The spectrum of an operator associated with $G_{2}-$instantons with $1-$dimensional singularities and Hermitian Yang-Mills connections with isolated singularities

Abstract

This is the first step in an attempt at a deformation theory for $G_{2}-$instantons with $1-$dimensional conic singularities. Under a set of model data, the linearization yields a self-adjoint first order elliptic operator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimension reduction, the operator $P$ 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 $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$ (denoted by $Spec P$). We show that $SpecP$ consists of finitely many integers induced by certain sheaf cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly. Using the representation theory of $SU(3)$ and the subgroup $S[U(1)\times U(2)]$, we show an example in which $SpecP$ and the multiplicities can be completely determined.