Black hole microstates in AdS$_4$ from supersymmetric localization (1511.04085v2)

Abstract: This paper addresses a long standing problem, the counting of the microstates of supersymmetric asymptotically AdS black holes in terms of a holographically dual field theory. We focus on a class of asymptotically AdS$_4$ static black holes preserving two real supercharges which are dual to a topologically twisted deformation of the ABJM theory. We evaluate in the large $N$ limit the topologically twisted index of the ABJM theory and we show that it correctly reproduces the entropy of the AdS$_4$ black holes. An extremization of the index with respect to a set of chemical potentials is required. We interpret it as the selection of the exact R-symmetry of the superconformal quantum mechanics describing the horizon of the black hole.

• The paper demonstrates that the extremized topologically twisted index exactly reproduces the Bekenstein-Hawking entropy of supersymmetric AdS4 black holes.
• It utilizes supersymmetric localization and index extremization to map R-symmetry parameters in ABJM theory to supergravity, confirming a deep holographic duality.
• The study’s methodology sets the stage for future research on quantum corrections and extended gauge theories within the AdS/CFT framework.

Analyzing Black Hole Microstates in AdS$_4$ via Supersymmetric Localization

The paper explores a significant challenge in theoretical physics: deciphering the microstates of supersymmetric black holes in asymptotically AdS$_4$ (Anti-de Sitter 4-dimensional) spaces through dual field theories. This work particularly emphasizes black holes compatible with a certain class of supersymmetries, dual to a version of the well-known ABJM (Aharony-Bergman-Jafferis-Maldacena) theory, deformed by topological twists.

The primary strategy involves evaluating the topologically twisted index of the ABJM theory in the large $N$ limit, where $N$ represents the number of M2-branes in the theoretical setup. This index, crucial for understanding the microstates, is computed using advanced mathematical techniques like supersymmetric localization. The paper remarkably shows that this index accurately mirrors the entropy of the black holes under consideration.

Key Numerical Insights and Claims

1. Entropy Calculation: The central finding is that the topologically twisted index's extremum with respect to chemical potentials perfectly matches the Bekenstein-Hawking entropy of supersymmetric black holes in AdS$_4$.
2. Index Extremization: The extremization principle, essential to this work, suggests that precise R-symmetry mixing selects the superconformal symmetry in the lower-dimensional quantum mechanics—a point where the index is evaluated to yield entropy.
3. R-symmetry and Supergravity: The work provides mappings from the field theory's R-symmetry parameters to the supergravity theory's sections at the black hole horizon, signifying a deep dual correspondence.
4. Holographic Interpretation: The paper interprets the interpolation of full spacetime, stretching between UV (ultraviolet) AdS$_4$ regime and IR (infrared) AdS$_2$ horizon geometry, as a dimensional flow in the holographic framework, providing a field theory analog in terms of RG flows in quantum mechanics.

Theoretical and Practical Implications

The theoretical implications are profound, promising new understandings in both string theory and quantum gravity. For practical applications, though currently abstract, these insights lay foundational work that may eventually impact fields requiring deep quantum computational understanding or gravitonic concerns in cosmology.

Furthermore, this theoretical framework suggests a broader principle of "I-extremization," potentially analogous to known extremization principles in other dimensions like $F$-maximization in three dimensions or $c$-extremization in two.

Future Prospects and Research Directions

• Extended Gauge Theories: The framework presents opportunities to explore black hole microstates in different gauge theories beyond ABJM, potentially revealing broader classes of dualities.
• Higher-Derivative Corrections: Future explorations might involve assessing the influence of quantum corrections on entropy beyond the leading large $N$ limit.
• Larger Mathematical Models: This methodology suggests new mathematical models exploring the index properties in complex dimensions, possibly integrative with other disciplines like algebraic geometry.

Overall, this research marks a step forward in understanding AdS/CFT correspondence in four dimensions, connecting macroscopic gravitational properties with microscopic quantum states through advanced field-theoretic calculations.