On a class of 4D Kahler bases and AdS_5 supersymmetric Black Holes

TL;DR

The paper develops a unifying geometric framework for $AdS_5$ supersymmetric black holes in $D=5$ minimal gauged SUGRA by using toric Kähler bases whose metric is governed by a single conformal factor $H(x)$. A near-horizon analysis shows the base becomes a Kähler cone, leading to a remarkably simple 6th-order equation $(H^2H'')'''=0$, reducible to $H^2H''=\alpha x+\beta$, with cubic polynomial solutions reproducing all known AdS_5 black holes. Full spacetime analysis confirms the same equation governs consistent solutions and reveals an infinite set of supersymmetric deformations that vanish at the horizon but modify asymptotics, sometimes yielding time-dependent descriptions in static coordinates. While the framework captures all known spherical-horizon black holes with a $U(1)^2$ isometry and accommodates non-compact horizons, it does not yet include black rings, which require relaxing the cone near-horizon assumption; future work may extend the approach to less symmetric bases. Overall, the work provides a systematic, geometric method to construct and classify AdS_5 SUSY black holes and their deformations, with potential implications for holography and horizon microphysics.

Abstract

We construct a class of toric Kahler manifolds, M_4, of real dimension four, a subset of which corresponds to the Kahler bases of all known 5D asymptotically AdS_5 supersymmetric black-holes. In a certain limit, these Kahler spaces take the form of cones over Sasaki spaces, which, in turn, are fibrations over toric manifolds of real dimension two. The metric on M_4 is completely determined by a single function H(x), which is the conformal factor of the two dimensional space. We study the solutions of minimal five dimensional gauged supergravity having this class of Kahler spaces as base and show that in order to generate a five dimensional solution H(x) must obey a simple sixth order differential equation. We discuss the solutions in detail, which include all known asymptotically AdS_5 black holes as well as other spacetimes with non-compact horizons. Moreover we find an infinite number of supersymmetric deformations of these spacetimes with less spatial isometries than the base space. These deformations vanish at the horizon, but become relevant asymptotically.