Black Holes in Supergravity with Applications to String Theory Carlos Shahbazi Alonso Instituto de Física Teórica UAM-CSIC A thesis submitted for the degree of Doctor of Philosophy June 2013 Universidad Consejo Superior de Autónoma de Madrid Investigaciones Científicas Departamento de Física Teórica Instituto de Física Teórica Facultad de Ciencias A GUJEROS NEGROS EN SUPERGRAVEDAD Y TEORÍA DE CUERDAS Memoria de Tesis Doctoral presentada ante el Departamento de Física Teórica de la Universidad Autónoma de Madrid para la obtención del título de Doctor en Ciencias Físicas Tesis Doctoral dirigida por: Profesor D. Tomás Ortín Miguel Profesor de Investigación, Consejo Superior de Investigaciones Científicas Junio, 2013 5 Agradecimientos En esta tesis se recoge parcialmente el resultado de tres años de trabajo en el Instituto de Física Teórica, pertenecientealConsejoSuperiordeInvestigacionesCientíficas,bajolatuteladelProfesorD.TomásOrtínMiguel. No tengo palabras para expresar mi agradecimiento a Tomás: de él he aprendido básicamente todo lo que sé sobre Física y Matemáticas. Me ha enseñado a abordar problemas y a resolverlos de la manera más efectiva, así como la importancia de dominar los cálculos matemáticos y utilizar a la vez la intuición; en definitiva, me ha enseñado lo que significa investigar en Física Teórica. Ha sido un tutor inmejorable y es sin duda es uno de los físicos más brillantes que conozco. Gracias a él además, he tenido la oportunidad de viajar, conocer y colaborar con otros físicos excepcionales, como Eric Bergshoeff, Renata Kallosh o Patrick Meessen, de los que también he aprendido mucho, a los que estoy muy agradecido y con los que espero seguir en contacto en el futuro. RenatafueunaanfitrionainmejorabledemivisitademedioañoalStanford Insititute of Theoretical Physics; meayudócontododesdeelprimermomento,tantofueracomodentrodelámbitoacadémico,sepreocupósiempre de que estuviera bien acomodado y me dio la oportunidad de colaborar con ella y su grupo de investigación, de lo que obtuve un conocimiento y experiencia que de otra manera no hubiera sido posible. Gracias Renata. Una mención de agradecimiento merecen todos los colaboradores con los que he tenido la oportunidad de trabajar durante este tiempo; todos me han aportado algo y de todos he aprendido. Me refiero a Antonio de AntonioMartín,EricBergshoeff,JohannesBroedel,PabloBueno,WissamChemissany,FrederikCoomans,Álvaro delaCruz-Dombriz,RhysDavies,PietroGalli,MechthildHuebscher,AntonioL.Maroto,PatrickMeessen,Tomás Ortín, Jan Perz, Antoine Van Proeyen y Marco Zambón. Gracias también a Ángel Uranga y Marco Zambón por atenderme tan pacientemente con mis interminables dudas y preguntas. Debo agradecer al Consejo Superior de Investigaciones Científicas por la financiación recibida, en concepto tanto de mi salario como de la estancia realizada. Al personal administrativo, tanto del IFT, Isabel, Roxanna, Chabely, como del SITP, Karen y Julie, les doy las gracias: sin personas como ellas la investigación científica no sería posible. Fuera del ámbito académico hay muchas personas a las que tengo que agradecer el poder estar, en el momento de escribir estas líneas, en disposición de obtener el título de doctor en Física Teórica. Primero, a mis padres, Maria del Carmen y Mahmood; sin la eduación que me dieron y el posterior apoyo nunca hubiera llegado a donde hoy estoy, y por ello, entre otras cosas, estaré eternamente agradecido. Un lugar especial ocupa también mi tía Maria José, que ya desde pequeño me pusiera en contacto con la ciencia y me mostrara las visicitudes de la vida del científico, siendo ella una gran científica. Debo agradecer asimismo todo el apoyo recibido a mi tía Teresa, siempre preocupada por mí, a mi hermana Marta, y a toda mi familia en Irán; aunque la distancia es grande, siempre se han preocupado por mí como si estuvieran aquí me han deseado lo mejor: ¡muchas gracias!, espero poder visitaros algún día en vuestro tierra, Irán, que también es la mía. Debo también mencionar a mis abuelos españoles,GenerosoyGloria,descansenenpaz,quesiemprequisieronquemededicaraaunaocupaciónintelectual. Agradezco en especial el apoyo y cariño diario, así como la comprensión, a mi novia Sara, con ella los buenos momentos son aún mejores, y los malos momentos son menos amargos: ¡gracias por todo! Gracias también a mis amigos, los que siempre habéis estado ahí, dando el callo: Santi, Charli, Iker, Roberto: ¡qué buenos ratos hemos pasado y los que aún nos quedan! Finalmente, mencionar a Pablo Bueno y Alberto Palomo-Lozano, compañeros de despacho con los que he pasado buenos ratos que van más allá de lo meramente profesional. Agradezco también a Pablo Bueno y Diego Regalado por estimulantes discusiones sobre Física y Matemáticas, así como por leer y comentar las versiones iniciales de este manuscrito. Madrid, Junio 2013. 