Scuola Internazionale Superiore di Studi Avanzati - Trieste Doctoral Thesis A Tale of Two Indices or How to Count Black Hole Microstates in AdS/CFT Candidate : Paolo Milan Supervisor : Prof. Francesco Benini Academic Year 2018 – 2019 SISSA - Via Bonomea 265 - 34136 TRIESTE - ITALY Abstract In this thesis we investigate on the microscopic nature of the entropy of black holes in string theory in the context of AdS/CFT correspondence. We focus our attention on two particularly interesting cases in different dimensions. First, we consider static dyonic BPS black holes in AdS4 in 4d N = 2 gauged supergravities with vector and hyper 6 multiplets. More precisely, we focus on the example of BPS black holes in AdS4 × S in massive Type IIA, whose dual three-dimensional holographic description is known and simple. To provide a microscopic counting of the black hole entropy we employ a topologically twisted index in the dual field theory, which can be computed exactly and non-perturbatively with localization techniques. We find perfect match at leading order. 5 Second, we turn to rotating electrically-charged BPS black holes in AdS5 × S in Type IIB. Here, the microscopic entropy counting is done in terms of the superconformal index of the dual N = 4 Super-Yang-Mills theory. We proceed by deriving a new formula, which expresses the index as a finite sum over the solution set of certain transcendental equations, that we dub Bethe Ansatz Equations, of a function evaluated at those solutions. Then, using the latter formula, we reconsider the large N limit of the superconformal index. We find an exponentially large contribution which exactly reproduces the Bekenstein- Hawking entropy of the black holes of Gutowski-Reall. Besides, the large N limit exhibits a complicated structure, with many competing exponential contributions and Stokes lines, hinting at new physics. iii iv Acknowledgements Completing this thesis represents the end of a four years long path in which I learned so much and lived many experiences that made me the person I am now. Here is the place where I can thank all the people that have been crucial in the completion of this journey. First, I would like to thank my supervisor Francesco. His patience and attention greatly helped me grow as a physicist, especially through some difficult times that hap- pened here and there. Needless to say, I will always be indebted to him for his dedication, for the effort he put in the works we have completed together and for sharing his vast knowledge with me. Next, I would like to thank the many excellent people I have had the opportunity to discuss with and to learn from: Arash Arabi Ardehali, Giulio Bonelli, Atish Dabholkar, Nima Doroud, Francesca Ferrari, Seyed Morteza Hosseini, Shiraz Minwalla, Kyriakos Papadodimas, Arnab Rudra and Alberto Zaffaroni. During these wonderful four years I met many people whom I want to thank for the friendship and support. A special mention goes to Luca Arceci, Francesco Ferrari, Juraj Hašik and Jacopo Marcheselli, for all the experiences we have lived together and for being there in the good as well as difficult moments. I would also like to thank Ric- cardo Bergamin, Lorenzo Bordin, Marco Celoria, Alessandro Davoli, Fabrizio Del Monte, Harini Desiraju, Daniele Dimonte, Laura Fanfarillo, Matteo Gallone, Marco Gorghetto (especially for the rants and all the Gorghettism moments), Maria Laura Inzerillo, J, Diksha Jain, Hrachya Khachatryan, Francesco Mancin, Shani Meynet, Giulia Panat- toni, Andrea Papale, Mihael Petac, Michele Petrini, Matteo Poggi, Chiara Previato, Giulio Ruzza, Paolo Spezzati, Martina Teruzzi, Arsenii Titov, Maria Francesca Trom- bini, Lorenzo Ubaldi, Elena Venturini and Ludovica Zaccagnini. Last but not least, my most heartfelt gratitude goes to my family—Carla, Renato, Enrico, Andrea, Francesca, Elena, Gabriele, Alessandro, Matteo, Davide and Marta— for their love and unconditional support. I would not have been able to achieve this accomplishment without them. v vi Preface This thesis is based on the following publications and preprints, listed in chronological order: [1] F. Benini, H. Khachatryan, and P. Milan, “Black hole entropy in massive Type IIA,” Class. Quant. Grav. 35 no. 3, (2018) 035004, arXiv:1707.06886 [hep-th] [2] F. Benini and P. Milan, “A Bethe Ansatz type formula for the superconformal in- dex,” arXiv:1811.04107 [hep-th]. Submitted to Communications in Mathemat- ical Physics. [3] F. Benini and P. Milan, “Black holes in 4d N = 4 Super-Yang-Mills,” arXiv:1812.09613 [hep-th] vii viii Contents Abstract iii Acknowledgements v Preface vii Introduction and summary 1 I The topologically twisted index and AdS4 black holes 9 1 Black hole entropy in massive Type IIA 11 1.1 Dyonic black holes in massive Type IIA . 11 1.1.1 Black hole horizons . 