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Exotic Gauge Theories of Spin-2 Fields Imperial College of Science, Technology and Medicine Department of Physics Exotic Gauge Theories of Spin-2 Fields Nicholas Andrew Ondo Supervised by Dr. Andrew J. Tolley Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in in Physics of Imperial College London and the Diploma of Imperial College, October 2017 Abstract There has been a resurgence of interest in theories of massive spin-2 fields, owing to the recent discovery of ghost-free self-interactions. In addition to reviewing the historical and recent progress in this subject, I discuss my contributions, including the derivation of the complete decoupling limit of dRGT ghost-free massive gravity, proving no-go theorems on ghost-free interactions for charged spin-2 fields, updating the method of Dimensional Deconstruction for fermions to obtain massive supersym- metric gauge theories, and my progress towards supergravity theories with non-zero graviton mass. i ii Statement of Originality All of original research presented in this thesis are works that were done either by myself or alongside contributors. My original contributions to these works represent a substantial contribution. Any non-original works are appropriately cited to the best of the author's knowledge. iii iv Publications (*)\ The Complete Decoupling Limit of Ghost-free Massive Gravity" by Nicholas A. Ondo, Andrew J. Tolley. Published in JHEP 1311 (2013) 059. [ARXIV:1307.4769] (*)\ Interactions of Charged Spin-2 Fields" by Claudia de Rham, Andrew Matas, Nicholas A. Ondo, and Andrew J. Tolley. Published in Class.Quant.Grav. 32 (2015) no.17, 175008. [ARXIV:1410.5422] (*)\ Deconstructing Supergravity, I: Massive Supermultiplets" by Nicholas A. Ondo, Andrew J. Tolley. Submitted to JHEP. [ARXIV:1612.08752] (*)\ Deconstructing Supergravity, II: Massive RS Fields" by Nicholas Ondo, An- drew J. Tolley. To appear. v vi Copyright Declaration The copyright of this thesis rests with the author and is made available under a Cre- ative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they at- tribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. vii viii Dedication I dedicate this work to my mother Deborah Ondo, my father John Ondo, and my brother Jeremy Ondo, all of whom have given me endless support and encourage- ment. ix x Acknowledgements I would like to express thanks to: • My supervisor, Andrew Tolley, my unofficial secondary supervisor Claudia de Rham, and my close friend Andrew Matas for the countless hours of conversa- tion, tutelage, and intellectual engagement that they have given me. I would also like to thank Andrew Matas and Alexander Harrold for looking over an early manuscript of this thesis. xi xii Contents Abstracti Statement of Originalityi Publications iii Copyright Declarationsv Acknowledgements xi 1 Introduction to IR-Modifications of Gravity1 1.1 GR as a Quantum Theory . .1 1.2 Cosmological Constant: Observation and Theory . .3 1.2.1 New IR Gravitational Physics from Non-Zero Graviton Mass4 1.2.2 De-Electrification and Degravitation . .5 1.3 Objectives . .6 1.3.1 Outline . .7 2 Introduction to Massive Gravity9 2.1 Massive Spin-2 Theories? . .9 2.2 Massive Gravity without Self-Interactions . 11 2.2.1 Representation-Theoretic Aspects . 11 2.2.2 Fierz-Pauli Theory . 11 2.2.3 An Aside on Degravitation . 13 xiii xiv CONTENTS 2.2.4 Proca Theory in St¨uckelberg Formalism . 14 2.2.5 FP Theory in St¨uckelberg Formalism . 16 2.2.6 Fierz-Pauli Tuning and Ghostly Modes . 18 2.2.7 Origin of the vDVZ Discontinuity . 20 2.2.8 Self-Interactions and Vainshtein's Conjecture . 21 2.3 Self-Interacting Massive Gauge Theories . 23 2.3.1 Implications of Nonabelian Gauge Redundancy . 23 2.3.2 Proca-Yang-Mills . 23 2.3.3 PYM in St¨uckelberg Formalism . 25 2.3.4 The Decoupling Limit of PYM . 26 2.3.5 PYM and the vDVZ Discontinuity . 27 2.4 Self-Interacting Massive Gravity . 29 2.4.1 Boulware-Deser Theories . 29 2.4.2 The Boulware-Deser Ghost . 30 2.4.3 All 6 Helicities of a D = 4 Spin-2 Field . 31 2.5 Ghost-Free, Self-Interacting Massive Gravity . 32 2.5.1 Towards Ghost-Freedom . 32 2.5.2 dRGT Theory of a Ghost-Free Massive Graviton . 34 2.5.3 dRGT in Einstein-Cartan Formalism . 35 2.5.4 Partial Decoupling Limit of dRGT . 36 2.5.5 Galileon Theory and the Vainshtein Mechanism . 38 3 The Complete Decoupling Limit of dRGT Massive Gravity 40 3.1 The Λ3 Decoupling Limit . 