6 Contents 1 Introduction 9 1.1 From String Theory to Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Remarks about Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Supergravity solutions and the attractor mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Mathematical preliminaries 19 2.1 Special Kähler Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Extended ungauged Supergravity in four dimensions 31 3.1 Extended electromagnetic duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 =2; d=4 ungauged Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 N 3.3 >2; d=4 ungauged Supergravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 N 4 Supergravity black holes 39 4.1 The general form a Supergravity black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Hidden symmetry and the microscopic description of the entropy . . . . . . . . . . . . . . . . . . . 41 5 All the supersymmetric black holes of extended Supergravity 45 5.1 The mathematical formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 The supersymmetric black hole solution of =8 Supergravity . . . . . . . . . . . . . . . . . . . . 46 N 5.3 Supersymmetric black holes and groups of Type E . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7 5.4 =2 Supergravity supersymmetric black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 N 6 The H-F.G.K. formalism 55 6.1 H-FGK for =2, d=4 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 N 7 Quantum black holes in String Theory 65 7.1 Type-IIA String Theory on a Calabi-Yau manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Perturbative quantum black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.3 Non-perturbative quantum black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A Resumen 75 B Conclusiones 77 References 79 8 INTRODUCTION Chapter 1 Introduction The aim of this thesis is to study black holes in String Theory (ST) through their classical description as Super- gravity solutions. ST [1, 2] is a framework that attempts to offer a unified quantum description of all the known fundamental interactions and, in particular, of gravity. It would solve, if correct, the long-standing problem of Quantum Gravity. In the present day there is no experimental evidence of ST, and there is no hope in the ST community that such an evidence will soon be found. Despite the lack of experimental results, ST has passed a few self-consistency checks that are expected to hold in the right unifying theory of nature [3, 4], if it exists. As a consequence, an enormous effort has been devoted over the last decades to develop ST, leading to beautiful and important advances and insights in modern Theoretical Physics [5–7] and Mathematics [8–16]. The ramifications of ST-inspired results are nowadays virtually everywhere in Theoretical Physics, and thus even if the right theory of nature was not ST, we can expect that they will have some ingredients in common. Hundreds of thesis, books and reviews have been written over the years dealing with the principal aspects of ST, such as supersymmetry, perturbativeST,conformalfieldtheory,D-branes,STdualities,aDS/CFTcorrespondence,M-Theory... andhence we refer the interested reader to the existing literature, for instance [17–21] and references therein. I will focus instead (see chapters 2 and 3), for reasons that will become apparent later, on a different area, sometimes forgotten in ST applications: the precise mathematical structure of the ST effective actions, i.e. field theories that describe the dynamics of the massless modes of the ST spectrum and are used to make contact with four-dimensional low-energy physics. The mathematical structure of these effective actions is crucial in order to ensure the consistency of the theory and it is the necessary background that we will need to pursue our goal, namely the study of black holes in ST. Let us see first how to go from the full-fledged ST to four-dimensional Supergravity. 1.1 From String Theory to Supergravity ST lives in ten dimensions, that is, the mathematical object that represents the space-time in ST is a ten- dimensionalmanifold. Sinceexperimentally,thatis,atlowenergies,weobservefourdimensions,somemechanism must be used to reconcile theory and experiment. The standard way to proceed is to assume that the space-time manifold 1 has the following fibre bundle structure M (cid:25) ; (1.1) 4 M(cid:0)!M where is the base space manifold (which represents the space-time that we observe at low energies) and the 4 M fibre (p)=(cid:25) 1(p) at each p is a compact manifold, small enough to not be accessible in current high- 6 (cid:0) 4 M 2M energyexperiments. Sotosay, (p)issosmallthatwecannotsee it withtheavailabletechnology. Noticethat 6 M (p)= (q) ; p;q , so we can denote the typical fibre simply by . M6 (cid:24)M6 8 2M4 M6 The general mechanism described above can be put in practice in several different ways, on which the precise size limits of depend [22–25]. Unfortunately, it is currently not known how to compactify ST on a non-trivial 6 M compact manifold , except for some particular cases. By non-trivial manifold here we mean a Riemannian 6 M manifold with a curved metric. The usual procedure to deal with this situation is to cook up an effective field theory action that encodes the dynamics of the massless modes present in the ST spectrum, which are the ones relevanttolowenergyphysics. Thereasonisthatthefirstmassivestatesinthespectrumhavemassesoftheorder of the Planck mass, which is way out of reach for current particle accelerators. 