Deformations of $G_2$ instantons on nearly $G_2$ manifolds

In this talk we will discuss the deformation theory for $G_2$ instantons on $7$-dimensional manifolds with nearly parallel $G_2$-structure. We prove that the space of infinitesimal deformations of a $G_2$ instanton on such manifolds can be identified with the eigenspaces of a $1$-parameter family of Dirac operators and as a result prove that the abelian instantons are rigid, yielding a different proof of a recent result of Ball-Oliveira. This identification along with some other properties of these instantons can then be further used to gather more information about the deformation space in the non abelian case. This talk is based on a work in progress and we aim in future to determine the deformation space of some particular instantons on some known examples of nearly $G_2$ manifolds.Non UBCUnreviewedAuthor affiliation: University of WaterlooGraduat