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 $SpecP$). 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.

Some typos corrected. Still 70 pages. Comments are welcome

Cited by in corpus (1)

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