# Quantizations of local surfaces and rebel instantons

## TL;DR

This paper constructs explicit deformation quantizations of certain noncompact complex surfaces and explores their impact on moduli spaces of vector bundles and instantons, introducing the concept of rebel instantons that misbehave under quantization.

## Contribution

It introduces explicit deformation quantizations of noncompact surfaces and the novel concept of rebel instantons that exhibit irregular behavior under these quantizations.

## Key findings

- Quantum instanton moduli space is an étale space over the classical moduli space.
- Rebel instantons are identified as those that misbehave under quantization.
- Quantizations affect the structure of moduli spaces of vector bundles and instantons.

## Abstract

We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ 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 \'etale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.