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

TL;DR

This paper uses the ADHM method to explicitly construct and analyze the moduli space of two U(1) instantons on noncommutative R^4 and R^3×S^1, revealing the Eguchi-Hanson metric and connections to gauge theory Coulomb branches.

Contribution

It provides the first explicit hyperK"ahler quotient construction of the two-instanton moduli space on noncommutative spaces and explores its relation to gauge theory Coulomb branches.

Findings

The relative metric of two instantons on R^4 is the Eguchi-Hanson metric.

A unique threshold bound state for two instantons is identified.

The asymptotic metric for calorons on R^3×S^1 is derived and conjectured to complete.

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 hyperK\"ahler 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.