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Extremal Black Holes in Supergravity

Laura Andrianopoli, Riccardo D'Auria, Sergio Ferrara, Mario Trigiante

TL;DR

This work surveys extremal black holes in four-dimensional supergravity, focusing on the attractor mechanism that fixes horizon data via charges independent of asymptotic moduli. It develops a unified, symplectic framework for BPS and non-BPS solutions across $N=2$ to $N=8$ theories, deriving first-order attractor equations from Killing-spinor conditions (and equivalently from the geodesic potential $V_{ ext{BH}}$). Central to the analysis are the central charges, their dressed expressions, and U-duality invariants (such as quartic $I_4$ and quadratic $I_2$) that determine the horizon entropy $S_{ ext{BH}}$ in a moduli-independent way. The results classify attractors by charge orbits on symmetric spaces, reveal the near-horizon Bertotti–Robinson geometry, and connect macroscopic entropy to microscopic dualities and invariants across all extended supergravities in four dimensions.

Abstract

We present the main features of the physics of extremal black holes embedded in supersymmetric theories of gravitation, with a detailed analysis of the attractor mechanism for BPS and non-BPS black-hole solutions in four dimensions.

Extremal Black Holes in Supergravity

TL;DR

This work surveys extremal black holes in four-dimensional supergravity, focusing on the attractor mechanism that fixes horizon data via charges independent of asymptotic moduli. It develops a unified, symplectic framework for BPS and non-BPS solutions across to theories, deriving first-order attractor equations from Killing-spinor conditions (and equivalently from the geodesic potential ). Central to the analysis are the central charges, their dressed expressions, and U-duality invariants (such as quartic and quadratic ) that determine the horizon entropy in a moduli-independent way. The results classify attractors by charge orbits on symmetric spaces, reveal the near-horizon Bertotti–Robinson geometry, and connect macroscopic entropy to microscopic dualities and invariants across all extended supergravities in four dimensions.

Abstract

We present the main features of the physics of extremal black holes embedded in supersymmetric theories of gravitation, with a detailed analysis of the attractor mechanism for BPS and non-BPS black-hole solutions in four dimensions.

Paper Structure

This paper contains 24 sections, 283 equations, 4 tables.

Table of Contents

  1. Introduction: Extremal Black Holes from Classical General Relativity to String Theory
  2. Extremal Black Holes as massive representations of supersymmetry
  3. Massive representations of the supersymmetry algebra
  4. Massive BPS multiplets
  5. Massive non-BPS multiplets
  6. The general form of the supergravity action in four-dimensions and its BPS configurations
  7. Duality Rotations and Symplectic Covariance
  8. Duality symmetries and central charges
  9. The $N=2$ theory
  10. Supersymmetric black holes: General discussion
  11. BPS extremal black holes
  12. BPS and non-BPS attractor mechanism: The geodesic potential
  13. Extremal black holes in supergravity
  14. B-H entropy as a U-invariant for symmetric spaces
  15. Detailed analysis of attractors in extended supergravities: BPS and non-BPS critical points
  16. ...and 9 more sections