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Magnetic monopoles in hyperbolic spaces. (English) Zbl 0722.53063
Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 1-33 (1987).
[For the entire collection see Zbl 0653.00006.]
In recent years, the Penrose twistor transform has been extensively and successfully used to convert certain problems arising in physics into problems of algebraic geometry [the author, Geometry of Yang-Mills fields (Pisa 1979; Zbl 0435.58001)]. More precisely, solutions of the self-dual Yang-Mills equations on \({\mathbb{R}}^ 4\) (describing ‘instantons’) convert into holomorphic bundles on the complex projective 3-space \({\mathbb{P}}^ 3\). Similarly, solutions of the Bogomolny equation in \({\mathbb{R}}^ 3\) (describing ‘magnetic monopoles’) convert into holomorphic bundles on \(T{\mathbb{P}}_ 1\) (the tangent bundle of \({\mathbb{P}}_ 1)\) [N. J. Hitchin, Commun. Math. Phys. 83, 579-602 (1982; Zbl 0502.58017)]. In this talk, I shall consider the analogous problem, for magnetic monopoles, when the Euclidean 3-space \({\mathbb{R}}^ 3\) is replaced by the hyperbolic 3-space \(H^ 3\). Twistor methods still apply and so then ‘hyperbolic monopoles’ can also be described by holomorphic bundles.

53C56 Other complex differential geometry
32L81 Applications of holomorphic fiber spaces to the sciences
53C80 Applications of global differential geometry to the sciences
78A25 Electromagnetic theory (general)