Two species of semi-local cosmic strings in Abelian gauged ${\mathbb{C}\mathbb{P}}^2$-sigma models coupled to gravity

We study a semi-local Abelian Higgs sigma model with ${\mathbb{C}\mathbb{P}}^2$-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 ${\mathbb{C}\mathbb{P}}^2$. In any of the other two charts of a minimum atlas, the transition functions break the global $SU(2)$ symmetry to the Abelian subgroup $e^{i{\sigma }^3\alpha }$. In this context, using the transition functions a new self-dual structure can be constructed, which defines a second species of cosmic strings. Because of the presence of a potential energy for the scalar fields, each species takes values in a different bounded region of the target space and is most naturally described using a particular chart of ${\mathbb{C}\mathbb{P}}^2$.