F-term strings in the Bogomol'nyi limit are also BPS states

We derive the Bogomol'nyi equations for supersymmetric Abelian F-term cosmic strings in four-dimensional flat space and show that, contrary to recent statements in the literature, they are BPS states in the Bogomol'nyi limit, but the partial breaking of supersymmetry is from N=2. The second supersymmetry is not obvious in the N=1 formalism, so we give it explicitly in components and in terms of a different set of N=1 chiral superfields. We also discuss the appearance of a second supersymmetry in D-term models, and the relation to N=2 F-term models. The analysis sheds light on an apparent paradox raised by the recent observation that D-term strings remain BPS when coupled to N=1 supergravity, whereas F-term strings break the supersymmetry completely, even in the Bogomol'nyi limit. Finally, we comment on their semilocal extensions and their relevance to cosmology.