Monopoles and instantons on partially compactifiedD-branes

Motivated by the recent $D$-brane constructions of world-volume monopoles and instantons, we study the supersymmetric SU$\mathrm{}\left(N\right)\mathrm{}$ 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 Bogomol'nyi-Prasad-Sommerfield 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-multimonopole moduli space is explicitly constructed, and seen to be identical to a Coulomb phase moduli space of a U${\mathrm{}\left(1\right)\mathrm{}}^N$ gauge theory in $2+1$ dimensions related to Kronheimer's gauge theory of SU$\mathrm{}\left(N\right)\mathrm{}$-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 parametrizes a centered SU$\mathrm{}\left(N\right)\mathrm{}$ instanton. We also elaborate on the case of restored SU$\mathrm{}\left(N\right)\mathrm{}$ symmetry.