Bogomolny equations

Summary

In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation

where is the curvature of a connection on a principal -bundle over a 3-manifold , is a section of the corresponding adjoint bundle, is the exterior covariant derivative induced by on the adjoint bundle, and is the Hodge star operator on . These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.[1][2]

The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If is closed, there are only trivial (i.e. flat) solutions.

See also edit

References edit

  1. ^ Atiyah, Michael; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, ISBN 978-0-691-08480-0, MR 0934202
  2. ^ Hitchin, N. J. (1982), "Monopoles and geodesics", Communications in Mathematical Physics, 83 (4): 579–602, Bibcode:1982CMaPh..83..579H, doi:10.1007/bf01208717, ISSN 0010-3616, MR 0649818, S2CID 121082095