Deformations of Asymptotically Conical $G_2$-Instantons

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 $R^7$.