Bogomolny equations

Short description: Equations describing magnetic monopoles

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

$F_A=\star d_A\mathrm{\Phi },$

where $F_A$ is the curvature of a connection $A$ on a principal $G$-bundle over a 3-manifold $M$, $\mathrm{\Phi }$ is a section of the corresponding adjoint bundle, $d_A$ is the exterior covariant derivative induced by $A$ on the adjoint bundle, and $\star$ is the Hodge star operator on $M$. These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.

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 $M$ 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

• Hazewinkel, Michiel, ed. (2001), "Magnetic_monopole", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page