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

Abstract

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 AdS_{5}×S^{5}, 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 S^{3} 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.