40 3.2 St¨uckelberg Procedure for dRGT . 41 3.2.1 dRGT Interactions as a Deformed Determinant . 41 3.2.2 St¨uckelberg Fields in Einstein-Cartan Variables . 43 3.2.3 Auxiliary Equations for Lorentz St¨uckelberg Field . 44 3.2.4 Equivalence to Metric dRGT . 47 CONTENTS xv 3.2.5 dRGT and the Deser-van Nieuwenhuizen Gauge . 49 3.3 Surviving Helicity-1 Interactions . 50 3.3.1 Scaling of Lorentz St¨uckelberg . 50 3.3.2 Solution via Decoupling Limit Action . 51 3.3.3 Solution via Limit of DvN Condition . 53 3.3.4 Integral Form of the Lorentz St¨uckelberg Field . 54 3.4 The Complete dRGT Action in the Decoupling Limit . 55 3.5 Comments on the Helicity-1 Modes . 58 4 Review of Dimensional Deconstruction 60 4.1 Massive Theories from Higher-Dimensional Massless Theories . 60 4.1.1 Embeddings of Reps . 61 4.2 Deconstructing Free Theories . 63 4.2.1 Deconstructing Maxwell to Proca Theory . 64 4.2.2 Deconstructing LGR to FP Theory . 65 4.3 Deconstruction and Interactions . 66 4.3.1 Obstructions to Interactions . 66 4.3.2 Deconstructing YM to PYM . 67 4.4 dRGT from Deconstruction . 68 4.4.1 Deconstructing EC Gravity to dRGT Gravity . 68 4.4.2 2-Site Deconstruction and Bigravity . 71 5 Obstructions to Interacting Charged Spin-2 Fields 73 5.1 Charged Spin-2 Fields? . 73 5.2 Federbush Theory of Massive Charged Spin-2 Fields . 75 5.2.1 Acausalities and Velo-Zwanziger . 77 5.3 Charged Deconstruction . 78 5.3.1 Deconstruction Prescription . 78 5.3.2 Charged spin-2 Fields from Deconstruction . 81 xvi CONTENTS 5.3.3 Charged Deconstruction Spectrum has Complex Spin-2 Fields 82 5.4 Deconstructed Action in Fourier Variables . 83 5.4.1 Interactions in Fourier Variables . 83 5.4.2 U(1) symmetry in the N ! 1 limit . 84 5.5 Obstructions to Restoring the U(1) Symmetry . 86 5.5.1 Projecting out the U(1)-Violating Interactions . 89 5.5.2 Generic non-linear completions . 89 5.5.3 Reproducing Federbush . 92 5.5.4 Deconstruction-Motivation is the Unique Ansatz . 93 5.6 Hamiltonian Analysis of the Ansatz . 93 5.6.1 Hamiltonian Setup . 94 5.6.2 Reappearance of the Boulware-Deser Mode . 96 5.7 Obstructions to Self-Interacting Charged Spin-2 from Group Theory 97 5.7.1 Non-Existence of Specific [ISO(1; d) × ISO(1; d)] o U(1) . 98 5.7.2 Checking the Jacobi identity . 99 5.8 Comments about No-Go Theorem on Charged Spin-2 Fields . 99 6 Deconstructing Supermultiplets 102 6.1 Review of Supersymmetry . 103 6.1.1 Massless N = 1 Supermultiplets . 106 6.1.2 Massless N = 2 Supermultiplets . 107 6.1.3 Massive N = 1 Supermultiplets . 108 3 6.1.4 Towards Massive N = 1;Y = 2 Supermultiplets . 109 6.2 The N = 1 Zinoviev Theory . 112 6.2.1 Gauge Symmetries of the Zinoviev Lagrangian . 113 6.2.2 The Supersymmetry Transformations . 115 6.3 Dimensional Deconstruction of Fermions . 116 6.3.1 Incorporating Fermions . 116 6.3.2 Engineering Fermion Mass . 117 CONTENTS xvii 6.3.3 Group-Theoretic Perspective . 119 6.3.4 Prescription for Deconstructing Fermions . 119 6.4 Super-Proca Theory a l´aDeconstruction . 120 6.4.1 5-D N = 2 Super-Maxwell Theory . 120 6.4.2 Deconstructing 5-D Super-Maxwell . 121 6.4.3 4-D Supercharges from Deconstructed 5-D Supercharges . 123 6.5 Deconstructing D = 5 SUGRA to the Zinoviev Theory . 125 6.5.1 Review of Linearised 5-D N = 2 Supergravity . 125 6.5.2 Deconstructing to the N = 1 Zinoviev Action . 126 6.5.3 Deriving the Zinoviev SUSY Transformations . 128 6.6 Discussion of Fermionic Deconstruction . 131 7 Decontructing Supergravity 133 7.1 Review of Supergravity . 134 7.1.1 N = 1 and N = 2 4-D Supergravity . 134 7.1.2 N = 2 5-D Supergravity . 135 7.2 Properties of the Conjectured Non-Linear Theory . 137 7.3 1-Site Deconstruction of N = 2 5-D SUGRA . 139 7.3.1 Spin-2 Sector . 140 7.3.2 Spin-1 Sector . 141 7.3.3 Deconstructing the Fermions . 141 3 7.3.4 The Action for a Massive Spin- 2 on a Curved Space . 145 7.4 Difficulties Deconstructing the Non-Linear Superalgebra . 146 7.4.1 Superalgebra for D = 5 SUGRA . 146 7.4.2 Solving for the Residual Gauge Symmetries . 148 7.4.3 Obstructions to Deconstructing the SUSY Transformations . 149 7.5 Comments on the Non-Linear Theory . 150 8 Conclusion 152 xviii CONTENTS 8.1 Summary of Thesis Achievements . 152 8.2 Outlook . 155 A Conventions and Nomenclature 172 A.1 Nomenclature . 172 A.2 Conventions and Notations . 173 A.2.1 Treatise on Spinors .
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