The cohomology of the space of magnetic monopoles. (English) Zbl 0854.57029

The authors’ abstract: “Denote by \(X_q\) the reduced space of \(\text{SU}_2\) monopoles of charge \(q\) in \(\mathbb{R}^3\). In this paper the cohomology of \(X_q\), the cohomology with compact supports of \(X_q\), and the image of the latter in the former are all calculated as representations of \(\mathbb{Z}/q\mathbb{Z}\) which acts on \(X_2\). This provides a nontrivial “lower bound” for the \(L^2\) cohomology of \(X_q\) which is compatible with the conjectures of Sen. It is also shown that, granted some assumptions about the metric on \(X_q\), its \(L^2\) cohomology does not exceed this bound in the situation referred to in the paper as the ‘coprime case’ ”.


57R99 Differential topology
58Z05 Applications of global analysis to the sciences
Full Text: DOI


[1] [A-H] Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. Oxford: Oxford Univ. Press, 1988 · Zbl 0671.53001
[2] [de R] de Rham, G.: Differential manifolds. English edition. Berlin, Heidelberg, New York: Springer, 1984 · Zbl 0534.58003
[3] [S] Segal, G.B.: The topology of spaces of rational functions. Acta Mathematica143, 39–72 (1979) · Zbl 0427.55006
[4] [Sen] Sen, A.: To appear (HEP preprints: hep-th/9402002 and hep-th/9402032)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.