructed instantons on $\mathbb{R}^7$.

Findings

Calculated the virtual dimension of moduli spaces of $G_2$-instantons.

Proved unobstructed instantons are $G_2$-invariant.

Established uniqueness of unobstructed $G_2$-instantons on $\mathbb{R}^7$.

Abstract

We develop the deformation theory of instantons on asymptotically conical $G_2$-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted Dirac operator on the $G_2$-manifold and to the eigenvalues of a twisted Dirac operator on the nearly K\"ahler link. This framework is then used to calculate the virtual dimension of the moduli spaces of $G_2$-instantons on which several known examples live. One such example considered is the $G_2$-instanton of G\"unaydin-Nicolai, which lives on $R^7$. As an application of the deformation theory, we show how knowledge of the virtual dimension of the moduli space allows us to prove that unobstructed connections in the moduli space are $G_2$-invariant. By classifying such connections we prove a uniqueness result for unobstructed $G_2$-instantons on the principal $G_2$-bundle over $R^7$.