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Benefits of Supersymmetry: from Scattering Amplitudes to Higher-Spin Benefits of supersymmetry: from scattering amplitudes to higher-spin gravity by Cheng Peng A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2013 Doctoral Committee: Associate Professor Henriette Elvang, Chair Professor Gordon Kane Professor Finn Larsen Professor Yongbin Ruan Professor Bing Zhou c Cheng Peng 2013 All Rights Reserved For all the people ii ACKNOWLEDGEMENTS First and foremost, I would like to express my deepest gratitude to my adviser, Prof Henriette Elvang, for her continuous and patient guidance in my research and career development. It would not have been possible for me to finish this thesis without the knowledge she teaches me, the time she spends in reading the draft and her critical comments on my work. She also makes me realise the importance of communication with people and shares with me many useful skills to clearly present my work to others. Furthermore, she provides me financial support which allows me to put more effort on my research. I owe Gordon Kane, Finn Larsen, and James Wells for several useful courses and the vast amount knowledge they have willingly and enthusiastically shared with me. I also would like to thank all other great teachers during my graduate study at the University of Michigan: Ratindranath Akhoury, Luming Duan, Leopoldo Pando- Zayas, Yongbin Ruan, Leonard Sander, Joel Smoller, and Bing Zhou. I am also indebted to fellow graduate students and postdocs-both those at Michi- gan and colleagues from afar-from whom I learned countless knowledge and invaluable experience to maintain a good status in my graduate study. Let me thank Kentaro Hanaki and Yu-tin Huang in particular for their advices and very nice examples of how to conduct independent research as a graduate student and a junior researcher, and also David McGady for helpful discussions and collaboration. Last, but not the least, I would also like to thank my friends and family, especially Zhixian Zhu, for their constant support and encouragement. iii TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::: iii LIST OF FIGURES ::::::::::::::::::::::::::::::: vii ABSTRACT ::::::::::::::::::::::::::::::::::: ix CHAPTER I. Introduction ..............................1 II. On-shell superamplitudes in N < 4 SYM ............. 19 2.1 Introduction and summary of results.............. 19 2.2 On-shell formalism for pure SYM: Φ-Φy formalism...... 22 2.2.1 On-shell superfields and MHV superamplitudes in N = 4 SYM...................... 22 2.2.2 N = 1 SYM on-shell superfields Φ and Φy and MHV superamplitude.................... 25 2.2.3 MHV vertex expansion................ 28 2.3 Pure N = 0; 1; 2; 3; 4 SYM: Φ-Ψ formalism........... 30 2.3.1 MHV superamplitudes for 0 ≤ N ≤ 4........ 32 2.3.2 NMHV superamplitudes for 0 ≤ N ≤ 4....... 34 2.3.3 On the range of N in SYM.............. 35 2.4 Super-BCFW........................... 36 2.5 Solution to the SUSY Ward identities in N < 4 SYM..... 39 2.6 Pure N < 8 SG amplitudes................... 43 2.7 Outlook.............................. 44 III. One-loop renormalization and the S-matrix ........... 46 3.1 Introduction and summary of results.............. 46 iv 3.2 Bubble coefficients in scalar field theories........... 51 3.3 Bubble coefficients for MHV (super) Yang-Mills amplitude.. 56 3.3.1 Extracting bubble coefficients in (N = 0; 1; 2 super) Yang-Mills....................... 60 3.3.2 MHV bubble coefficients in N = 1; 2 super Yang- Mills theory...................... 62 3.3.3 MHV bubble coefficients for pure Yang-Mills.... 71 3.4 Towards general cancellation of common collinear poles... 73 3.5 Conclusion and future directions................ 76 IV. Symmetries of Holographic Super-Minimal Models ...... 78 4.1 Introduction........................... 78 4.2 Higher-spin supergravity as a Chern-Simons Theory..... 80 4.2.1 Supersymmetric higher-spin algebra shs[λ]..... 81 4.2.2 Boundary conditions, constraints and gauge fixing. 83 4.2.3 Global symmetry................... 84 4.3 Super-W1[λ] algebra as the asymptotic symmetry...... 86 4.3.1 Boundary condition for super-W1[λ] algebra.... 87 4.3.2 Super-W1[λ] symmetries............... 89 4.4 Identification with the CPn chiral algebras........... 96 4.4.1 Large n limits of the CPn chiral algebra...... 97 4.4.2 Degenerate representations.............. 101 4.5 Conclusion............................ 107 V. Dualities from higher-spin supergravity ............. 109 5.1 Introduction........................... 