RevModPhys.84.987 - REVIEWS OF MODERN PHYSICS VOLUME 84...

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How higher-spin gravity surpasses the spin-two barrier Xavier Bekaert * Laboratoire de Mathe ´matiques et Physique The ´orique, Unite ´ Mixte de Recherche 7350 du CNRS, Fe ´de ´ration de Recherche 2964 Denis Poisson, Universite ´ Franc ¸ois Rabelais, Parc de Grandmont, 37200 Tours, France Nicolas Boulanger and Per A. Sundell Service de Me ´canique et Gravitation, Universite ´ de Mons—UMONS, 20 Place du Parc, 7000 Mons, Belgium (published 3 July 2012) Aiming at nonexperts, the key mechanisms of higher-spin extensions of ordinary gravities in four dimensions and higher are explained. An overview of various no-go theorems for low-energy scattering of massless particles in flat spacetime is given. In doing so, a connection between the S -matrix and the Lagrangian approaches is made, exhibiting their relative advantages and weaknesses, after which potential loopholes for nontrivial massless dynamics are highlighted. Positive yes-go results for non-Abelian cubic higher-derivative vertices in constantly curved backgrounds are reviewed. Finally, how higher-spin symmetry can be reconciled with the equivalence principle in the presence of a cosmological constant leading to the Fradkin-Vasiliev vertices and Vasiliev’s higher-spin gravity with its double perturbative expansion (in terms of numbers of fields and derivatives) is outlined. DOI: 10.1103/RevModPhys.84.987 PACS numbers: 04.60. ± m, 11.15.Pg, 11.30.Ly, 14.80. ± j CONTENTS I. Introduction 987 II. No-go Theorems in Flat Spacetime 990 A. Preamble: The gauge principle and the Fronsdal program 990 B. The Weinberg low-energy theorem 990 1. Charge conservation: The spin-one case 990 2. Equivalence principle: The spin-two case 991 3. Higher-order conservation laws: The higher-spin cases 991 C. Coleman-Mandula theorem and its avatar: No higher-spin conserved charges 991 D. Generalized Weinberg-Witten theorem 992 E. Velo-Zwanziger difficulties 992 III. Possible Ways Out 993 A. Masslessness 993 B. Asymptotic states and conserved charges 993 C. Lorentz minimal coupling 994 D. Flat background 994 E. Finite dimensionality of spacetime 994 IV. Various Yes-go Examples 994 A. Consistent cubic vertices in Minkowski spacetime 995 B. Cubic vertices in AdS spacetime 996 C. Main lessons 997 D. Higher-spin symmetry breakings 998 1. Higher-spin gauge symmetries are broken at the infrared scale 999 2. Dynamical symmetry breaking: Spin one versus higher spin 999 V. Fully Interacting Example: Vasiliev’s Higher-Spin Gravity 999 A. Examples of non-Abelian gauge theories 999 B. The need for a complete theory 1000 C. Vasiliev’s equations 1001 D. AdS/CFT correspondence: Vasiliev’s theory from free conformal fields 1001 E. Emergence of extended objects 1003 VI. Conclusions and Outlook 1004 Appendix A: Weinberg Low-energy Theorem: S matrix and Lagrangian Dictionary 1004 1. Emission of a massless particle: Lorentz versus gauge invariances 1004 2. Cubic vertices 1005 Appendix B: Weinberg-Witten Theorem: A Lagrangian Reformulation 1005 1. Weinberg-Witten theorem 1005 2. Refinement of Weinberg-Witten theorem 1006 I. INTRODUCTION This review is an attempt at a nontechnical summary of how higher-spin gravity 1
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