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Killing Spinors Universidad Consejo Superior de Aut´onomade Madrid Investigaciones Cient´ıficas Departamento de F´ısicaTe´orica Instituto de F´ısicaTe´orica Facultad de Ciencias KILLING SPINORS - BEYOND SUPERGRAVITY arXiv:1212.4111v2 [hep-th] 6 Mar 2013 Alberto R. Palomo Lozano Universidad Consejo Superior de Aut´onomade Madrid Investigaciones Cient´ıficas Departamento de F´ısicaTe´orica Instituto de F´ısicaTe´orica Facultad de Ciencias ESPINORES DE KILLING - MAS´ ALLA´ DE LA SUPERGRAVEDAD Memoria de Tesis Doctoral presentada ante el Departamento de F´ısicaTe´orica de la Universidad Aut´onomade Madrid para la obtenci´ondel t´ıtulode Doctor en Ciencias Tesis Doctoral dirigida por: Dr. D. Patrick A. A. Meessen Investigador Ram´ony Cajal, Universidad de Oviedo y tutelada por: Dr. D. Tom´asOrt´ınMiguel Cient´ıficoTitular, Consejo Superior de Investigaciones Cient´ıficas 2012 \For the things we have to learn before we can do, we learn by doing" `Aristotèlhc El trabajo de un f´ısicote´oricoes probablemente uno de los trabajos m´assolitarios que existen. La elaboraci´onde esta memoria es el resultado de numerosas horas en soledad dedicadas primero a entender/ reproducir literatura y c´alculosprevios, frutos del esfuerzo de otras personas, y sobre los que se cimentan los trabajos que forman el n´ucleode mis a~nos como doctorando, que han sido finalmente plasmados en estas p´aginas.Dicho esto, hubiera sido imposible llegar hasta aqu´ıde haber trabajado exclusivamente por mi cuenta, sin haber contado con la ayuda de varias personas. Quisiera por tanto agradecerles su apoyo durante estos a~nos. Un lugar privilegiado ocupan mis supervisores de tesis, Tom´asy Patrick, a quienes quiero agradecer primero el haberme ofrecido la oportunidad de cumplir uno de mis sue~nos,y tambi´enel haberme dedicado un tiempo y esfuerzo sin el que no podr´ıahaber culminado esta tesis. Patrick, con qui´enhe trabajado m´asdirectamente, ha sido mucho m´asque un director acad´emico;me ha instruido en aspectos t´ecnicosrelacionados con las construcciones matem´aticasque figuran en estas p´aginas,pero tambi´enhemos hablado de la vida en la Academia, y las implicaciones personales que conlleva esta profesi´on.Es una de las personas acad´emicamente m´ascapacitadas que jam´ashe conocido1, y tiene adem´asun sentido del humor y una capacidad de auto-desaprobaci´onque encuentro muy refrescante, y de lo que espero haberme empapado un poco. De Tom´ashe aprendido, o al menos as´ıcreo que lo ha intentado ´el,a ser m´aspr´actico,y no andarme por las nubes. Este trabajo representa el final de mi andadura como estudiante de posgrado, pero no quisiera olvidarme de tres buenos amigos que me orientaron en los comienzos en Inglaterra, y a quienes debo en gran parte la que ser´ıami futura vinculaci´oncon la Universidad Aut´onoma de Madrid, Daniel Cremades, Manuel Donaire y Javier Fdz. Alcazar. As´ımismo, debo dar las gracias a mis colaboradores, con quienes he compartido ideas y e-mails estos a~nos,y de los que he aprendido otra forma de trabajar. Estos son: Jai Grover, Jan Gutowski, Carlos Herdeiro y Wafic Sabra. No puedo tampoco olvidarme de Ulf Gran, bajo cuya direcci´one instrucci´onpas´eseis fant´asticosmeses en la Chalmers Tekniska H¨ogskola de Gotemburgo, Suecia, y que me instruy´oen el m´etodo espinorial de atacar la clasificaci´on de soluciones a teor´ıasde Supergravedad. Aunque he hecho poca vida social en el IFT durante estos a~nos,he llegado a entablar unos v´ınculosque sobrepasaban lo puramente acad´emicocon mis compa~nerosde doctorado, a los que quisiera dar las gracias por los buenos momentos pasados. Tambi´ena Paco, de Limcamar, con quien he compartido reflexiones sobre f´utbol y el devenir de la Liga 2011- 2012. As´ımismo, no podr´ıahaber llevado a buen culmen toda la parte log´ısticasin la ayuda de un fant´asticoequipo administrativo, encabezado por Isabel P´erez. <Muchas gracias a todos! Quisiera tambi´enagradecer al director del IFT, Alberto Casas, el haber sido siempre 1Hace algunos a~nosdije de organizar una fiesta cuando descubriera un fallo significativo en alguno de los articulos en que hemos trabajado juntos. Aunque logr´eencontrar uno, seguramente era de transcripcci´on,y no fundamental como yo hubiese querido. tan coagente con mis peticiones relacionadas con el nuevo edificio, y a Carlos Pena lo f´acilque resulta hablar con ´el,que aprend´ıcuando estaba organizando el minicurso sobre branas. Es un placer tambi´enmencionar la ayuda inform´aticarecibida por parte de Giovanni Ram´ırez, Andr´esD´ıaz-Gily Charo Villamariz, as´ıcomo de la familia Carton en los ´ultimosd´ıasantes de la entrega de esta tesis, porque