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Interactions in Higher-Spin Gravity: a Holographic Interactions in Higher-Spin Gravity: A Holographic Perspective Dissertation by Charlotte Sleight Interactions in Higher-Spin Gravity: A Holographic Perspective Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Charlotte Sleight aus Hull, Großbritannien M¨unchen,den 2. Mai 2016 Dissertation Submitted to the faculty of physics of the Ludwig{Maximilians{Universit¨atM¨unchen by Charlotte Sleight Supervised by Prof. Dr. Johanna Karen Erdmenger Max-Planck-Institut f¨urPhysik, M¨unchen 1st Referee: Prof. Dr. Johanna Karen Erdmenger 2nd Referee: Prof. Dr. Dieter L¨ust Date of submission: May 2nd 2016 Date of oral examination: July 26th 2016 To my dad, sister, and the wonderful memory of my mum. \Die Probleme werden gel¨ost,nicht durch Beibringen neuer Erfahrungen, sondern durch Zusammenstellung des l¨angstBekannten." | Ludwig Wittgenstein Zusammenfassung Die vorliegende Dissertation widmet sich der Untersuchung der H¨ohereSpin-Gravitation auf einem Anti-de-Sitter-Raum (AdS). Es gibt Hinweise darauf, dass dem Hochen- ergiebereich der Quantengravitation eine große (H¨ohereSpin-) Symmetriegruppe zu- grundeliegt und somit eine m¨ogliche effektive Beschreibung dieses Beriechs durch eine H¨ohereSpin-Theorie gegeben ist. Die Reaslisierung einer Quantengravitation mit dieser Symmetrie ist insbesondere in der String Theorie bei hohen Energien gegeben, wenn die Stringspannung vernachl¨assigtwerden kann. Die Untersuchung der H¨ohereSpin-Theorien ist von physikalischer Relevanz, weil ihre unendlichdimensionale Symmetrie als Leitprinzip verwendet werden kann, um neue Erkenntnisse ¨uber die Quantengravitation bei Energien jenseits der heute experimen- tall erreichbaren Energieskalen zu gewinnen. Es muss dabei ber¨ucksichtigt werden, dass die verf¨ugbarennichtlinearen Formulierungen der H¨ohere-Spin-Theorien unendlich viele Hilfsfelder mit einschließen, so dass selbst grundlegende Eigenschaften wie Unitarit¨at oder Kausalit¨atnoch nicht vollst¨andigverstanden sind. Zielsetzung dieser Arbeit ist es, die Wechselwirkungen zwischen h¨oherenSpin-Feldern im Rahmen der Lagrangeformulierung zu untersuchen, da dies ohne Hilfsfelder m¨oglich ist. Um dies zu erm¨oglichen, wird die vermutete Dualit¨atzwischen der H¨ohereSpin- Gravitation im AdS-Raum und integrablen konformen Feldtheorien benutzt. Durch Anwendung von Techniken der konformen Feldtheorie werden in der Lagrangefunktion der H¨oheren Spin-Theorie alle Wechselwirkungsterme bis zur dritten Ordnung sowie die Selbstwechselwirkung des Skalarfeldes bis zur vierten Ordnung bestimmt. Zu diesem Zweck werden neue Methoden entwickelt: Diese umfassen eine neue Herangehensweise vii zur Berechnung von Dreipunktamplituden mit externen Feldern beliebigen Spins und die explizite Form von Propagatoren f¨urEichfelder mit beliebigem ganzzahligen Spin im AdS-Raum. Weiterhin werden Techniken zur Berechnung konformer partieller Wellen- zerlegungen von Vierpunktamplituden in H¨ohereSpin-Theorien entwichelt. Die α0-Korrekturen in der Stringtheorie spiegeln sich in der Nichtlokalit¨atder Wech- selwirkungen zwischen H¨ohereSpin-Feldern wider: Es treten beliebig hohe Ableitun- gen der Felder auf. In der Stringtheorie bei endlicher Stringspannung bedeuten diese h¨oherenAbleitungsterme offensichtlich keine Einschr¨ankung von fundamentalen Eigen- schaften wie etwa der Kausalit¨at. In der H¨ohere-Spin-Theoriejedoch ist dies Gegen- stand aktueller Forschung. Im Kontext der Ergebnisse dieser Arbeit wird die Bedeu- tung der AdS/CFT Dualit¨atf¨urdie Lokalit¨atder Wechselwirkungen zwischen H¨ohere Spin-Feldern diskutiert. Insbesondere wird auf die holographische Interpretation der Korrelatoren in der konformen Feldtheorie als Streuamplituden der dualen Gravita- tionstheorie im AdS-Raum eingegangen. Abstract In this thesis we investigate aspects of higher-spin gravity on an anti-de Sitter (AdS) background. Such theories are believed to give an effective description of quantum gravity in its high energy regime, where there have been hints that a large underlying (higher-spin) symmetry may become restored. In particular, through considerations of string theory in the high-energy (tensionless) limit. The study of higher-spin gauge theories is thus of physical relevance, for their infinite dimensional symmetry may serve as a guiding principle to help shed light on the nature of quantum gravity at energies which are beyond the reach of present day experiments. To fulfil this potential, there is much still to be understood about higher-spin the- ories themselves. To date, the fully non-linear formulations of higher-spin theories include infinitely many additional auxiliary fields. With this work we take steps to- wards clarifying the nature of interactions amongst higher-spin