Quantizations of local surfaces and rebel instantons

Abstract

We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k:=\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^1}(-k))$ and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.