14 1.1.2 The Bekenstein-Hawking entropy . 18 1.2 Microscopic counting in field theory . 19 1.2.1 The topologically twisted index . 24 1.2.2 The large N limit . 26 1.3 Entropy matching through attractor equations . 30 1.4 Conclusions . 32 II The superconformal index and AdS5 black holes 33 2 A Bethe Ansatz formula for the superconformal index 35 2.1 The 4d superconformal index . 35 2.2 A new Bethe Ansatz formula . 38 ix 2.2.1 Continuation to generic fugacities . 41 2.2.2 Proof of the formula . 42 2.3 Conclusions . 51 3 Black holes in 4d N = 4 Super-Yang-Mills 53 3.1 BPS black holes in AdS5 ............................ 53 3.2 The dual field theory and its index . 55 3.2.1 Exact solutions to the BAEs . 59 3.3 The large N limit . 61 3.3.1 Contribution of the basic solution . 62 3.3.2 Contribution of SL(2; Z)-transformed solutions . 68 3.3.3 Final result and Stokes lines . 69 3.3.4 Comparison with previous literature . 71 3.4 Statistical interpretation and I-extremization . 72 3.5 Black hole entropy from the index . 75 3.5.1 Example: equal charges and angular momenta . 79 3.6 Conclusions . 83 Appendices 85 A Special functions 87 B Numerical semigroups and the Frobenius problem 91 C A dense set 95 D Weyl group fixed points 97 D.1 The rank-one case . 97 D.2 The higher-rank case . 101 E Real fugacities 107 E.1 The case 0 < q < 1 ............................... 107 x E.2 The case −1 < q < 0 .............................. 112 Bibliography 117 xi xii Introduction and summary One of the fascinating aspects of the physics of black holes is its connection with the laws of thermodynamics. Of particular importance is the fact that black holes carry a macroscopic entropy [4–8], semi-classically determined in terms of the horizon area by the famous Bekenstein-Hawking formula: 3 kB c Area SBH = ; (1) ~ GN 4 where c is the speed of light, kB is the Boltzmann constant, ~ is the reduced Planck constant and GN is the Newton constant (to simplify the notation, throughout the rest of this thesis we will work in natural units and set c = kB = ~ = 1). In this formula, the 1 4 proportionality factor is very important and it has been very precisely determined by Hawking [8], using arguments involving the black hole thermal radiation—the Hawking radiation. In the search for a theory of quantum gravity, explaining the microscopic origin of black hole thermodynamics is a fundamental but challenging test. As string theory is proposed to embed gravity in a consistent quantum system, it should in particular provide a microscopic description of (1) in terms of a degeneracy of string states. Evidences that such expectation is correct have been put forward by appealing to the nature of black holes arising in string theory. The latter can indeed be formed from systems of branes [9] which, in turn, admit a description in terms of a worldvolume gauge theory [10–15]. This provides a powerful alternative point of view, besides the gravitational description, much more amenable to a quantum treatment. This framework was first used by Strominger and Vafa [16] to show that, within string theory, one can give a microscopic statistical interpretation to the thermodynamic Bekenstein-Hawking entropy (1) of BPS black holes in flat space. Many different setups have been analyzed since then, including quantum corrections, to an impressive precision [17–21] (see e.g. [22] for more references). In essentially all examples, the microscopic counting is performed in a 2d conformal field theory (CFT) that appears—close to the black hole horizon—as one plays with the moduli available in string theory (and takes advantage of various dualities). In fact, also the entropy of BPS black holes in AdS3 is well understood, since the microstate counting can be performed in 1 A Tale of Two Indices the 2d CFT related to AdS3 by the AdS/CFT correspondence [23–25] (see e.g. [26,27] for reviews and further references). It is important to stress here that many of the derivations of the black hole entropy in these contexts involve the use of the Cardy formula [28], which gives the asymptotic growth of CFT states for large charges. In the case of asymptotically-AdS black holes in dimension d ≥ 4 the situation is different, because in general there is no regime in which the black holes are described by a 2d CFT. On the other hand, the AdS/CFT duality constitutes a natural and wonderful framework to study their properties at the quantum level. The duality provides a non- perturbative definition of quantum gravity, in terms of a CFT living at the boundary of AdS space. Therefore, one would expect the black holes to appear as ensembles of states with exponentially large degeneracy in the dual field theory.1 The problem of offering a microscopic account of the black hole entropy is then rephrased into that of counting particular CFT states.