Bogomolny equations

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

${\displaystyle F_{A}=\star d_{A}\Phi ,}$

where ${\displaystyle F_{A}}$ is the curvature of a connection ${\displaystyle A}$ on a principal ${\displaystyle G}$-bundle over a 3-manifold ${\displaystyle M}$, ${\displaystyle \Phi }$ is a section of the corresponding adjoint bundle, ${\displaystyle d_{A}}$ is the exterior covariant derivative induced by ${\displaystyle A}$ on the adjoint bundle, and ${\displaystyle \star }$ is the Hodge star operator on ${\displaystyle M}$. 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 ${\displaystyle M}$ is closed, there are only trivial (i.e. flat) solutions.

See also

• Monopole moduli space
• Ginzburg–Landau theory
• Seiberg–Witten theory
• Bogomol'nyi–Prasad–Sommerfield bound

References

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

External links

• Bogomolny equation on nLab
• "Magnetic_monopole", Encyclopedia of Mathematics, EMS Press, 2001 [1994]