Deformations of Asymptotically Conical G2-Instantons

This thesis develops the deformation theory of instantons on asymptotically conical G2-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 G2-manifold and to the eigenvalues of a twisted Dirac operator on the nearly Kähler link. As an application, we use this framework to study the moduli spaces of known examples of G2-instantons living on the Bryant-Salamon manifolds and on R7. We develop two methods for determining eigenvalues of twisted Dirac operators on nearly Kähler 6- manifolds and apply this to calculate the virtual dimension of the moduli spaces that we study. In the case of the instanton of Günaydin-Nicolai, which lives on R7; we show how knowledge of the virtual dimension of the moduli space can be used to study uniqueness properties of this instanton.