Bogomolny equations

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

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

where $F_{A}$ is the curvature of a connection $A$ on a principal $G$-bundle over a 3-manifold $M$, $\Phi$ 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

• "Magnetic_monopole", Encyclopedia of Mathematics, EMS Press, 2001 [1994]