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Solutions and Democratic Formulation of Supergravity Theories in Four Dimensions

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Solutions and Democratic Formulation of Supergravity Theories in Four Dimensions Universidad Consejo Superior Aut´onoma de Madrid de Investigaciones Cient´ıficas Facultad de Ciencias Instituto de F´ısica Te´orica Departamento de F´ısica Te´orica IFT–UAM/CSIC Solutions and Democratic Formulation of Supergravity Theories in Four Dimensions Mechthild Hubscher¨ Madrid, May 2009 Copyright c 2009 Mechthild Hubscher.¨ Universidad Consejo Superior Aut´onoma de Madrid de Investigaciones Cient´ıficas Facultad de Ciencias Instituto de F´ısica Te´orica Departamento de F´ısica Te´orica IFT–UAM/CSIC Soluciones y Formulaci´on Democr´atica de Teor´ıas de Supergravedad en Cuatro Dimensiones Memoria de Tesis Doctoral realizada por Da Mechthild Hubscher,¨ presentada ante el Departamento de F´ısica Te´orica de la Universidad Aut´onoma de Madrid para la obtenci´on del t´ıtulo de Doctora en Ciencias. Tesis Doctoral dirigida por Dr. D. Tom´as Ort´ın Miguel, Investigador Cient´ıfico del Consejo Superior de Investigaciones Cient´ıficas, y Dr. D. Patrick Meessen, Investigador Contratado por el Consejo Superior de Investigaciones Cient´ıficas. Madrid, Mayo de 2009 Contents 1 Introduction 1 1.1 Supersymmetry, Supergravity and Superstring Theory . ....... 1 1.2 Gauged Supergravity and the p-formhierarchy . 10 1.3 Supersymmetric configurations and solutions of Supergravity ..... 18 1.4 Outlineofthisthesis............................ 23 2 Ungauged N =1, 2 Supergravity in four dimensions 25 2.1 Ungauged matter coupled N =1Supergravity. 25 2.1.1 Perturbative symmetries of the ungauged theory . ... 27 2.1.2 Non-perturbative symmetries of the ungauged theory . ..... 33 2.2 Ungauged matter coupled N =2Supergravity. 35 2.2.1 N = 2, d =4SupergravityfromStringTheory . 42 3 Gauging Supergravity and the four-dimensional tensor hierarchy 49 3.1 Theembeddingtensorformalism . 52 3.2 The D =4tensorhierarchy ........................ 54 3.2.1 Thesetup.............................. 54 3.2.2 The vector field strengths F M .................. 58 3.2.3 The 3-form field strengths HA .................. 60 M 3.2.4 The 4-form field strengths GC ................. 62 3.3 The D =4dualityhierarchy.. .. .. .. .. .. .. .. .. .. 66 3.3.1 Purelyelectricgaugings . 66 3.3.2 Generalgaugings.......................... 70 3.3.3 Theunconstrainedcase . 74 3.4 The D =4action.............................. 76 4 Applications: Gauging N =1, 2 Supergravity 81 4.1 Gauged N =1Supergravity........................ 81 4.1.1 Electric gaugings of perturbative symmetries . .... 81 iv Contents 4.1.2 General gaugings of N =1, d =4supergravity . 85 4.1.3 Supersymmetric tensor hierarchy of N = 1, d =4 supergravity 87 4.1.4 Thegauge-invariantbosonicaction . 101 4.1.5 Possible couplings of the hierarchy p-form fields to (p 1)-branes103 − 4.2 N =2Einstein-Yang-MillsSupergravity . 103 5 Supersymmetric solutions 107 5.1 Ungauged N = 2 SUGRA coupled to vector and hypermultiplets . 107 5.1.1 Supersymmetric configurations: generalities . ..... 107 5.1.2 Thetimelikecase. 108 5.1.3 Thenullcase ............................ 116 5.2 N =2Einstein-Yang-MillsSupergravity . 122 5.2.1 Supersymmetric configurations: general setup . .. 122 5.2.2 Thetimelikecase. 126 5.2.3 Thenullcase ............................ 148 6 Coupling of higher-dimensional objects to p-forms: Cosmic strings in N =2 Supergravity 159 6.1 The1-forms................................. 161 6.2 World-lineactionsfor0-branes . 162 6.3 The2-forms:thevectorcase . 164 6.3.1 TheNoethercurrent . 164 6.3.2 DualizingtheNoethercurrent. 169 6.3.3 The 2-form supersymmetry transformation . 169 6.4 World-sheetactions: thevectorcase . 172 6.5 Supersymmetricvectorstrings. 173 6.6 The2-forms:thehypercase. 176 6.6.1 TheNoethercurrent . 176 6.6.2 DualizingtheNoethercurrent. 177 6.6.3 The 2-form supersymmetry transformation . 178 6.7 World-sheetactions: thehypercase . 179 6.8 Supersymmetrichyperstrings . 180 7 Summary 183 8 Resumen 185 9 Conclusions 187 10 Conclusiones 189 Contents v 11 Agradecimientos/Danksagungen/ Acknowledgements 191 A Conventions 193 A.1 Tensors ................................... 194 A.2 Gammamatricesandspinors . 197 B K¨ahler geometry 199 B.1 Gauging holomorphic isometries of K¨ahler-Hodge manifolds . 202 B.1.1 Complexmanifolds. 202 B.1.2 K¨ahler-Hodgemanifolds . 205 B.2 K¨ahler weights of certain frequently used objects . ........ 209 C Special K¨ahler geometry 211 C.1 Prepotential: Existence and more formulae . .. 214 C.2 Gauging holomorphic isometries of special K¨ahler manifolds . 215 C.3 Some examples ofquadratic prepotentials . .. 217 C.4 The [2,n]models ............................ 220 ST D Quaternionic K¨ahler geometry 223 D.1 Gauging isometries of quaternionic K¨ahler manifolds . ......... 