Black Holes in 4D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> 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 (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>AdS</a:mi><a:mo>/</a:mo><a:mi>CFT</a:mi></a:mrow></a:math>): type IIB string theory on <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>AdS</c:mi></c:mrow><c:mrow><c:mn>5</c:mn></c:mrow></c:msub><c:mo>×</c:mo><c:msup><c:mi>S</c:mi><c:mn>5</c:mn></c:msup></c:mrow></c:math>, equivalent to maximally supersymmetric Yang-Mills theory in four dimensions. We thus resolve a long-standing question: Does the four-dimensional <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="script">N</e:mi><e:mo>=</e:mo><e:mn>4</e:mn></e:math> <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mrow><h:mi>SU</h:mi></h:mrow><h:mo stretchy="false">(</h:mo><h:mi>N</h:mi><h:mo stretchy="false">)</h:mo></h:math> Super-Yang-Mills theory on <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>S</l:mi><l:mn>3</l:mn></l:msup></l:math> at large <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>N</n:mi></n:math> 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 <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>N</p:mi></p:math> 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 <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>N</r:mi></r:math> 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. Published by the American Physical Society 2020

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