Near-horizon symmetries of extremal black holes

TL;DR

The paper proves that extremal black holes with $D-3$ rotational symmetries in $D=4,5$ possess a near-horizon symmetry enhancement from the intrinsic $G_2$ to $SO(2,1) imes U(1)^{D-3}$ (or its orientation-preserving subgroup for Maxwell fields), and that this enhanced symmetry persists under general two-derivative theories with abelian vectors and uncharged scalars, as well as under higher-derivative corrections given analyticity in the coupling. The authors provide a constructive proof via the near-horizon limit, lemmas, and energy-momentum constraints, and extend the results with a detailed treatment of examples in five dimensions, including $S^3$ and $S^1 imes S^2$ horizon topologies, as well as analytic continuations to SU(2)-symmetric, non-extremal solutions. A key contribution is the demonstration that NH geometries can be determined and classified through a set of explicit NH data and that the general vacuum NH geometry reduces to a 3D sigma-model problem solved by Maison’s formalism, yielding a broad, robust framework for extremal NH structures. This work provides a rigorous basis for the universality of attractor behavior in extremal black holes and clarifies how NH symmetries constrain both the geometric and gauge-field content, with implications for holography and higher-dimensional gravity.

Abstract

Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a near-horizon SO(2,1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional solutions (including black rings). The result is valid for a general two-derivative theory of gravity coupled to abelian vectors and uncharged scalars, allowing for a non-trivial scalar potential. We prove that it remains valid in the presence of higher-derivative corrections. We show that SO(2,1)-symmetric near-horizon solutions can be analytically continued to give SU(2)-symmetric black hole solutions. For example, the near-horizon limit of an extremal 5D Myers-Perry black hole is related by analytic continuation to a non-extremal cohomogeneity-1 Myers-Perry solution.