Bianchi Attractors: A Classification of Extremal Black Brane Geometries

TL;DR

This work extends the holographic dictionary by classifying extremal black brane horizons under generalized translational symmetries, organizing spatial homogeneity via the Bianchi nine-class classification. It demonstrates that simple gravity + massive gauge-field systems with a negative cosmological constant naturally realize a rich set of near-horizon, scale-invariant geometries across many Bianchi classes (including VII_0, II, VI/III/V, IX, and VIII). A concrete Type VII_0 solution is presented, along with several other class realizations and an explicit extremal brane that interpolates from Type VII in the IR to AdS_5 in the UV, illustrating RG-like flows between homogeneous phases. The paper also explores the intriguing possibility of time-involving generalized translations, including constructions that produce closed timelike curves and cautionary notes about their physical interpretation. Overall, the findings reveal a large zoo of homogeneous yet anisotropic extremal horizons with potential applications to gauge/gravity duality and condensed-matter systems, and lay out clear avenues for stability analyses, finite-temperature deformations, and UV completions.

Abstract

Extremal black branes are of interest because they correspond to the ground states of field theories at finite charge density in gauge/gravity duality. The geometry of such a brane need not be translationally invariant in the spatial directions along which it extends. A less restrictive requirement is that of homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. In this paper, we find large new classes of homogeneous but anisotropic extremal black brane horizons, which could naturally arise in gauge/gravity dual pairs. In 4+1 dimensional spacetime, we show that such homogeneous black brane solutions are classified by the Bianchi classification, which is well known in the study of cosmology, and fall into nine classes. In a system of Einstein gravity with negative cosmological term coupled to one or two massive Abelian gauge fields, we find solutions with an additional scaling symmetry, which could correspond to the near-horizon geometries of such extremal black branes. These solutions realize many of the Bianchi classes. In one case, we construct the complete extremal solution which asymptotes to AdS space.

Figures (6)

• Figure 1: Numerical solution for metric component $\log f(r)$.
• Figure 2: Numerical solution for metric component $\log g(r)$.
• Figure 3: Numerical solution for metric component $\log h(r)$.
• Figure 4: Numerical solution for metric component $\log j(r)$.
• Figure 5: Numerical solution for gauge field $\log A_{1t}(r)$.