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.

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.