Two species of semilocal cosmic strings in Abelian gauged CP2 -sigma models coupled to gravity

We study a semi-local Abelian Higgs sigma model with $\mathbb{CP}^2$-valued scalar fields coupled to gravity. We establish the conditions needed for self-duality in this system and obtain cosmic strings, both of self-dual character and out of the self-dual point, in the reference chart of $\mathbb{CP}^2$. In any of the other two charts of a minimum atlas, the transition functions break the global $SU(2)$ symmetry to the Abelian subgroup $e^{i \sigma^3 \alpha}$ and a new kind of semi-local Abelian gauge theory arises. The new system admits also a self-dual structure, and when coupled to gravity a second species of cosmic strings enters the scene.