The Moduli Space of Many BPS Monopoles for Arbitrary Gauge Groups

TL;DR

The paper develops a unified framework for the moduli spaces of many BPS monopoles in gauge theories with maximal symmetry breaking to $U(1)^k$, deriving the asymptotic metric from dyon interactions and identifying the two-monopole relative moduli space as Taub-NUT. Using Atiyah–Hitchin techniques, it proves the asymptotic two-monopole metric is exact and argues the generalized multi-monopole metric is smooth and hyperkähler, with strong symmetry constraints guiding its structure. It further shows that the moduli space for an arbitrary number of distinct fundamental monopoles decomposes into center-of-mass and a smooth relative part, suggesting the asymptotic form is likely exact for these cases. The results have implications for nonperturbative dynamics and dualities in supersymmetric gauge theories, including the existence of bound states related to dual vector mesons and potential simplifications in non-Abelian unbroken sectors. Overall, the work provides a geometric, hyperkähler treatment of multi-monopole dynamics and supports duality-based expectations in a broad class of gauge theories.

Abstract

We study the moduli space for an arbitrary number of BPS monopoles in a gauge theory with an arbitrary gauge group that is maximally broken to $U(1)^k$. From the low energy dynamics of well-separated dyons we infer the asymptotic form of the metric for the moduli space. For a pair of distinct fundamental monopoles, the space thus obtained is $R^3 \times(R^1\times {\cal M}_0)/Z$ where ${\cal M}_0$ is the Euclidean Taub-NUT manifold. Following the methods of Atiyah and Hitchin, we demonstrate that this is actually the exact moduli space for this case. For any number of such objects, we show that the asymptotic form remains nonsingular for all values of the intermonopole distances and that it has the symmetries and other characteristics required of the exact metric. We therefore conjecture that the asymptotic form is exact for these cases also.