Deformation of singular connections I: $G_{2}-$instantons with point singularities

In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. $1$. over a closed 7-manifold, under some natural conditions, a $G_{2}-$instanton and its point singularities can still be "seen" when the $G_{2}-$structure is properly perturbed. $2$. over a ball in $\R^{7}$, for any almost-Euclidean $G_{2}-$structure, there exists a $G_{2}-$monopole asymptotic to an arbitrary Hermitian Yang-Mills connection on $S^{6}$.