Charges from Attractors

TL;DR

This work develops a comprehensive framework to extract the conserved charges of extremal black holes from their near-horizon geometries using gravitational Noether-Wald charges, with explicit results for type IIB supergravity in 10 dimensions and 5d minimal/gauged supergravity. By performing dimensional reduction and relating NHG charges to Kaluza-Klein charges, the authors show how to recover full angular momenta and electric charges from NHG data, even in the presence of Chern-Simons terms. They extend the entropy-function formalism to account for CS terms, derive a NHG-based entropy and a first-law-like relation that holds intrinsically in the NHG, and connect these to the Euclidean action and a NHG Smarr formula. The approach is validated on Gutowski–Reall and BMPV black holes and supersymmetric non-equal angular momentum cases, and it clarifies the conditions under which NHG charges agree with asymptotic charges, while highlighting notable exceptions such as black rings where topology obstructs a straightforward reduction. Overall, the paper provides a practical, gauge-consistent method to obtain black hole charges from near-horizon data, with broad relevance for attractor studies and holographic duals to NHG dynamics.

Abstract

We describe how to recover the quantum numbers of extremal black holes from their near horizon geometries. This is achieved by constructing the gravitational Noether-Wald charges which can be used for non-extremal black holes as well. These charges are shown to be equivalent to the U(1) charges of appropriately dimensionally reduced solutions. Explicit derivations are provided for 10 dimensional type IIB supergravity and 5 dimensional minimal gauged supergravity, with illustrative examples for various black hole solutions. We also discuss how to derive the thermodynamic quantities and their relations explicitly in the extremal limit, from the point of view of the near-horizon geometry. We relate our results to the entropy function formalism.