Moduli Spaces of Instantons on Toric Noncommutative Manifolds

TL;DR

This paper investigates the structure and properties of moduli spaces of instantons on toric noncommutative four-manifolds, providing explicit dimension formulas and demonstrating smoothness and Hausdorff properties.

Contribution

It explicitly characterizes the moduli spaces of U(2) instantons on toric noncommutative manifolds, including the four-sphere, showing they are smooth manifolds with computable dimensions.

Findings

Moduli spaces are either empty or smooth Hausdorff manifolds.

Dimension of moduli space on S^4_θ is 8k-3 for instantons with second Chern number k.

Explicit dimension formulas are derived for these moduli spaces.

Abstract

We study analytic aspects of U(n) gauge theory over a toric noncommutative manifold $M_\theta$. We analyse moduli spaces of solutions to the self-dual Yang-Mills equations on U(2) vector bundles over four-manifolds $M_\theta$, showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere $S^4_\theta$ we find that the moduli space of U(2) instantons with fixed second Chern number $k$ is a smooth manifold of dimension $8k-3$.