The spectrum of an operator associated with G 2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities

This is the first step in an attempt at a deformation theory for G₂-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator P on a certain bundle over S⁵, called the link operator. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold. Using the quaternion structure in the Sasakian geometry of S⁵, we describe the set of all eigenvalues of P, denoted by P. We show that P consists of finitely many integers induced by certain sheaf cohomologies on P², and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over S⁵. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on S⁵ to certain sheaf cohomologies on~P². Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of (3) and the subgroup S[U(1)× U(2)], we show an example in which P and the multiplicities can be completely determined.