Monopoles and clusters (Q1006304)

The moduli space \(\mathcal{M}_{n}\) of framed SU(2)-monopoles of charge \(n\) in \(\mathbb{R}^{3}\) is a complete Riemannian manifold; the topological infinity corresponds to monopoles of charge \(n\) breaking down into monopoles of lower charges. This asymptotic picture may be stated as: Given an infinite sequence of points in \(\mathcal{M}_{n}\), there exist a subsequence \((m_{r})\), a partition \(n=\sum_{i=1}^{s}n_{i}\) (\(n_{i}>0\)), sequences of points \(( x_{r}^{i}) \) of \(\mathbb{R}^{3}\), \(1\leq i\leq s\), such that {\parindent=5mm \begin{itemize}\item[1.] the sequence \(( m_{r}^{i}) \) of monopoles translated by \(-x_{r}^{i}\) converges weakly to a monopole of charge \(n_{i}\) with centre at origin; \item[2.] as \(r\to\infty\), the distance \(| x_{r}^{i}-x_{r} ^{j}| \) tends to \(\infty\) for any pair \((i,j)\) and the direction of the line \(x_{r}^{i}x_{r}^{j}\) converges to a fixed direction. \end{itemize}} The aim of the paper is to describe this asymptotic picture in metric terms. The paper generalises the construction of an asymptotic metric given by \textit{Gibbons} and \textit{Manton} [Phys. Lett. B 356, 32--38 (1995)] for the case when all \(n_{i}\) are \(1\); this asymptotic metric is an exponentially good approximation of the monopole metric. For any partition \(n=\sum_{i=1}^{s}n_{i}\) (\(n_{i}>0\)), the author defines a space of (framed) clusters \(\mathcal{M}_{n_{1},\dots,n_{s}}\) with a natural pseudo-hyper-Kähler metric. When the size of clusters is bounded by \(K\) and the distances between their centres \(x^{i}\) are larger than some \(R_{0} =R_{0}(K)\), then the cluster metric in this region is \(Ce^{-\alpha R}\)-close to the monopole metric in the corresponding region of \(\mathcal{M}_{n}\), where \(R=\min\{|x^{i}-x^{j}|\); \(1\leq i<j\leq r\}\). The paper deals essentially with the case of two clusters. From the introduction of the paper: ``Apart from notational complications when \(s>2\), the chief difficulty (also for \(s=2\)) is that unlike in the case of the Gibbons-Manton metric, we have not found a description of \(\mathcal{M}_{n_{1},\dots,n_{s}}\) as a moduli space of Nahm's equations. For \(s=2\) we have such a description of the smooth (and complex) structure of \(\mathcal{M}_{n_{1},n_{2}}\), but not of its metric nor of its hypercomplex structure.''