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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.

MSC:
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)