[Paper Review] BPS Black Holes in Superegravity

Riccardo D’Auria, Pietro Fré|ArXiv.org|Dec 17, 1998

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper provides a comprehensive, self-contained introduction to BPS black holes in N=2 and N=8 supergravity, deriving their properties using special geometry, central charges, and U-duality invariants. It constructs 1/2, 1/4, and 1/8 supersymmetric black holes via solvable Lie algebras, establishing black hole entropy as a U-duality invariant, offering a geometric and algebraic framework for understanding extremal black holes in extended supergravity theories.

ABSTRACT

In these lectures we explain the concept of supergravity p-branes and BPS black holes. Introducing an audience of general relativists to all the necessary geometry related with extended supergravity (special geometry, symplectic embeddings and the like) we describe the general properties of N=2 black holes, the structure of central charges in extended supergravity and the description of black hole entropy as an invariant of the U duality group. Then, after explaining the concept and the use of solvable Lie algebras we present the detailed construction of 1/2, 1/4 and 1/8 supersymmetry preserving black holes in the context of N=8 supergravity. The Lectures are meant to be introductory and self contained for non supersymmetry experts but at the same time fully detailed and complete on the subject.

Motivation & Objective

• To introduce general relativists to the geometric structures of extended supergravity, including special geometry and symplectic embeddings.
• To explain the role of central charges in N=2 supergravity and their connection to black hole entropy.
• To demonstrate how black hole entropy arises as an invariant under the U-duality group in extended supergravity.
• To present a systematic construction of 1/2, 1/4, and 1/8 supersymmetric black holes in N=8 supergravity using solvable Lie algebras.
• To provide a fully self-contained, pedagogical framework for non-experts in supersymmetry to understand BPS black hole solutions.

Proposed method

• Utilizes the formalism of special geometry to describe the moduli space of N=2 supergravity and its relation to central charges.
• Applies symplectic embeddings to relate the gauge group structure to the central charge matrix in extended supergravity.
• Employs solvable Lie algebras to systematically construct the BPS black hole solutions in N=8 supergravity.
• Derives the entropy of BPS black holes as a U-duality invariant expression, linking it to the central charge and the attractor mechanism.
• Uses the attractor mechanism to fix moduli at the horizon, ensuring extremality and supersymmetry preservation.
• Constructs explicit solutions for 1/2, 1/4, and 1/8 BPS states by analyzing the nilpotent orbits and representation theory of the U-duality group.

Experimental results

Research questions

• RQ1 How do central charges in N=2 supergravity determine the properties of BPS black holes?
• RQ2 What is the geometric and algebraic structure underlying the U-duality invariant entropy of BPS black holes?
• RQ3 How can solvable Lie algebras be used to classify and construct BPS black hole solutions in N=8 supergravity?
• RQ4 What role does special geometry play in the moduli stabilization of BPS black holes?
• RQ5 How do the fractions of preserved supersymmetry (1/2, 1/4, 1/8) relate to the structure of the U-duality group and the solution's quantum numbers?

Key findings

• The entropy of BPS black holes in N=8 supergravity is shown to be a U-duality invariant, expressed as a quartic form in the central charge.
• The attractor mechanism stabilizes moduli at the black hole horizon, leading to a fixed value of the entropy independent of initial moduli values.
• 1/2, 1/4, and 1/8 BPS black holes in N=8 supergravity are classified via the nilpotent orbits of the U-duality group E7(7), corresponding to different representations.
• The central charge matrix in N=2 supergravity is derived from the holomorphic prepotential and symplectic structure of special geometry.
• The use of solvable Lie algebras allows for a systematic and explicit construction of the BPS black hole solutions in N=8 supergravity.
• The paper establishes a complete correspondence between the supersymmetry fraction and the rank of the nilpotent orbit in the U-duality group.