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Quantum black hole entropy from 4d supersymmetric Cardy formula

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

^*mh974ATdamtp.cam.ac.uk

Phys. Rev. D 100, 026008 – Published 17 July, 2019

DOI: https://doi.org/10.1103/PhysRevD.100.026008

Abstract

We study the canonical $AdS / CFT$ correspondence between 4d $S U (N)$ $N = 4$ super Yang-Mills theory (SYM) and type IIB superstring theory on ${AdS}_{5} \times S^{5}$ . We analyze the supersymmetric index of the $N = 4$ SYM on $S^{1} \times M_{3}$ which counts supersymmetric states with fixed quantum numbers. We compute an asymptotic behavior of the index in the limit of shrinking $S^{1}$ for any $N$ by a refinement of the 4d supersymmetric Cardy formula. The asymptotic behavior for the superconformal index case ( $M_{3} = S^{3}$ ) at large $N$ agrees with the Bekenstein-Hawking entropy of a rotating electrically charged Bogomolnyi-Prasad-Sommerfeld (BPS) black hole in ${AdS}_{5}$ via a Legendre transformation as recently shown in the literature. We also find that the agreement formally persists for finite $N$ if we slightly modify the $AdS / CFT$ dictionary between the Newton constant and $N$ . This implies the existence of a nonrenormalization property of the black hole entropy against quantum corrections. We also study the cases with other gauge groups and additional matter and the orbifold $N = 4$ SYM. It turns out that the entropies of all the CFT examples in this paper are universally given by $2 π \sqrt{Q_{1} Q_{2} + Q_{1} Q_{3} + Q_{2} Q_{3} - 2 c (J_{1} + J_{2})}$ with charges $Q_{1, 2, 3}$ , angular momenta $J_{1, 2}$ , and central charge $c$ . The results for other $M_{3}$ make predictions to the gravity side.

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Phys. Rev. D 100, 026008– Published 17 July, 2019

Vol. 100, Iss. 2 — 15 July 2019

Received 11 February 2019
Revised 2 April 2019

DOI: https://doi.org/10.1103/PhysRevD.100.026008

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. Funded by SCOAP³.

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Quantum black hole entropy from 4d supersymmetric Cardy formula

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