Supersymmetry
In supersymmetry, the BPS bound is saturated when half (or a quarter or an eighth) of the SUSY generators are unbroken. This happens when the mass is equal to the central extension, which is typically a topological charge.
In fact, most bosonic BPS bounds actually come from the bosonic sector of a supersymmetric theory and this explains their origin.
Read more about this topic: Bogomol'nyi-Prasad-Sommerfield Bound