The moduli space of conically singular instantons over an SU(3)-manifold

TL;DR

This paper develops a deformation theory for conically singular SU(3)-instantons on 6-manifolds, establishing a Kuranishi structure and computing the virtual dimension of their moduli space.

Contribution

It introduces a Fredholm deformation framework fixing tangent connections while allowing the bundle to vary, and provides formulas for the moduli space dimension.

Findings

Established a Kuranishi structure for the moduli space.

Derived a formula for the dimension of the cokernel of the deformation operator.

Applied results to instantons with structure group PU(n) and expressed the virtual dimension via sheaf cohomology.

Abstract

In this article we study the moduli space of conically singular instantons (or Hermitian Yang--Mills connections) with prescribed tangent connections over a 6-manifold equipped with an $\mathrm{SU}(3)$-structure. That is, we develop a Fredholm deformation theory for such $\mathrm{SU}(3)$-instantons in which we fix the tangent connection but allow the underlying principal bundle (and, in particular, the singular set) to vary. This leads to the existence of a Kuranishi structure for this moduli space. Moreover, we investigate the cokernel of the instanton deformation operator and give under certain assumptions a formula for its dimension. Ultimately, we apply our results to conically singular instantons with structure group $\mathbb{P}\mathrm{U}(n)$ and give a formula for the virtual dimension of their moduli space in terms of sheaf cohomology of certain vector bundles over $\mathbb{P}^2$.