City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 6-2016 New Classical Solutions in Supergravity Zhibai Zhang Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1298 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] New Classical Solutions in Supergravity by Zhibai Zhang A dissertation submitted to the Graduate Faculty in Physics in partial ful- fillment of the requirements for the degree of Doctor of Philosophy, The City University of New York. 2016 ii This manuscript has been read and accepted for the Graduate Faculty in Physics in satisfaction of the dissertation requirements for the degree of Doc- tor of Philosophy. (required signature) Date Chair of Examining Committee (required signature) Date Executive Officer Andrea Ferroglia Daniel Kabat James Liu Giovanni Ossola Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iii Abstract New Classical Solutions in Supergravity by Zhibai Zhang Advisor: Justin V´azquez-Poritz In this Ph.D. thesis we construct three classes of new solutions to su- pergravity theories in various dimensions and study their properties. The first class is reduction ansatz of 10D and 11D supergravity on Ricci-flat and noncompact manifolds. These reductions are from a scaling limit of the fa- mous spherical reductions, and can be solely supported by warp factors. The second class contains a large number of String/M theory solutions that have Lifshitz or Schr¨odinger scaling symmetry, obtained from marginally deform- ing the geometry of internal dimensions of previous solutions. We propose that these new solutions are dual to marginal deformations of certain holo- graphic non-relativistic field theories. The last class are singly spinning non- spherical black holes named black rings in 5D U(1)3 supergravity with three dipole charges and three electric charges, which lie in the classification of iv 5D nonsupersymmetric black holes. We analyze their thermodynamic and global properties. As a byproduct, we embed the three dipole black ring into spacetime that contains background magnetic fields. Acknowledgments First I would like to thank my Ph.D. advisor Justin V´azquez-Poritz for his guidance and support, without which this thesis could not have been com- pleted. I would also like to thank my undergraduate advisor Bo Feng, and Hong Lu for their help which enabled me to study in the U.S. and the projects I have done with them. I am grateful to Andrea Ferroglia and Giovanni Os- sola for our collaborations on particle phenomenology and introducing me to scientific computing. Many thanks to Roman Kezerashvili and all the faculty members for welcoming me to the Physics Department at New York City College of Technology, where I have enjoyed teaching during my Ph.D. study. I would like to thank Daniel Kabat and James Liu for being in my Ph.D. committee and their valuable comments on this thesis. I have also benefited a lot from my interactions and collaborations with Philip Argyres, Mboyo Esole, Eoin O´ Colg´ain,Chris Pope and S. T. Yau. Thanks to Mirjam Cvetic, v vi Martin Kruczenski, David Kutasov, David Kastor, Sebastian Franco and many others for giving me the opportunities of giving talks at various places. The five years at the Graduate Center, CUNY have been wonderful, thanks to the faculty and the Ph.D. students in the program for all the in- teresting courses, seminars and discussion groups. I would also like to thank my friends outside CUNY: Zhigang Bao, Zhihao Fu, Zhen Liu, Junqi Wang and Yihong Wang for chatting mathematics on late night phone calls and deriving physics in coffee stores. Last but not least, I am indebted to my family for their support and unconditional love. I am grateful to my wife Ashley for not complaining too much when I do physics at all ungodly hours. I would like to express my deepest gratitude to my parents for their inspiration and helping their son discover one of the most intriguing and exciting paths one can have in life. The bulk of this thesis contains three of my papers [17, 70, 111], which were previously published on Physical Review D and Journal of High Energy Physics. Contents 1 Introduction 1 1.1 General Relativity . .1 1.2 String Theory and Supergravity . .4 1.3 Classical solutions in GR and SUGRA . .6 1.4 AdS/CFT correspondence . .8 1.5 Summary . 10 2 Ricci-flat reductions and Holography 12 2.1 Introduction . 12 2.2 Ricci-flat solutions . 17 2.3 KK Reduction on the solutions . 20 2.3.1 KK Reduction on the R6 ................. 21 2.3.2 Origin of the KK reduction . 25 2.3.3 KK reductions on R4 ................... 28 2.3.4 KK reductions on R3 ................... 