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This Electronic Thesis Or Dissertation Has Been Downloaded from the King’S Research Portal At This electronic thesis or dissertation has been downloaded from the King’s Research Portal at https://kclpure.kcl.ac.uk/portal/ Dynamical supersymmetry enhancement of black hole horizons Kayani, Usman Tabassam Awarding institution: King's College London The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without proper acknowledgement. END USER LICENCE AGREEMENT Unless another licence is stated on the immediately following page this work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ You are free to copy, distribute and transmit the work Under the following conditions: Attribution: You must attribute the work in the manner specified by the author (but not in any way that suggests that they endorse you or your use of the work). Non Commercial: You may not use this work for commercial purposes. No Derivative Works - You may not alter, transform, or build upon this work. Any of these conditions can be waived if you receive permission from the author. Your fair dealings and other rights are in no way affected by the above. Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 24. Apr. 2021 Dynamical supersymmetry enhancement of black hole horizons Usman Kayani A thesis presented for the degree of Doctor of Philosophy Supervised by: Dr. Jan Gutowski Department of Mathematics King's College London, UK November 2018 Abstract This thesis is devoted to the study of dynamical symmetry enhancement of black hole horizons in string theory. In particular, we consider supersymmetric horizons in the low energy limit of string theory known as supergravity and we prove the horizon conjecture for a number of supergravity theories. We first give important examples of symmetry enhancement in D = 4 and the mathematical preliminaries required for the analysis. Type IIA supergravity is the low energy limit of D = 10 IIA string theory, but also the dimensional reduction of D = 11 supergravity which itself the low energy limit of M-theory. We prove that Killing horizons in IIA supergravity with compact spatial sections preserve an even number of supersymmetries. By analyzing the global properties of the Killing spinors, we prove that the near-horizon geometries undergo a supersymmetry enhancement. This follows from a set of generalized Lichnerowicz-type theorems we establish, together with an index theory argument. We also show that the symmetry algebra of horizons with non-trivial fluxes includes an sl(2; R) subalgebra. As an intermediate step in the proof, we also demonstrate new Lichnerowicz type theorems for spin bundle connections whose holonomy is contained in a general linear group. We prove the same result for Roman's Massive IIA supergravity. We also consider the near-horizon geometry of supersymmetric extremal black holes in un-gauged and gauged 5-dimensional supergravity, coupled to abelian vector multiplets. We consider important examples in D = 5 such as the BMPV and supersymmetric black ring solution, and investigate the near-horizon geometry to show the enhancement of the symmetry algebra of the Killing vectors. We repeat a similar analysis as above to prove the horizon conjecture. We also investigate the conditions on the geometry of the spatial horizon section S. i Acknowledgements Firstly, I would like to express my gratitude to Jan Gutowski for his guidance in my research, and for his patience and understanding. I would also like to thank my PhD mentor, Geoffrey Cantor, who has guided me through the trials and tribulations and Jean Alexandre for all his support in helping me cross the finish line. Finally, I would like to thank my family and friends, especially my parents, for all their support over the years and throughout my education and life. ii \Black holes provide theoreticians with an important theoretical laboratory to test ideas. Conditions within a black hole are so extreme, that by analyzing aspects of black holes we see space and time in an exotic environment, one that has shed important, and sometimes perplexing, new light on their fundamental nature." Brian Greene iii Table of Contents Abstract i Acknowledgements ii 1 Introduction 1 1.1 Black holes........................................1 1.1.1 Other solutions: Reisser-Nordstr¨ommetric (1918)..............7 1.1.2 Kerr metric (1963)................................9 1.1.3 The Kerr-Newman geometry (1965)...................... 10 1.1.4 The no-hair theorem............................... 11 1.1.4.1 Higher dimensional black holes.................... 11 1.1.5 The laws of black hole mechanics and entropy................. 12 1.2 Symmetry in physics................................... 