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Non-Supersymmetric Black Holes: a Step Toward Quantum Gravity

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Non-Supersymmetric Black Holes: a Step Toward Quantum Gravity Non-Supersymmetric Black Holes: A Step Toward Quantum Gravity by Yangwenxiao Zeng A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2020 Doctoral Committee: Professor Finn Larsen, Chair Professor Henriette Elvang Professor James Liu Professor Jon Miller Assistant Professor Liuyan Zhao Yangwenxiao Zeng [email protected] ORCID: 0000-0002-5195-0458 © Yangwenxiao Zeng 2020 All Rights Reserved To my grandpa and grandma. ii ACKNOWLEDGEMENTS I would like to thank my advisor, Prof. Finn Larsen, for all his thoughtful guidance, support and understanding throughout my graduate career. This work would have never been possible without his help and I extremely appreciate it. I also want to thank my collaborators Prof. Alejandra Castro, Victor Godet and Jun Nian for fruitful discussions and collaborations. I want to thank Prof. Henriette Elvang, Prof. James Liu, Callum Jones, Brian McPeak for great discussions and companionship. Ad- ditional thanks to Prof. Jon Miller and Prof. Liuyan Zhao, for being part of my dissertation committee. I want to thank all the members of the High Energy Theory Group at the University of Michigan, for providing me a friendly and pleasant environment to live and study for the past five years. I would also like to thank all of my friends whose company made this journey warm and diversified. Special thanks to Jia Li, Shang Zhang and Zhi Zheng. I would like to thank my family. The past few years have been a hard time for all of us, but I know I would never be who I am today without the love I received. Finally, I would like to thank my love, Xiaoyan Shen, for bringing me sunshine and making me happy when skies are grey. iii TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::::::: iii LIST OF TABLES :::::::::::::::::::::::::::::::::::: vii ABSTRACT ::::::::::::::::::::::::::::::::::::::: viii CHAPTER I. Introduction ..................................1 1.1 Black Holes . .1 1.1.1 Black Hole Thermodynamics . .2 1.1.2 Macroscopic and Microscopic Descriptions of Black Holes .3 1.2 Supersymmetry and Supergravity . .4 1.2.1 BPS and nonBPS Black Holes . .5 1.3 The AdS/CFT Correspondence . .6 1.4 Overview of Results . .7 II. Logarithmic Corrections to Black Hole Entropy: the NonBPS Branch .............................9 2.1 Introduction and Summary . .9 2.2 The Kaluza-Klein Black Hole . 13 2.3 The KK Black Hole in N = 8 SUGRA . 16 2.3.1 N = 8 Supergravity in Four Dimensions . 16 2.3.2 The Embedding into N = 8 SUGRA . 19 2.4 Quadratic Fluctuations in N = 8 SUGRA . 20 2.4.1 Global Symmetry of Fluctuations . 20 2.4.2 The Decoupled Fluctuations . 23 2.4.3 Summary of Quadratic Fluctuations . 26 2.5 Consistent Truncations of N = 8 SUGRA . 27 2.5.1 The N = 6 Truncation . 27 2.5.2 The N = 4 Truncation . 29 iv 2.5.3 The N = 2 Truncation . 32 2.5.4 More Comments on Consistent Truncations . 33 2.6 The General KK Black Hole in N = 2 SUGRA . 34 2.6.1 N = 2 SUGRA with Cubic Prepotential . 34 2.6.2 The Embedding into N = 2 SUGRA . 36 2.6.3 Decoupled Fluctuations: General Case . 37 2.6.4 Decoupled Fluctuations: Constant Dilaton . 39 2.7 Logarithmic Corrections to Black Hole Entropy . 43 2.7.1 General Framework: Heat Kernel Expansion . 44 2.7.2 Local Contributions . 46 2.7.3 Quantum Corrections to Black Hole Entropy . 48 2.8 Discussion . 50 III. Black Hole Spectroscopy and AdS2 Holography ............ 53 3.1 Introduction and Summary . 53 3.2 Black Holes and Their Fluctuations . 57 2 3.2.1 The AdS2 × S Backgrounds in N = 8 Supergravity . 57 3.2.2 Adaptation to N = 4 Supergravity . 59 3.2.3 Structure of Fluctuations . 60 3.3 Mass Spectrum . 61 2 3.3.1 Partial Wave Expansion on S and Dualization of AdS2 Vec- tors . 61 3.3.2 Bulk Modes of the Scalar Block . 63 3.3.3 Bulk Modes of the Vector Block . 64 3.3.4 Bulk Modes of the KK Block . 66 3.3.5 Bulk Modes of the Gaugino Block . 71 3.3.6 Bulk Modes of Gravitino Block . 73 3.3.7 Boundary Modes . 77 3.3.8 Summary of the Mass Spectrum . 80 3.4 Heat Kernels . 82 3.4.1 Heat Kernel Preliminaries . 82 3.4.2 Bulk Modes . 83 3.4.3 Boundary Modes . 86 3.4.4 Zero Mode Corrections . 88 3.4.5 Summary of Anomaly Coefficients . 89 3.5 Compactifications with an AdS3 Factor . 90 2 3.5.1 String Theory on AdS3 × S × M .............. 90 3.5.2 nNull Reduction: Thermodynamics . 91 3.5.3 nNull Reduction: Kinematics . 92 2 2 3.5.4 Explicit Comparison Between AdS3 × S and AdS2 × S . 94 3.6 Global Supersymmetry . 98 3.6.1 Global Supercharges: the BPS Branch of N=8 Theory . 