1FormoredetailsabouttheprecisepropertiesofM,seechapter2. 9 10 INTRODUCTION As a consequence of ST having local space-time supersymmetry, the ST massless states action is a very special one: a Supergravity [26–30]. The key point is that, even though it is not know how to compactify ST, it is known how to compactify a field theory, and in particular a Supergravity. Therefore, although the procedure described above is sometimes called “ST compactification” what is actually compactified is a ten-dimensional Supergravity. We can add to it ST corrections, that is, modifications to the tree level result as prescribed by ST, which is not a field theory but a theory of extended objects. The resulting corrected action is, again, a Supergravity but, tipically, it has higher order terms in the Lagrangian and modified couplings. If we compactify now low energy ST, that is, ten-dimensional Supergravity, on a particular compact six- dimensional manifold 2, we obtain a four-dimensional effective action which will also be a Supergravity, four- 6 M dimensional in this case, if some technical requirements are obeyed by the compact manifold . I will come 6 M back to this point later, but for the moment I refer the reader to [25] and references therein. To summarize, we have started with ten-dimensional ST and we have ended up with four-dimensional Super- gravity,whichisatheoryofgravityandmatterthatcanbeembeddedinST,andthereforeseemstobetheperfect starting point to study four-dimensional black holes in ST. 1.2 Supergravity Supergravity is a locally supersymmetric theory of gravity, that is, a field theory invariant under local supersym- metry transformations. Supersymmetry is a not so new, and yet to be observed, hypothetical symmetry between bosons and fermions [28, 31–33]. If we consider a field theory content with spin less or equal than two, there are seven different types of four-dimensional Supergravity = 1; ;6;8, depending on the amount of supersymmetry of the theory. N (cid:1)(cid:1)(cid:1) N Supersymmetry transformations are generated by a set of spinors (cid:15) (p);I = 1; ; , where is the number I (cid:1)(cid:1)(cid:1) N N of supersymmetries of the given Supergravity. In four dimensions, the minimal spinors (cid:15) (p) can be taken to be I WeylorMajorana,andthereforewehaverespectively2 complexor4 realassociatedcharges. Supersymmetry N N transformations can be schematically written as (cid:14) (cid:30) (cid:15)(cid:22)(p) (cid:30) +(cid:30)(cid:22) (cid:30) (cid:30) ; (cid:14) (cid:30) @(cid:15)(p)+ (cid:30) +(cid:30)(cid:22) (cid:30) (cid:15)(p)(cid:30) ; (1.2) (cid:15) b b f f f (cid:15) f b f f f (cid:24) (cid:24) where (cid:30) denotes the bosonic field(cid:0)s and (cid:30) de(cid:1)notes the fermionic fields(cid:0). (cid:1) b f Since Supergravity is a supersymmetric theory, and supersymmetry relates bosonic a fermionic fields, every Supergravity contains bosons and fermions. Truncating the fermions is always consistent, thanks to the following Z symmetry, present in every Supergravity Lagrangian 2 (cid:30) (cid:30) ; (cid:30) (cid:30) : (1.3) b b f f ! !(cid:0) The bosonic sector of four-dimensional Supergravity is a particular instance of General Relativity, as formulated by Albert Einstein in 1915 [34]. That is, it is a metric theory of gravity coupled to a particular matter content, which includes scalars and vector fields, and where the equation of motion for the metric g is given by R( )=T: (1.4) r Here R( ) is the Ricci tensor of the Levi-Civita connection associated to g on the space-time tangent bundle r r [35], and T is the geometrized energy-momentum tensor corresponding to the matter content of the theory. General Relativity cannot be, in principle, coupled to fermions, since it is formulated in a way on which only the diffeomorphisms group Di(cid:11)( ) acts naturally on the matter content. We need to make manifest the local M actionoftheLorentzgroupSO(1;3)3 onthemattercontentofthetheory,sincefermionsareassociatedtospinorial representationsofSO(1;3). Therefore,ifwewanttoconsiderthecompleteSupergravityaction,wehavetochange the set-up and use a more general formalism, which turns out to be the Cartan-Sciama-Kibble theory [19, 36–38], a generalization of Einstein’s General Relativity. Just as the bosonic sector of Supergravity is a particular case of General Relativity, the complete Supergravity theory is a particular case of the Cartan-Sciama-Kibble theory, which is an extension of General Relativity that can accommodate fermions. Before introducing the Cartan-Sciama-Kibble theory it is necessary to modify a bit the geometric set-up, in order to geometrically introduce fermions. General Relativity coupled to matter can be described in terms of objects that transform as tensors under space-time diffeomorphisms (such as sections over the tensor products of T and T or connections on principal bundles over ). For instance the metric, which describes the (cid:3) M M M 2Noticethatitispossibletocompactifyinmanifoldswithdimensionotherthansix, obtainingaseffectiveactionsSupergravities indimensionsotherthanfour. 3Rather,theactionofitsdouble-cover,thespingroupSpin(1;3),seebelow.