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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