The moduli space of two U(1) instantons on noncommutative $R^4$ and $R^3\times S^1$

TL;DR

The authors determine the moduli spaces of two U(1) instantons on noncommutative $R^4$ and $R^3\times S^1$ via the ADHM construction and a hyperKähler quotient, showing the relative moduli space on $R^4$ is the Eguchi–Hanson manifold with a unique threshold bound state. They extend the analysis to calorons on $R^3\times S^1$, deriving the asymptotic metric and conjecturing a full completion, with a deep link to the Coulomb branches of 3D ${\cal N}=4$ gauge theories and mirror symmetry. The work connects noncommutative instanton/caloron moduli spaces to Coulomb branches for ADE groups, and identifies the SU(2) theory with a single massive adjoint hypermultiplet as reproducing the two-caloron moduli space in the strong coupling/decompactification limit. Overall, the paper provides a unified HK-quotient, ADHM framework for these moduli spaces and highlights their physical realizations in lower-dimensional gauge dynamics.

Abstract

We employ the ADHM method to derive the moduli space of two instantons in U(1) gauge theory on a noncommutative space. We show by an explicit hyperKähler quotient construction that the relative metric of the moduli space of two instantons on $R^4$ is the Eguchi-Hanson metric and find a unique threshold bound state. For two instantons on $R^3\times S^1$, otherwise known as calorons, we give the asymptotic metric and conjecture a completion. We further discuss the relationship of caloron moduli spaces of A, D and E groups to the Coulomb branches of three dimensional gauge theory. In particular, we show that the Coulomb branch of SU(2) gauge group with a single massive adjoint hypermultiplet coincides with the above two caloron moduli space.