Bogomolny equations

In mathematics, the Bogomolny equations for magnetic monopoles are the equations F_A = *D_Aφ, where F_A is the curvature of a connection A on a G-bundle over a 3-manifold M, φ is a section of the corresponding adjoint bundle and * is the Hodge star operator on M. These equations are named after E. B. Bogomolny.

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

See also

• Monopole moduli space
• Ginzburg–Landau theory
• Seiberg–Witten theory

References

• Hitchin, N. J. (1982), "Monopoles and geodesics", Communications in Mathematical Physics, 83 (4): 579–602, doi:10.1007/bf01208717, ISSN 0010-3616, MR 0649818
• Hazewinkel, Michiel, ed. (2001) [1994], "Magnetic_monopole", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4