prefiero tirarme por la ventana antes de que falle el ordenador. En este sentido, me gustar´ıahacer menci´onde las comunidades de open source GNU/ Linux/ LATEX, que se encargan de producir y mantener software de muy alta calidad, gracias al cual me ha resultado mucho m´asf´acildesarrollar mis tareas. Si todav´ıausas un sistema operativo propietario, realmente vale la pena hacer la transici´ona una distribuci´onLinux. Juan Jover me ayud´ocuando yo la hice a Ubuntu, all´apor la release 7.10 Gutsy Gibbon. Debo tambi´enagradecer el apoyo econ´omicorecibido durante estos a~nospor el Consejo Superior de Investigaciones Cient´ıficas, en el contexto de las ayudas para la investigaci´on `Junta para la Ampliaci´onde Estudios' JAE 2007. Probablemente la persona m´asfeliz por este trabajo sea mi padre, que lleva cuatro a~nos pregunt´andomesi todav´ıano hab´ıaacabado la tesis. \<Tesis, tesis, tesis!", ha repetido en numerosas ocasiones. Es pues que quiero dedicarle estas p´aginas,al igual que a mi madre y hermana, por proveeer el balance adecuado, y por soportar mi tendencia a estar despierto a altas horas de la noche, as´ıcomo a mi t´ıaLuz, por estar siempre ah´ı, y a mi re-t´ıoMat´ıas, que ahora D.E.P. junto a su Se~nor. Por ´ultimo,quisiera terminar recordando una peque~nafrase que se atribuye a Arist´oteles, pero que a mi me transmiti´oel que fuera mi profesor de M´etodos num´ericos, el Dr. Ventzeslav Valev, que dec´ıaque \aprendemos haciendo, no viendo". Es una m´aximaque en su momento no llegu´ea comprender del todo, pero que durante el proceso de este doctorado he llegado a asimilar y reconocer. Este´ ha sido para mi un viaje que me ha instruido m´asall´ade lo puramente acad´emico,tambi´enacerca de mis propias capacidades, mis l´ımites,e incluso mis mecanismos epistemol´ogicos,que espero me sirvan en el futuro. Madrid, junio 2012. Contents 1 Introduction 9 1.1 Killing spinors . 10 1.2 Beyond Supergravity . 12 2 Classification of supersymmetric SUGRA solutions 15 2.1 Bilinear formalism . 16 2.2 Spinorial geometry techniques . 25 3 N = 2 d = 4 gauged fakeSUGRA coupled to non-Abelian vectors 35 3.1 Fake N = 2 Einstein-Yang-Mills . 36 3.2 Analysis of the timelike case . 38 3.3 Analysis of the null case . 45 3.4 Non-BPS solutions to N = 2 SUGRA . 52 3.5 Summary of the chapter . 55 4 N = 1 d = 5 minimal fakeSUGRA 59 4.1 Fake KSE and first implications . 59 4.2 Differential constraints . 61 4.3 Solving the EOMs . 65 4.4 Dimensional reduction and link with the solution to d = 4 fakeSUGRA . 66 4.5 Summary of the chapter . 67 5 N = 1 d = 5 fakeSUGRA coupled to Abelian vectors 69 5.1 N = 1 d = 5 fSUGRA and Killing spinors . 69 5.2 Analysis of gravitino Killing spinor equation . 71 5.3 Some simple examples . 76 5.4 The Berger sphere provides a solution . 79 5.5 Solutions with a recurrent vector field . 83 5.6 Summary of the chapter . 85 6 Classification of `Supersymmetric' Einstein-Weyl spaces 87 6.1 Covariant rule and the Einstein-Weyl condition . 87 6.2 Timelike solutions . 89 6.3 Null N = 1 d = 4 solutions . 90 6.4 Null N = (1; 0) d = 6 solutions . 93 6.5 Remaining null cases . 97 6.6 Summary of the chapter . 98 ANNEXE I Introducci´on 99 II Resumen 101 APPENDICES A Conventions 105 A.1 Tensor conventions . 105 A.2 Spinorial structures . 109 A.3 Spinor bilinears and Fierz identities . 112 B Scalar manifolds 117 B.1 K¨ahlergeometry . 118 B.2 Special geometry . 125 B.3 Hyper-K¨ahlerand quaternionic-K¨ahlergeometry . 129 B.4 Real special geometry . 134 C Geometrical data for null case solutions 137 C.1 Spin connection and curvatures in d = 4 fSUGRA . 137 C.2 Spin connection and curvatures in minimal d = 5 fSUGRA . 138 C.3 Kundt metrics . 139 D Weyl geometry 141 D.1 Einstein-Weyl and Gauduchon-Tod spaces . 142 E Similitude group and holonomy 145 E.1 Sim and ISim as subgroups of the Caroll and Poincar´egroups . 145 E.2 Holonomy . 146 E.3 Recurrency . 147 F The Lorentz and the Spin groups 149 References 153 Chapter 1 Introduction These pages are the result of the research undertaken in the Grupo de Gravitaci´ony Cuerdas, led by Prof. Tom´asOrt´ınat the Instituto de F´ısicaTe´oricaUAM/ CSIC, during the tenure of my doctoral degree. They are intended to give an account of the kind of problems attacked during this time, as well as an illustration of why they can prove of interest, and the methods considered to solve them. In particular, it is a study of various constructions which arise in the context of supergravity theories, and along those lines we have made use of methods developed during the programme of classification of supersymmetric solutions to said theories, in order to obtain our results.
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