stu Black Holes Unveiled

TL;DR

This work provides the complete, general solution to the radial attractor-flow equations for extremal black holes in the $stu$ model, covering 1/2-BPS, non-BPS $Z=0$, and non-BPS $Z\neq 0$ branches. It employs a harmonic-function ansatz, $U$-duality, and fake supergravity to derive explicit flows and their associated fake superpotentials, showing the BPS bound holds along the entire non-BPS flows and that, for non-BPS $Z\neq 0$, the marginal stability bound is saturated in general. The analysis reveals a two-parameter moduli space $(SO(1,1))^2$ along non-BPS $Z\neq 0$ flows and provides detailed results for D0-D6, D0-D2-D4-D6, D0-D4, and D2-D6 configurations, including entropy from the Cayley hyperdeterminant / $\mathcal{I}_4$ invariant and duality relations between charge configurations. These findings extend previous work (K2-bis, Hotta, GLS-1, Cai-Pang) and illuminate the integrability of scalar dynamics in extremal BH backgrounds, with implications for microstate counting, stability, and links to quantum information theoretic interpretations of the STU system.

Abstract

The general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing central charge Z and for Z=0, are obtained for the so-called stu model, the minimal rank-3 N=2 symmetric supergravity in d=4 space-time dimensions. Comparisons with previous results, as well as the fake supergravity (first order) formalism and an analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability of corresponding D-brane configurations, are given.