226 D.2 AllabouttheC-map............................ 230 D.2.1 Dual-Quaternionic metric and its symmetries . 230 D.2.2 TheuniversalqK-space . 232 D.2.3 Quadbein, su(2)-connection and momentum maps . 233 E Projectors, field strengths and gauge transformations of the 4d ten- sor hierarchy 237 E.1 Projectors of the d =4tensorhierarchy . 237 E.2 Properties of the W tensors ........................ 238 E.3 Transformations and field strengths in the D =4 tensor hierarchy . 239 E.4 Gauge transformations in the D = 4 duality hierarchy and action . 241 F The Wilkinson-Bais monopole in SU(3) 245 F.1 AhairydeformationoftheW&Bmonopole . 247 G Publications 249 Bibliography 251 Chapter 1 Introduction This thesis deals with four-dimensional Supergravity theories and solutions thereto. Supergravities are very interesting theories for many reasons. In this introduction we shall give a short overview of the main motivations for introducing Supersymmetry, Supergravity and, last but not least, Superstring Theory. We will briefly describe how supersymmetry might help to address some “problems” of the Standard Model, then we shortly summarize some basic facts about Superstring Theory and its low energy limit, Supergravity. In the second part of this introduction we will discuss gaugings of Supergravity and its implications, focussing on the so-called tensor hierarchy. In the section 1.3 we will describe schematically how to find supersymmetric solutions to a given Supergravity theory. The outline of this thesis is given in the last section of this introduction. 1.1 Supersymmetry, Supergravity and Superstring Theory In the last decades of the past century a new theory, Superstring Theory, arose. There are two basic ingredients of Superstring Theory. First, there is the assumption that the fundamental constituents of matter are not pointlike particles, but oscillating one-dimensional objects: strings. The second basic ingredient of Superstring theory is Supersymmetry (SUSY). We start by giving an overview of some open open questions which supersymmetry, especially in the framework of Superstring Theory, might help to answer. The Standard Model (SM) of elementary particle physics is a spectacularly suc- cessful theory of the known particles and their electroweak and strong interactions [1]. Experiments have verified its predictions with incredible precision, and all the parti- 2 Introduction cles predicted by this theory have been found apart from the Higgs boson, which is expected to be detected soon at particle accelerators, such as e.g. at LHC at CERN. However, the Standard Model does not explain everything. For example, gravity is not included in the Standard Model of particle physics. Due to its weakness (at 25 a typical energy-scale of particle physics, it is about 10− times weaker than the 38 1 weak force and 10− times than the strong nuclear force ) gravity is irrelevant for describing the interactions of the matter studied by particle physicists. While the electroweak and strong forces are transmitted by spin-1 particles, gravity is supposed to be transmitted by a particle which carries spin 2, and in contrast to the other forces, it acts on every particle. On the one hand, Quantum Field Theory is used to explain the fundamental interactions at small distances, while on the other hand the large scale structure of the universe is governed by gravitational interactions described accurately by Einstein’s General Relativity. Trying to add gravity to the Standard Model and in particular to combine General Relativity with Quantum Mechanics leads to inconsistencies [2]. From a theoretical and conceptual point of view this is fairly unsatisfactory since we assume that there should be a way to describe the four fundamental forces within the framework of a unique underlying theory. The biggest problems of the Standard Model, as recognized by its practitioners, are: The SM is a Yang-Mills gauge theory, in which the gauge group SU(3) • c × SU(2)L U(1)Y is spontaneously broken to SU(3)c U(1)EM by the non- vanishing× vacuum expectation value (VEV) of a fundamental× scalar field, the Higgs field. Phenomenologically, the mass of the Higgs boson associated with electroweak symmetry breaking must be in the electroweak range h 246 GeV. However, the contribution of radiative corrections to the Higgs bosonh i ∼ mass is nonzero, divergent and positive. While the corrections to the electron mass are themselves proportional to the electron mass and quite small, even if we use the Planck scale as cut-off the mass of Higgs particles is very sensitive to the scale.the (mass)2 of the Higgs boson receives radiative corrections from higher- order terms in perturbation theory and a fine tuning of 28 orders of magnitude is necessary in order to obtain a phenomenologically viable Higgs mass. This is possible but very unnatural. This is the so-called hierarchy problem and it is the main motivation for introducing
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