Monopoles and Instantons on Partially Compactified D-Branes

TL;DR

The work analyzes monopoles and Wilson-loop instantons in supersymmetric SU($N$) Yang–Mills on $S^1\times \mathbb{R}^{3+1}$ with a Wilson-loop breaking, showing the $N$-th KK monopole completes the set of $N$ fundamental monopoles and that a fully monopole configuration can reinterpret as a Wilson-loop instanton of unit Pontryagin number. It derives the exact moduli-space metric, identifies it with the Coulomb branch of a 3D $U(1)^N$ theory (Kronheimer-type), and demonstrates that in the symmetric phase the relative moduli space tends to the Calabi metric, matching the Calabi/Kronheimer picture. The results extend Intriligator–Seiberg to finite couplings and provide a unifying D-brane and field-theory perspective on solitons in partially compactified gauge theories. This work clarifies the interrelation between monopoles, instantons, and three-dimensional gauge dynamics, suggesting further exploration of S-duality, Nahm transform, and higher-torus compactifications.

Abstract

Motivated by the recent D-brane constructions of world-volume monopoles and instantons, we study the supersymmetric SU(N) Yang-Mills theory on $S^1 \times R^{3+1}$, spontaneously broken by a Wilson loop. In addition to the usual N-1 fundamental monopoles, the N-th BPS monopole appears from the Kaluza-Klein sector. When all N monopoles are present, net magnetic charge vanishes and the solution can be reinterpreted as a Wilson-loop instanton of unit Pontryagin number. The instanton/multi-monopole moduli space is explicitly constructed, and seen to be identical to a Coulomb phase moduli space of a U(1)^N gauge theory in 2+1 dimensions related to Kronheimer's gauge theory of SU(N) type. This extends the results by Intriligator and Seiberg to the finite couplings that, in the infrared limit of Kronheimer's theory, the Coulomb phase parameterizes a centered SU(N) instanton. We also elaborate on the case of restored SU(N) symmetry.