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Holography and Holomorphy in Quantum Field Theories and Gravity

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Holography and Holomorphy in Quantum Field Theories and Gravity Departamento de F´ısica de Part´ıculas HOLOGRAPHY AND HOLOMORPHY IN QUANTUM FIELD THEORIES AND GRAVITY Eduardo Conde Pena Santiago de Compostela, abril de 2012. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas HOLOGRAPHY AND HOLOMORPHY IN QUANTUM FIELD THEORIES AND GRAVITY Eduardo Conde Pena Santiago de Compostela, abril de 2012. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas HOLOGRAPHY AND HOLOMORPHY IN QUANTUM FIELD THEORIES AND GRAVITY Tese presentada para optar ao grao de Doutor en F´ısica por: Eduardo Conde Pena Abril, 2012 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas Alfonso V´azquez Ramallo, Catedr´atico de F´ısica Te´orica da Universidade de Santiago de Compostela, CERTIFICO: que a memoria titulada “Holography and holomorphy in quantum field theories and gravity” foi realizada, baixo a mi˜na direcci´on, por Eduardo Conde Pena, no departamento de F´ısica de Part´ıculas desta Universidade e constit´ue o traballo de Tese que presenta para optar ao grao de Doutor en F´ısica. Asinado: Alfonso V´azquez Ramallo. Santiago de Compostela, abril de 2012. A mis padres What I cannot create, I cannot understand. Became famous when found in Feynman’s last chalkboard. Agradecementos En primer lugar me gustar´ıaagradecer a Alfonso Ramallo la oportunidad de hacer el doctorado bajo su tutela. Es alguien de quien un estudiante de doctorado tiene mucho que aprender: por ejemplo su metodolog´ıade trabajo, su dedicaci´on y su continua ilusi´on son excelentes valores para “un joven aprendiz”. Le tengo que agradecer las muchas veces que nos hemos sentado a hacer cuentas codo con codo, algo nada com´un y que alivia sobremanera la tortuosa iniciaci´on al mundo de la investigaci´on. Espero no haber agotado su paciencia con mi ignorancia y algunas de mis man´ıas: aunque sigo aprovechando demasiado la celulosa, ¡logr´eacabar el paquete de los 500 folios en blanco! La atm´osfera agradable en nuestro grupo, con el “Big Three” que forma junto a Javier Mas y Jose Edelstein, ha sido un muy agradable caldo de cultivo para mi formaci´on. Junto a Alfonso, las personas que mas me han influido en mi formaci´on como f´ısico, y a quien tambi´en debo unas enormes “gracias” son (curiosamente hispano-hablantes las dos) Carlos N´u˜nez y Freddy Cachazo. Carlos ha sido poco menos que mi co-advisor durante pr´acticamente todo mi doctorado y, aunque haya sido por ´osmosis, algo me ha conseguido pasar de su inagotable saber de Field Theory. Carlos, tu entusiasmo perpetuo es altamente contagioso ¡Te debo mucho en la F´ısica y tambi´en en lo personal! A Freddy lo he conocido hace poco, y por un breve tiempo durante mi estancia en Canad´a; pero le ha sobrado para marcarme tambi´en con su visi´on preclara de la F´ısica. Gracias tambi´en por dedicarme una buena parte de tu tiempo a trabajar juntos en Canad´a. Of course, my physics personality (and my personal character as well!) cannot be un- derstood without all the people (students, post-docs, colleagues, etc. ) that I have met along the way. If this were an advertisement, I should make a disclaimer: I may contain traces of all these people. In the first place, I would like to thank all the physicists, and also friends, with whom I have shared appreciable time (and some enjoyable collaborations!) in Santiago, like Daniel Are´an, Javier Tarr´ıo, John Shock, Dimitrios Zoakos, Xi´an Camanho, Paolo Benincasa (who explained a lot of new things to me), etc. From all of them I have learnt Physics, and spent great times both in Santiago and in meetings abroad. During my different stays abroad, all of them very amusing, I have met many more people that I would also like to recall here: of course J´erˆome Gaillard, from Swansea, who also spent a very fructiferous year in Santiago, and with whom it is a pleasure to collaborate; in Swansea there were actually many nice people: Johannes Schmude, Agostino Patella, Joyce Myers, Prem Kumar, Maurizio Piai, Biaggio Lucini, Adi Armoni, . all constituted a group whose nicety, as Carlos would say, is difficult to match. Dalla mia permanenza a Trieste, voglio ringraziare un’ altra volta a Daniel, che fece che le due mesi e mezzo che fu l´afurono molto belli (e con molta epica sportiva!). Discussing some Physics with Chethan Krishnan was very stimulating, e mi porto un sacco di ricordi della gente italiana: Alessandro Lovato e Andrea Prudenziati, i due guerreri sulla bici, Luca Lepori (la sua storia della chiave e il Taurus ´ela pi´udivertente che mai ho sentito in inglese), e tantissima altra