14 1.2.1 Noether's theorem................................ 15 1.2.2 Poincar´esymmetry................................ 16 1.3 Quantum gravity..................................... 16 1.3.1 Kaluza-Klein unification and compactification................. 18 1.3.2 Supersymmetry.................................. 19 1.3.3 Super-poincar´esymmetry............................ 20 1.3.4 Superstring theory................................ 21 1.3.4.1 The first superstring revolution.................... 23 1.3.4.2 The second superstring revolution.................. 23 1.4 Supergravity....................................... 25 1.4.1 D = 4;N = 8 supergravity............................ 28 1.4.2 D = 11;N = 1 supergravity........................... 29 1.5 Summary of Research.................................. 31 1.5.1 Plan of Thesis.................................. 35 1.6 Statement of Originality................................. 37 1.7 Publications........................................ 37 2 Killing Horizons and Near-Horizon Geometry 38 2.1 Killing horizons...................................... 38 iv TABLE OF CONTENTS v 2.2 Gaussian null coordinates................................ 40 2.2.1 Extremal horizons................................ 42 2.3 The near-horizon limit.................................. 43 2.3.1 Examples of near-horizon geometries...................... 45 2.3.2 Curvature of the near-horizon geometry.................... 48 2.3.3 The supercovariant derivative.......................... 49 2.4 Field strengths...................................... 50 2.5 The maximum principle................................. 51 2.6 The classical Lichnerowicz theorem........................... 51 3 Supergravity 54 3.1 D = 11 to IIA supergravity............................... 54 3.2 IIA to Roman's massive IIA............................... 58 3.3 D = 11 to D = 5;N = 2 supergravity.......................... 60 3.3.1 Ungauged..................................... 60 3.3.2 Gauged...................................... 64 4 D = 10 IIA Horizons 67 4.1 Horizon fields and KSEs................................. 67 4.1.1 Near-horizon fields................................ 67 4.1.2 Horizon Bianchi identities and field equations................. 68 4.1.3 Integration of KSEs along the lightcone.................... 70 4.1.4 Independent KSEs................................ 72 4.2 Supersymmetry enhancement.............................. 73 4.2.1 Horizon Dirac equations............................. 73 4.2.2 Lichnerowicz type theorems for D(±) ...................... 74 4.2.3 Index theory and supersymmetry enhancement................ 75 4.3 The sl(2; R) symmetry of IIA horizons......................... 76 4.3.1 Construction of η+ from η− Killing spinors.................. 76 4.3.2 Killing vectors.................................. 78 4.3.3 sl(2; R) symmetry of IIA-horizons........................ 79 4.4 The geometry and isometries of S ............................ 80 5 Roman's Massive IIA Supergravity 81 5.1 Horizon fields and KSEs................................. 81 5.1.1 Horizon Bianchi identities and field equations................. 81 5.1.2 Integration of KSEs along lightcone...................... 84 5.1.3 The independent KSEs on S .......................... 86 5.2 Supersymmetry enhancement.............................. 87 5.2.1 Horizon Dirac equations............................. 87 5.2.2 Lichnerowicz type theorems for D (±) ..................... 87 5.2.3 Index theory and supersymmetry enhancement................ 88 TABLE OF CONTENTS vi 5.3 The sl(2; R) symmetry of massive IIA horizons.................... 89 5.3.1 η+ from η− Killing spinors........................... 89 5.3.2 Killing vectors.................................. 90 5.3.3 sl(2; R) symmetry................................ 90 5.4 The geometry and isometries of S ............................ 91 6 D = 5 Supergravity Coupled to Vector Multiplets 92 6.1 Near-horizon geometry of the BMPV black holes and black rings.......... 93 6.2 Horizon fields and KSEs................................. 99 6.2.1 Near-horizon fields................................ 99 6.2.2 Horizon Bianchi indentities and field equations................ 100 6.2.3 Gauge field decomposition............................ 101 6.2.4 Integration of the KSEs along the lightcone.................. 103 6.2.5 The independent KSEs on S .......................... 105 6.3 Supersymmetry enhancement.............................. 106 6.3.1 Horizon Dirac equation............................. 106 6.3.2 Lichnerowicz type theorems for D (±) ...................... 107 6.3.3 Index theory and supersymmetry enhancement...............
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