Black holes and black strings in the Einstein SU(N) nonlinear sigma model

Abstract

We construct analytical solutions describing black holes and black strings in the Einstein $SU(N)$-nonlinear sigma model in ($3+1$) dimensions. This construction is carried out using the maximal embedding ansatz of $SU(2)$ together with the Euler parametrization of the $SU(N)$ group, in such a way that the nonlinear 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.