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

TL;DR

This paper presents analytical solutions for black holes and black strings within the Einstein $SU(N)$-non-linear sigma model, revealing how the flavor number influences their geometry and thermodynamics.

Contribution

It introduces a novel analytical construction of black hole and black string solutions in the Einstein $SU(N)$-non-linear sigma model using a maximal embedding ansatz.

Findings

Constructed black holes with spherical and flat horizons.

Developed black strings with BTZ black hole geometry.

Analyzed the impact of flavor number on thermodynamics.

Abstract

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.