30 vii CONTENTS viii 2.4 Stability of AdS vacua . 31 6 2.4.1 D = 11; p = 5; ΣD−p = R ................ 34 4 2.4.2 D = 10; p = 6; ΣD−p = R ................ 35 3 2.4.3 D = 11; p = 8; ΣD−p = R ................ 37 2.5 de Sitter vacua . 40 2.6 Conclusion . 46 3 Lifshitz and Schr¨odingertype solutions 51 3.1 Introduction . 51 3.2 Marginal deformations of (0; 2) Landau-Ginsburg theory . 54 3.3 Marginal deformations of theories with Schr¨odingersymmetry 59 3.3.1 An example with a five-sphere . 59 3.3.2 Countably-infinite examples with the Lp;q;r spaces . 63 3.4 Marginal deformations of Lifshitz vacua . 68 3.4.1 Lifshitz-Chern-Simons gauge theories . 68 3.4.2 Countably-infinite Lifshitz vacua with dynamical ex- ponent z =2 ....................... 71 3.4.3 An example with general dynamical exponent . 77 3.5 Conclusions . 82 4 Tri-dipole black rings in supergravity 84 CONTENTS ix 4.1 Introduction . 84 4.2 Black rings in five-dimensional U(1)3 supergravity . 88 4.2.1 From C-metrics to dipole black rings . 88 4.2.2 Adding electric charges . 92 4.2.3 Adding background magnetic fields . 94 4.3 Global analysis of Ricci-flat solutions . 96 4.3.1 D = 5 Ricci-flat metric . 96 4.3.2 D = 6 Ricci-flat metric . 100 4.4 Global properties and thermodynamics of black rings . 104 4.4.1 Black rings with triple dipole charges . 104 4.4.2 Electrically-charged black rings and naked CTC's . 113 4.4.3 In background magnetic fields . 115 4.5 Conclusions . 117 Appendices 120 A. Dimensional Reduction . 121 B. Solution generating techniques . 123 Bibliography 127 List of Tables 1.1 Some prototypical examples of the AdS/CFT correspondence . 10 x List of Figures 2.1 The mass-squared eigenvalues for scalar fluctuations around 3 2 the geometry AdS3× T × H .................. 39 2.2 A sample plot of the potential function in the lower dimen- sional theory . 43 xi Chapter 1 Introduction 1.1 General Relativity It has been an entire century since Einstein published the theory of Gen- eral Relativity (GR) [1]. As a successor theory to Special Relativity (SR), which only concerns inertial reference frames, GR extends the scope of SR and considers covariant physics in generic reference frames, especially non- inertial reference frames. This is achieved by incorporating the Equivalence Principle which views acceleration in non-inertial frames equivalent as grav- itational forces. Therefore GR is essentially a covariant theory of gravity. It further revolutionizes our view of gravity by relating the gravitational field to the metric of spacetime which is a pseudo-Riemannian manifold. In 4-dimensions, if only the gravity field is considered then the Lagrangian den- sity is the simplest covariant function of the metric: the scalar curvature, 1 CHAPTER 1. INTRODUCTION 2 and the action is 1 Z p S = d4x −gR; (1.1) 16πG4 where G4 is the 4-dimensional Newton constant. Many other terms can be added covariantly for different scenarios such as a cosmological constant, electromagnetic field, scalar fields and so on. Since its birth, GR has passed many nontrivial experimental tests includ- ing the newly announced direct observation of gravitational waves [2] and is well accepted as the modern theory of gravity. However, fundamental prob- lems also manifest themselves along with the successes. A lot of solutions found in GR suffer from severe pathologies, such as singularities. Among them the most crucial ones are those at which some curvature invariant be- comes infinite, and thus can not be resolved by any classical techniques. For instance, this happens inside of the horizon of a black hole. And it was proved that at general circumstances these singularities are unavoidable [3]. The existence and definiteness of these singularities demonstrates that GR as a theory of gravity is incomplete. Nevertheless this is compatible with the fact that GR is a classical theory. Modern physics has shown that once we focus on physics of small regions the quantum effects will not be negligible and a full quantum theory has to be used. In the case of singularities, they CHAPTER 1. INTRODUCTION 3 happen precisely when the gravity field and matter are compressed into a small region and involves an extremely small length scale. In this sense GR functions as well as Maxwell theory in which Coulomb's law is a solution (in Coulomb's gauge); it has a singularity at r = 0 and this is resolved by the quantum theory i.e. Quantum Electrodynamics (QED). It is reasonable to expect that a quantum theory of gravity would resolve these singularities in GR, since they appear as extremely short scale (the Planck scale) phenomena.