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Thesis Is Presented for the Degree of Doctor of Philosophy of the University of Western Australia

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Thesis Is Presented for the Degree of Doctor of Philosophy of the University of Western Australia Off-shell higher-spin gauge supermultiplets and conserved supercurrents Jessica Hutomo Supervisor: Prof. Sergei M. Kuzenko Co-supervisor(s): A/Prof. Evgeny I. Buchbinder Prof. Ian N. McArthur This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia Department of Physics March 2020 Abstract This thesis presents the general structure of non-conformal higher-spin supercurrent mul- tiplets in three and four spacetime dimensions. Such supercurrents are in one-to-one correspondence with off-shell massless higher-spin gauge supermultiplets, some of which are constructed in this thesis for the first time. Explicit realisations of these conserved current multiplets in various supersymmetric theories are worked out in detail. In the first part of the thesis, we begin by reviewing the key properties of known massless higher-spin N = 1 supermultiplets in four-dimensional (4D) Minkowski and anti-de Sitter (AdS) backgrounds. We then propose a new off-shell gauge formulation for the massless integer superspin multiplet. Its novel feature is that the gauge-invariant action involves an unconstrained complex superconformal prepotential, in conjunction with two types of compensators. Its dual version is obtained by applying a superfield Legendre transformation. Next, we deduce the structure of consistent non-conformal higher-spin N = 1 supercurrents associated with these massless supersymmetric gauge theories. Explicit closed-form expressions for such supercurrents are derived for various supersymmetric theories in 4D N = 1 Minkowski and AdS superspaces. These include a model of N massive chiral superfields with an arbitrary mass matrix, along with free theories of tensor and complex linear multiplets. The second part of the thesis is devoted to a detailed study of N = 1 and N = 2 supersymmetric higher-spin theories in three-dimensional AdS space. By analogy with our 4D N = 1 constructions, we derive two dually equivalent off-shell Lagrangian formu- lations for the massless multiplets of arbitrary superspin in (1,1) AdS superspace. These formulations allow us to determine the most general higher-spin supercurrent multiplets and provide their examples for models of chiral superfields. With regards to (2,0) AdS supersymmetry, our approach is to first identify a multiplet of conserved higher-spin cur- rents in simple models for a chiral superfield. This is then used to construct two series of a massless half-integer superspin multiplet in (2,0) AdS superspace. Finally, our (2,0) AdS higher-spin supermultiplets are reduced to (1,0) AdS superspace, which yield four series of N = 1 supersymmetric massless higher-spin models. We illustrate the duality trans- formations relating some of these dynamical systems. We also perform the component reduction of two new N = 1 higher-spin actions in flat superspace. Further applications of these off-shell N = 1 models are discussed, one of which is related to the construction of two new off-shell formulations for the massive N = 1 gravitino supermultiplet in AdS. ii Acknowledgements First and foremost, I would like to sincerely thank my supervisor, Prof. Sergei Kuzenko, with whom I have been working for the past 3.5 years. His expertise and passion in theoretical physics are truly inspiring, and I feel very fortunate to be one of his PhD students. The work presented in this thesis would not have been possible without his guidance and constant support at every stage of my research. I would also like to thank my co-supervisors, A/Prof. Evgeny Buchbinder and Prof. Ian McArthur, for their helpful suggestions. I am really grateful to Evgeny for his teachings and collaborations on our joint papers throughout my PhD candidature. His guidance and availability for discussions have made it a wonderful learning experience. I wish to thank my thesis examiners: Prof. Evgeny Ivanov, Prof. Ulf Lindstr¨omand Prof. Rikard von Unge, for their valuable comments and suggestions. I am grateful to Dr. Gabriele Tartaglino-Mazzucchelli, Daniel Ogburn and Daniel Hutchings for their collaborations on our joint papers. To my fellow PhD colleagues in Field Theory and Quantum Gravity group: Michael, Daniel H, Emmanouil, James, Ben. Thank you for a wonderful time and many interesting discussions. In particular, I would like to thank Daniel, Emmanouil and Michael for their comments on some parts of this thesis. Last, but certainly not least, my gratitude goes out to my family, especially my parents. Thank you very much for the endless support, love and prayers throughout my studies. This research was supported by an Australian Government Research Training Program (RTP) Scholarship, and a University Postgraduate Award for International Students. iii Contents 1 Introduction 1 2 Field theories in N = 1 Minkowski and AdS superspaces 11 2.1 Field theories in N = 1 Minkowski superspace . 