Primary scalar hair in Gauss-Bonnet black holes with Thurston horizons

TL;DR

This paper constructs new five-dimensional AdS black hole solutions with scalar hair supported by Thurston geometries, revealing spontaneous symmetry breaking and unique thermodynamic properties, and extends some solutions to six dimensions.

Contribution

It introduces novel asymptotically AdS black holes with primary scalar hair linked to Thurston horizon geometries in Einstein-Gauss-Bonnet theory at the Chern-Simons point.

Findings

Existence of black holes with scalar hair supported by Nil, Solv, and SL(2,R) geometries.

Solutions exhibit spontaneous isometry breaking away from special symmetric lines.

Mass and entropy of these solutions vanish, with only one integration constant representing true hair.

Abstract

In this work, we construct novel asymptotically locally AdS$_5$ black hole solutions of Einstein-Gauss-Bonnet theory at the Chern-Simons point, supported by a scalar field that generates primary hair. The strength of the scalar field is governed by an independent integration constant; when this constant vanishes, the spacetime reduces to a black hole geometry devoid of hair. The existence of these solutions is intrinsically tied to the horizon metric, which is modeled by three non-trivial Thurston geometries: Nil, Solv, and $SL(2,\mathbb{R})$. The quadratic part of the scalar field action corresponds to a conformally coupled scalar in five dimensions -an invariance of the matter sector that is explicitly broken by the introduction of a quartic self-interaction. These black holes are characterized by two distinct parameters: the horizon radius and the temperature. Notably, there exists a straight line in this parameter space along which the horizon geometry exhibits enhanced isometries, corresponding to solutions previously reported in JHEP 02, 014 (2014). Away from this line, for a fixed horizon radius and temperatures above or below a critical value, the metric's isometries undergo spontaneous breaking. Employing the Regge-Teitelboim approach, we compute the mass and entropy of these solutions, both of which vanish. Despite this, only one of the integration constants can be interpreted as hair, as the other modifies the local geometry at the conformal boundary. Finally, for Solv horizon geometries, we extend these hairy solutions to six dimensions.