Moduli spaces of instantons on toric noncommutative manifolds

Abstract

We study analytic aspects of $\mathrm{U}(n)$ gauge theory over a toric noncommutative manifold $M_\theta$. We analyse moduli spaces of solutions to the self-dual Yang-Mills equations on $\mathrm{U}(2)$ 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 $\mathrm{U}(2)$ instantons with fixed second Chern number $k$ is a smooth manifold of dimension $8k - 3$.