11 2.1.1 The Poincar´esuperalgebra . 11 2.1.2 N = 1 Minkowski superspace . 12 2.1.3 Superfields . 13 2.1.4 Supersymmetric action principle . 16 2.2 Linearised N = 1 Poincar´esupergravity and variant supercurrents . 18 2.3 Field theories in N = 1 AdS superspace . 23 2.4 Variant supercurrents in AdS space . 25 2.5 Off-shell higher-spin multiplets: a brief review . 26 2.5.1 Massless half-integer superspin multiplets . 26 2.5.1.1 Transverse formulation . 27 2.5.1.2 Longitudinal formulation . 28 2.5.2 Massless integer superspin multiplets . 30 2.5.2.1 Longitudinal formulation . 30 2.5.2.2 Transverse formulation . 30 3 Non-conformal higher-spin supercurrents in Minkowski space 32 3.1 Conformal gauge (super)fields and compensators . 32 3.2 The massless integer superspin multiplets revisited . 34 3.2.1 Reformulation of the longitudinal theory . 34 3.2.2 Dual formulation . 37 vii 3.2.3 Massless gravitino multiplet . 39 3.3 Higher-spin multiplets of conserved currents . 40 3.3.1 Non-conformal supercurrents: Half-integer superspin . 40 3.3.1.1 Examples of higher-spin supercurrents . 42 3.3.2 Non-conformal supercurrents: Integer superspin . 45 3.3.2.1 Examples of higher-spin supercurrents . 47 3.4 Discussion . 49 4 Higher-spin supercurrents in AdS space 51 4.1 Linear superfields . 52 4.2 Massless integer superspin multiplets . 53 4.2.1 New formulation . 54 4.2.2 Dual formulation . 57 4.2.3 Models for the massless gravitino multiplet in AdS . 59 4.3 Higher-spin supercurrents . 62 4.3.1 Massless half-integer superspin multiplets . 62 4.3.2 Non-conformal supercurrents: Half-integer superspin . 63 4.3.3 Improvement transformations . 64 4.3.4 Non-conformal supercurrents: Integer superspin . 65 4.3.5 Improvement transformation . 67 4.4 Higher-spin supercurrents for chiral superfields: Half-integer superspin . 67 4.4.1 Superconformal model for a chiral superfield . 68 4.4.2 Non-superconformal model for a chiral superfield . 69 4.4.3 Superconformal model with N chiral superfields . 70 4.4.4 Massive model with N chiral superfields . 71 4.5 Higher-spin supercurrents for chiral superfields: Integer superspin . 75 4.5.1 Massive hypermultiplet model . 75 4.5.2 Superconformal model with N chiral superfields . 77 4.5.3 Massive model with N chiral superfields . 78 4.5.3.1 Symmetric C ......................... 78 viii 4.5.3.2 Antisymmetric C ....................... 80 4.5.3.3 Massive hypermultiplet model revisited . 81 4.6 Summary and applications . 82 4.6.1 Higher-spin supercurrents for a tensor multiplet . 83 4.6.2 Higher-spin supercurrents for a complex linear multiplet . 85 4.6.3 Gauge higher-spin multiplets and conserved supercurrents . 86 5 N = 2 supersymmetric higher-spin gauge theories and current multiplets in three dimensions 88 5.1 Superconformal higher-spin multiplets . 90 5.1.1 Conformal supergravity . 90 5.1.2 Primary superfields . 92 5.1.3 Conformal gauge superfields . 94 5.2 Geometry of N = 2 AdS superspaces . 95 5.2.1 (1,1) AdS superspace . 95 5.2.2 (2,0) AdS superspace . 98 5.3 Massless half-integer superspin gauge theories in (1,1) AdS superspace . 100 5.4 Massless integer superspin gauge theories in (1,1) AdS superspace . 103 5.4.1 Longitudinal formulation . 103 5.4.2 Transverse formulation . 104 5.4.3 Reformulation of the longitudinal theory . 105 5.4.4 Massless gravitino multiplet . 108 5.4.5 N = 2 !N = 1 superspace reduction . 108 5.5 Higher-spin (1,1) AdS supercurrents . 113 5.5.1 Non-conformal supercurrents: Half-integer superspin . 113 5.5.1.1 Models for a chiral superfield . 116 5.5.1.2 Superconformal model with N chiral superfields . 117 5.5.2 Non-conformal supercurrents: Integer superspin . 118 5.6 Higher-spin supercurrents for chiral matter in (2,0) AdS superspace . 120 5.6.1 Massless models . 121 ix 5.6.2 Massive model . 123 5.7 Massless higher-spin gauge theories in (2,0) AdS superspace . 124 5.7.1 Type II series . 124 5.7.2 Type III series . 126 5.8 Summary and discussion . 128 6 Field theories with (2,0) AdS supersymmetry in N = 1 AdS superspace 131 6.1 (2,0) ! (1,0) AdS superspace reduction . 131 6.1.1 Geometry of (2,0) AdS superspace: Real basis . 132 6.1.2 Reduction from (2,0) to (1,0) AdS superspace . 133 6.1.3 The (2,0) AdS supersymmetric actions in AdS3j2 ........... 137 6.1.4 Supersymmetric nonlinear sigma models . 138 6.2 Massless higher-spin models: Type II series . 140 6.2.1 Reduction of the gauge prepotentials to AdS3j2 ............ 140 6.2.2 Massless higher-spin N = 1 supermultiplets in AdS3 ......... 143 1 6.2.2.1 Longitudinal formulation for massless superspin-(s + 2 ) multiplet . 143 6.2.2.2 Transverse formulation for massless superspin-s multiplet .
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