98 3.6.2 Global Supercharges: the nonBPS Branch of N=8 Theory . 99 3.6.3 Global Supercharges in AdS3 ................. 100 v 3.6.4 Global Supersymmetry in the N = 4 Theory: the nonBPS Branch . 101 IV. AdS5 Black Hole Entropy near the BPS Limit ............. 103 4.1 Introduction and Summary . 103 4.2 Black Hole Thermodynamics . 107 4.2.1 General AdS5 Black Holes . 107 4.2.2 BPS AdS5 Black Holes . 109 4.2.3 Near-Extremal Limit . 112 4.2.4 Extremal NearBPS Limit . 115 4.2.5 General NearBPS Limit . 119 4.3 The BPS Partition Function . 122 4.3.1 The Partition Function . 122 4.3.2 Single Particle Enumeration . 124 4.3.3 Multiparticle Enumeration . 127 4.3.4 The N = 4 SYM Perturbative Matrix Model . 127 4.3.5 The BPS Limit . 128 4.3.6 Discussion . 130 4.4 Black Hole Statistical Physics . 130 4.4.1 Studying nearBPS using BPS data: Introduction . 130 4.4.2 Entropy Extremization for BPS Black Holes . 132 4.4.3 NearBPS Microscopics . 134 4.4.4 Extremizing Free Energy near BPS . 137 4.5 Summary and Outlook . 142 APPENDICES :::::::::::::::::::::::::::::::::::::: 144 BIBLIOGRAPHY :::::::::::::::::::::::::::::::::::: 153 vi LIST OF TABLES Table 2.1 Decomposition of quadratic fluctuations. 11 2.2 The USp(8) representation content of the quadratic fluctuations. 23 2.3 Decoupled quadratic fluctuations in N = 8 supergravity around the KK black hole. 26 2.4 The degeneracy of multiplets in the spectrum of quadratic fluctuations around the KK black hole embedded in various theories. For N = 4, the integer n is the number of N = 4 matter multiplets. For N = 2, the integer nV refers to the ST (nV − 1) model. 27 2.5 Decoupled quadratic fluctuations in N = 2 SUGRA around a general KK black hole. 39 2.6 Decoupled fluctuations in N = 2 SUGRA around the KK black hole with constant dilaton. The decoupling in the bosonic sector holds for an arbitrary prepotential. The fermionic sector has been further decoupled by specializing to the ST (n) model. 43 2.7 Contributions to a4(x) decomposed in the multiplets that are natural to the KK black hole. 47 2.8 The degeneracy of multiplets in the spectrum of quadratic fluctuations around the KK black hole embedded in to various theories, and their respective val- ues of the c and a coefficients defined in (2.125). For N = 4, the integer n is the number of N = 4 matter multiplets. For N = 2, the recorded values of c and a for the gravitino and the gaugino blocks were only established for ST (nV − 1) models. 47 3.1 Decoupled quadratic fluctuations around the KK black hole in N = 8 and N = 4 supergravity. The columns # denote the multiplicity of the blocks. 61 3.2 Mass spectrum of bulk modes in nonBPS blocks. The label k = 0; 1;::: .. 80 3.3 Mass spectrum of boundary modes in nonBPS blocks. (Multiplicity −1 denotes contribution from ghosts.) . 82 3.4 Anomaly Coefficients of the NonBPS Blocks. 89 2 3.5 The spectrum of chiral primaries on AdS3 × S × M. The label k = 0; 1;::: 91 3.6 Bulk spectrum of BPS solutions. The integral label k ≥ 0. In each line the first entry is the chiral primary and the remaining entries reflect the structure (3.147) of a short multiplet. 95 vii ABSTRACT One of the greatest challenges of modern theoretical physics is to find a quantum theory of gravity that could unify general relativity and quantum mechanics. The microscopic understanding of black holes plays an essential role in pursuing this goal for its unique connection between gravity and quantum effects. Despite the remarkable progress that has been made towards this direction, most develop- ments rely heavily on supersymmetry and elude more realistic non-supersymmetric (nonBPS) black holes. This thesis is to improve this situation and develop both the macroscopic and microscopic sides of nonBPS black holes. In the first part of this thesis, we study a special class of nonBPS black hole solutions in supergravity theories and compute logarithmic corrections to black hole entropy. We find that the correction is universal and independent of black hole parameters only when the number of supercharges N ≥ 6. In the second part, we compute the spectrum of extremal nonBPS black holes introduced in the previous part by studying supergravity on their near-horizon geometry. We find that the spectrum exhibits significant simplifications even though supersymmetry is completely broken and we interpret our results in the framework of the AdS/CFT correspondence. In the final part, we analyze AdS5 black holes that are nearly supersymmetric. They depart from the BPS limit either by having nonzero temperature or by violating a constraint on potentials.
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