gente con chi ho viaggiato a Croazia, fatto diverse “meeting at 6pm, 6 floor, left wing”, bagnato nell Adriatico dopo la notte, etc. And I cannot forget either all the people I met in Perimeter Institute during my more recent stay: the many lunch discussions about anything with Jo˜ao Caetano and John Toledo, the collaboration with Sayeh Rajabi, y toda la comunidad hispana que all´ıconoc´ı, como Pedro Liendo (aunque ´el s´olo estuviera de paso) y Coco (Jorge Escobedo). Finalmente, en esta secci´on de F´ısica, me gustar´ıaagradecer en ´ultimo lugar a quien empez´oun poco todo y me introdujo, o aliment´o, el gusanillo por la F´ısica: mi profesor de bachillerato Fernando Botana; gracias por unas irreverentes y muy especiales clases de F´ısica. E por suposto, de f´ora da F´ısica (a´ında que moitos dos implicados sexan f´ısicos) l´evome os meus recordos m´ais doces destes cinco anos de doutorado. Moitos dos meus compa˜neiros dunha moi agrad´abel nosa promoci´on, e doutras promoci´ons e carreiras, ficaron por Com- postela e con eles segu´ına compartir tempo nas pedras da zona vella: Xi´an (outra vez), Alex,´ Lu, Jes´us, Ana, Celes, Gonza, Vane, os diversos Migueles matem´aticos, . Alg´uns logo marcharon, pero as conexi´ons ficaron: Delia, Xe (a t´ua casa en Madrid poder´ıadicirse que ´e a mi˜na segunda residencia oficial ;) ), a Xouba; moitas grazas polas numerosas reuni´ons que temos feito estes anos. Da facultade tam´en me levo o recordo de moita outra xente, como de Lu´ı˜no, que ao igual que Carlos tam´en posee un contaxioso entusiasmo, e particularmente de Dani, Brais e Breixo. Dani e Breixo aprend´eronme a esquiar tir´andome dende o curuto da pista m´ais alta de Andorra, e as nosas peregrinaxes anuais ´aneve est´an cheas de engrazadas an´ecdotas. Brais contribu´ıu´ami˜na introduci´on no mundo da bici, e con el e Breixo demos renda solta durante todos estes anos ´anosa paix´on polo bal´on e compartimos alegr´ıas e desilusi´ons nos “Autos Liouville”, que ficar´acomo o equipo do meu coraz´on. Por suposto te˜no que voltar a contar a Daniel outra vez aqu´ı;agardo que todas as boas tradici´ons que creamos estes anos sigan por moitos vindeiros! Outra xente que co˜nec´ınf´ora de Santiago e que me gustar´ıamencionar aqu´ıtam´en son Jacek e Alberto, proba de que as amizades “a distancia” son pos´ıbeis. Tampouco me quero esquecer de todo o grupo do CLM de italiano, con quen compart´ınunha viaxe m´axica ao sur de Italia. Moi especialmente, grazas a Ceres e ao Mapache/Orsetto, d´uas das persoas m´ais entretidas que co˜nec´ın na mi˜na vida, que non reparan en loucuras, e con quen compart´ınmomentos moi singulares durante as mi˜nas estad´ıas f´ora. Some special thanks go to “amazing Grace”, whose magical personality made my first stay abroad in Swansea unforgettable, and with whom I shared some more good times afterwards, and hopefully I will share more in times to come; 57 forever! E finalmente, as grazas m´ais especiais van para o Koala, que me leva apoiando dende tempos inmemoriais e que sempre estivo a´ıdurante todos estes anos en Compostela. Es a constante universal ;). Esta parte de agradecementos ter´ıaque ter sido escrita en Fontes. mais cando menos algo do que al´ıescrib´ınficou aqu´ıplasmado! Y para finalizar, en este caso es cierto que “los ´ultimos ser´an los primeros” (un dicho que nunca entend´ımuy bien, la verdad; pero este puede ser un buen ejemplo), me gustar´ıa agradecer a toda mi familia, productores de un entorno hogare˜no maravilloso, que me ha servido de “campo base” para llegar hasta donde me encuentro ahora. Obviamente, muy especialmente las gracias van para mi familia “m´as cercana”, Marta, mis padres y Lambda incluida, donde siempre he recibido un apoyo fundamental para m´ı;y en particular a mis padres, a quienes esta tesis va dedicada y que han generado de la nada mi (important´ısima) inercia en la vida. Y lo han hecho sin necesidad ninguna de saber de F´ısica. “Caminante, no hay camino, se hace camino al andar.” Contents 1 Introduction 5 1.1 Motivation..................................... 5 1.2 Contextofthisthesis............................... 6 1.3 Contentofthisthesis............................... 10 1.4 Fixthenotation.................................. 11 2 Scattering amplitudes on the complex plane 15 2.1 A briefing into scattering amplitudes . 15 2.1.1 The spinor-helicity formalism . 18 2.1.2 Three-pointamplitudes. .. .. 20 2.2 On-shell recursion relations . 21 2.2.1 TheBCFWconstruction......................... 21 2.2.2 Generalizing BCFW . 25 2.2.3 Commentsonzeroes ........................... 27 2.3 Thecomplex-UVbehavior ............................ 28 2.3.1 Factorization in the Pij channel ....................
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