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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)
##### Citations:
Zbl 0653.00006; Zbl 0435.58001; Zbl 0502.58017