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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’ ”.

MSC:

57R99 Differential topology
58Z05 Applications of global analysis to the sciences
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References:

[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)
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