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Black Holes in 4D N = 4 Super-Yang-Mills Field Theory

Francesco Benini and Paolo Milan
SISSA, via Bonomea 265, 34136 Trieste, Italy
INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy
ICTP, Strada Costiera 11, 34151 Trieste, Italy

     Black-hole solutions to general relativity carry a thermodynamic entropy, discovered by Bekenstein and Hawking to be proportional to the area of the event horizon, at leading order in the semiclassical expansion.
In a theory of quantum gravity, black holes must constitute ensembles of quantum microstates whose large number accounts for the entropy.We study this issue in the context of gravity with a negative cosmological constant.We exploit the most basic example of the holographic description of gravity (AdS=CFT): type IIB string theory on AdS5 ; S5 , equivalent to maximally supersymmetric Yang-Mills theory in four dimensions. We thus resolve a long-standing question: Does the four-dimensional N 4 SU N Super-Yang-Mills theory on S3 at large N contain enough states to account for the entropy of rotating
electrically charged supersymmetric black holes in 5D anti–de Sitter space? Our answer is positive. By reconsidering the large N limit of the superconformal index, using the so-called Bethe-ansatz formulation,
we find an exponentially large contribution which exactly reproduces the Bekenstein-Hawking entropy of the black holes. Besides, the large N limit exhibits a complicated structure, with many competing  exponential contributions and Stokes lines, hinting at new physics. Our method opens the way toward a quantitative study of quantum properties of black holes in anti–de Sitter space.
DOI: 10.1103/PhysRevX.10.021037 Subject Areas: Gravitation, Particles and Fields, String Theory
I. INTRODUCTION AND RESULTS
Black holes are possibly the most simple and featureless classical solutions to Einstein’s theory of gravitation, according to no-hair theorems, but simultaneously the most difficult objects to understand conceptually at the
quantum level. One of the fascinating aspects of black-hole physics is its connection with the laws of thermodynamics.
Of particular importance is the fact that black holes carry a microscopic entropy [1], semiclassically determined in terms of the area of the event horizon. In the search for a theory of quantum gravity, explaining the microscopic origin of black-hole thermodynamics—namely, the multi-
tude of classically indistinguishable quantum states responsible for it—is a challenging but fundamental test.
String theory was proposed to be a theory that embeds gravity in a consistent quantum system, hence it should, in particular, explain the black-hole entropy in terms of a degeneracy of string states. This was beautifully shown to be the case in the seminal paper [2] by Strominger and Vafa, where the Bekenstein-Hawking entropy of a class of supersymmetric asymptotically flat black holes was microscopically reproduced.
In the case of asymptotically anti–de Sitter (AdS) black holes—namely, black holes in a gravitational theory with negative cosmological constant—the AdS=CFT or gauge-gravity duality [3,4] constitutes a natural and wonderful
framework to study their properties at the quantum level. In fact, the duality provides a fully consistent nonperturbative definition of quantum gravity, in terms of a conformal field theory (CFT) living at the boundary of AdS space. The problem of offering a microscopic account of the black-hole
entropy is rephrased into that of counting particular states in the dual CFT. However, despite the very favorable setup, the problem remains challenging, for two reasons. First and most importantly, the dual CFT is strongly coupled, and thus performing computations is daunting. Second, the “phenom-
enologically” interesting regime of weak-curvature gravity corresponds to a largeN limit in the CFT, i.e., a limit inwhich the central charge goes to infinity. Indeed, this problemin four or more dimensions has remained unsolved for many years, and only recently a concrete examplewas successfully studied in Refs. [5,6]. There, the Bekenstein-Hawking entropy of a class of static dyonic Bogomol’nyi-Prasad-Sommerfield
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
PHYSICAL REVIEW X 10, 021037 (2020) Featured in Physics
2160-3308=20=10(2)=021037(21) 021037-1 Published by the American Physical Society