Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS$_{\bf 5}$ black holes (1810.11442v4)

Abstract: We present a holographic derivation of the entropy of supersymmetric asymptotically AdS$_5$ black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject to a constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual $\mathcal{N}=1$ SCFT on a twisted $S^1\times S^3$ with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large $N$ exactly reproduces the Bekenstein-Hawking entropy of the black hole.

• The paper presents a holographic derivation linking the Legendre transform of the on-shell gravitational action to the microscopic entropy of AdS5 black holes.
• It employs supersymmetric localization in the dual SCFT on S¹×S³ to compute an exact partition function that scales as O(N²).
• The study bridges macroscopic gravitational entropy with microscopic state degeneracies, offering new insights into quantum gravity via AdS/CFT correspondence.

Essay on "Microscopic Origin of the Bekenstein-Hawking Entropy of Supersymmetric AdS Black Holes"

The paper "Microscopic Origin of the Bekenstein-Hawking Entropy of Supersymmetric AdS Black Holes" explores the fundamental question of understanding the statistical origin of the entropy associated with supersymmetric asymptotically Anti-de Sitter (AdS) black holes. This work is grounded in the framework of string theory and utilizes the renowned AdS/CFT correspondence.

Overview of Contributions

The authors present a holographic approach to derive the entropy of these black holes, focusing on a supersymmetric subclass by imposing a novel BPS limit on black hole thermodynamics. The black hole entropy is shown to be the Legendre transform of the on-shell gravitational action when computed with respect to three chemical potentials under a specific constraint. This constraint emerges naturally from the requirement of supersymmetry and regularity in the Euclidean section of the bulk geometry. By performing a localization computation, the paper calculates the exact partition function for the dual $\mathcal{N}=1$ superconformal field theory (SCFT) living on a background $S^1 \times S^3$, which satisfies identical constraints on its chemical potentials, leading to the characterization of a generalized supersymmetric Casimir energy. The paper successfully demonstrates that the Legendre transform of this field-theoretic result at large $N$ precisely yields the classical Bekenstein-Hawking entropy observed in the black hole.

Computational Techniques and Results

The paper methodically applies localization techniques in supersymmetric quantum field theory to compute the partition function of the dual SCFT, a significant advancement allowing direct comparison with gravitational computations. The analysis leads to a formulation of the supersymmetric Casimir energy, which scales as $O(N^2)$ and closely matches the physical characteristics of the black hole, linking field theory calculations to the entropy of black holes within this supersymmetric regime.

Bold Claims and Implications

The paper makes a significant assertion regarding the role of the supersymmetric Casimir energy in large $N$ limits. It implies that this energy encapsulates not only the vacuum energy but also the degeneracies of states responsible for the gravitational entropy, thereby elucidating a microscopically understood quantity in terms of macroscopic black hole properties. Furthermore, this paper provides a clearer understanding of the relations underlying the extreme opposite limits—supersymmetry and extremality—in black hole solution space.

Future Directions

The findings in this research open several pathways for future exploration. The authors indicate potential in extending their results to four-dimensional SCFTs and other black hole families, including those not preserving supersymmetry. Additionally, the resemblance of the entropy extremization with attractor mechanisms could be further studied in diverse string theory setups, thereby cultivating deeper insights into the microstate geometries of black holes in AdS spaces and potentially identifying new invariants or indices pertinent to quantum gravity.

Overall, this work significantly contributes to the understanding of supersymmetric AdS black holes within the string theory context by providing a robust field-theoretic description of their entropy, placing longstanding theoretical conjectures on a concrete computational foundation. The methodologies and results presented have profound implications for both theoretical physics and mathematical physics, encouraging further endeavors into the holographic nature of gravity and its quantum underpinnings.