fields in a metric-like Lagrangian framework, free from such auxiliary fields. To this end, we employ the con- jectured holographic duality between higher-spin gravity theories on AdS and certain, solvable, conformal field theories (CFTs). In this way well-known methods in CFT may be applied, and with them we determine completely all cubic interactions and the quartic self-interaction of the scalar in the higher-spin Lagrangian on an AdS back- ground. For this purpose a number of further original results were established, which include: Tools to evaluate three-point amplitudes in AdS space involving external fields of arbitrary integer spin, propagators for gauge fields of arbitrary integer spin on an AdS background, and techniques to determine conformal partial wave expansions of four-point amplitudes in higher-spin theories. ix Reminiscent of the tail of α0 corrections in string theory, higher-spin interactions are non-local in the sense that they are unbounded in their number of derivatives. While in (finite tension) string theory this higher-derivative behaviour does not appear to pose a threat to fundamental properties like causality, in higher-spin theories this remains to be clarified. In the context of the results established in this work, we briefly discuss the implications of the AdS/CFT duality on the locality properties of interactions in higher-spin theory. In particular, by drawing on the holographic interpretation of CFT correlation functions as scattering amplitudes in the dual gravity theory on AdS. Thesis publications This thesis is based on some of the authors work that was carried out during the period of May 2013 - 2016 at the Max-Planck-Institut f¨urPhysik (Werner-Heisenberg-Institut) M¨unchen, under the supervision of Prof. Johanna Erdmenger. The corresponding pub- lications are: [1] Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange. Xavier Bekaert, Johanna Erdmenger, Dmitry Ponomarev and Charlotte Sleight. Published in JHEP: JHEP 1503 (2015) 170 [2] Quartic AdS Interactions in Higher-Spin Gravity from CFT Xavier Bekaert, Johanna Erdmenger, Dmitry Ponomarev and Charlotte Sleight. Published in JHEP: JHEP 1511 (2015) 149 [3] Bulk quartic vertices from boundary four-point correlators Xavier Bekaert, Johanna Erdmenger, Dmitry Ponomarev and Charlotte Sleight. arXiv:1602.08570 [hep-th], to be published in World Scientific. Proceedings: \International Workshop on Higher Spin Gauge Theorie" (4-6 Nov. 2015, Singapore). [4] Higher-spin Interactions from CFT: The Complete Cubic Couplings Charlotte Sleight and Massimo Taronna. Published in Physical Review Letters: Phys.Rev.Lett. 116 (2016) no.18, 181602 Contents Zusammenfassung vi Abstract viii Thesis publications x 1 Introduction 1 1.1 A tale of two theories ............................ 1 1.2 Higher-spins and strings ........................... 2 1.3 This thesis .................................. 5 1.3.1 Summary of main results ...................... 9 1.3.2 Outline ............................... 13 2 The AdS/CFT correspondence 15 2.1 Practicalities ................................. 15 2.1.1 Field-operator map ......................... 18 2.1.2 Gauge fields and conserved currents . 20 2.2 Higher-spin / vector model holography ................... 22 2.2.1 AdS higher-spin interactions from CFT . 26 3 Correlation functions in CFT 31 3.1 Conformal algebra and transformations ................... 31 3.2 Representations on operators ........................ 33 3.2.1 Primaries and descendants ..................... 35 xii CONTENTS 3.3 Radial quantisation .............................. 37 3.3.1 State-operator correspondence ................... 39 3.3.2 Unitarity bounds, conserved currents and free scalars . 41 3.4 Operator product expansion ......................... 42 3.5 Correlation functions of primary operators .................. 45 3.5.1 Two-point functions ......................... 46 3.5.2 Three-point functions ........................ 50 3.5.3 Four-point functions ........................ 55 3.6 Conformal block decomposition ....................... 57 3.6.1 Integral representation ....................... 59 4 Correlators in the free O (N) model 63 4.1 The free scalar O (N) vector model ..................... 63 4.1.1 Singlet sector ............................ 65 4.2 Singlet sector correlation functions ..................... 67 4.2.1 2-point functions .......................... 68 4.2.2 3-point functions of single-trace operators . 69 4.2.3 Scalar 4-point function ....................... 74 5 Witten diagrams in higher-spin theory 81 5.1 Witten diagrams ............................... 82 5.1.1 Warm-up example: Scalar field in AdS . 83 5.2 Propagators of arbitrary spin ......................... 90 5.2.1 Bulk-to-bulk propagators .....................
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