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Construction of Euclidian Monopoles
Author(s) -
Jarvis S
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000446
Subject(s) - magnetic monopole , mathematics , euclidean space , euclidean geometry , partial differential equation , ordinary differential equation , manifold (fluid mechanics) , descent (aeronautics) , gauge group , gauge theory , pure mathematics , mathematical analysis , algebra over a field , differential equation , geometry , mathematical physics , physics , quantum mechanics , mechanical engineering , engineering , meteorology
This paper describes a procedure for the construction of monopoles on three‐dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from CP 1 to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein. A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well‐known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not SU (2). Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation. A non‐linear flow equation, essentially just the path of steepest descent down the Yang‐Mills‐Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ ‐ constructed explicitly from the rational map ‐ a solution to the flow must exist, and converge to an exact monopole having the desired rational map. 1991 Mathematics Subject Classification : 53C07, 53C80, 58D27, 58E15, 58G11.