Black holes and black strings in the Einstein $SU(N)$-non-linear sigma model

We construct analytical solutions describing black holes and black strings in the Einstein $SU(N)$-non-linear sigma model in $(3+1)$ dimensions. This construction is carried out using the maximal embbeding ansatz of $SU(2)$ together with the Euler parameterization of the $SU(N)$ group, in such a way that the non-linear sigma model equations are automatically satisfied for arbitrary values of the flavor number $N$ while the Einstein equations can be solved analytically. In particular, we construct black holes with spherical and flat horizons as well as black strings that present the geometry of a three-dimensional charged Ba\~nados-Teitelboim-Zanelli black hole on the transverse section of the string. These configurations are not trivial embeddings of $SU(2)$ into $SU(N)$, which allow us to explicitly show the role that the flavor number plays on the geometry and thermodynamics of the black holes and black strings. Finally, we perform a thermal comparison between these configurations.