In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation F A = ⋆ d A Φ, {\displaystyle F_{A}=\star d_{A}\Phi,} where F A {\displaystyle F_{A}} is the curvature of a connection A {\displaystyle A} on a principal G {\displaystyle G} -bundle over a 3-manifold M {\displaystyle M}, Φ {\displaystyle \Phi } is a section of the corresponding adjoint bundle, d A {\displaystyle d_{A}} is the exterior covariant derivative induced by A {\displaystyle A} on the adjoint bundle, and ⋆ {\displaystyle \star } is the Hodge star operator on M {\displaystyle M}. Wikipedia