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.