109 5.2 3d higher-spin gravity, its AdS vacua and the RG flow.... 110 5.3 Higher-spin supergravity with different osp(1j2) embeddings. 112 5.3.1 Higher-spin supergravity as a Chern-Simons theory 113 5.3.2 Normalization..................... 114 5.3.3 osp(1j2) embeddings of sl(3j2)............ 116 5.4 The asymptotic symmetry.................... 119 5.4.1 The asymptotic symmetry of the principal embedding 119 5.4.2 The asymptotic symmetry of the non-principal em- bedding I........................ 120 5.4.3 The asymptotic symmetry of the non-principal em- bedding II....................... 122 5.5 Geometry corresponding to the different embeddings..... 126 5.5.1 AdS vacua corresponding to different embeddings. 127 5.5.2 RG flows between different AdS vacua........ 128 5.5.3 Comparison with the bosonic spin-3 gravity..... 134 5.5.4 Operators generating the RG flows.......... 136 5.5.5 Relations between UV and IR operators....... 138 v 5.5.6 A remark on the central charge............ 147 5.6 Discussion............................ 147 5.6.1 Relation with the Hamiltonian reductions of the WZW model.......................... 147 5.6.2 osp(1j2) embedding and sl(1j2) embedding..... 151 5.6.3 Generalizations to arbitrary superalgebra sl(n + 1jn) 151 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 153 vi LIST OF FIGURES Figure 1.1 Two examples of the Feynman diagram. The solid lines represent fermions and the wavy lines represent gauge bosons. Each vertex in the diagram represents a 3-point interaction between two fermions and a gauge boson. The diagram on the left is an example of tree-level amplitude, and the diagram on the right illustrates a 1-loop ampli- tude, which means there is one closed loop in the diagram. When considering the time direction to go from left to right, these diagrams can be used to describe the physical process where an electron and a positron annihilate into a virtual photon which then decays to a pair of muon and anti-muon.........................6 3.1 A schematic representation of the cancellation of common collinear poles (CCP). The bubble coefficient of the cut in figure (a), receives contributions from the four collinear poles indicated by colored ar- rows. Each collinear pole is also present in the corresponding adjacent cut indicated in figures (b), (c), (d), and (e) respectively. In the sum of bubble coefficients such contributions cancel in pairs....... 49 3.2 The terminal channels that gives non-trivial contribution to the sum of bubble coefficients. Note the helicity configurations of the loop legs of the n-point tree amplitude is identical with the two external legs on the 4-point tree amplitude in the cut............. 50 3.3 For any given tree-diagram in the φ4 theory, each vertex can be blown up into 4-point one-loop subdiagrams in three distinct ways, while preserving the tree graph propagators. Each case contributes a factor of α4 times the original tree diagram to the bubble coefficient. In this figure we show the example of 6-point amplitude........... 54 3.4 For pure φk theory, the only one-loop diagrams that gives non-trivial contribution to the bubble integrals are those with only two loop propagator. The contribution is simply the product of the tree dia- grams on both side of the cut, connected by a new 2(k − 2) vertex.. 56 vii 3.5 An illustration of the cancellation between adjacent channels. The contribution to the bubble coefficient coming from the dLIPS integral evaluated around the collinear pole hl1ii ! 0, indicated by the (red) arrows, of the two diagrams cancels as indicated in eq. (3.28).... 58 3.6 The \terminal" pole that contributes to the bubble coefficient. Such poles appear in the two particle cuts that have two legs on one side of the cut and one of the legs has to be a minus helicity. Note that, at this point in phase-space, l1 = −pn−1 and l2 = −pn......... 58 3.7 A two-particle cut for a generic n-point amplitude.......... 62 3.8 Illustration of two internal helicity configurations of the cut loop mo- menta.................................. 63 (i;j) 3.9 The two-particle cut that gives Sbfa;bg; N ................. 63 3.10 A graphical representation of eq. (3.28). The bubble coefficient of a given channel is separated into four terms, each having a different collinear pole as the origin of the holomorphic anomaly that gives a non-zero dLIPS integral. The 4-contributions can be grouped into two channels, the i- and the j-channel................. 65 3.11 A schematic representation of the cancellation of CCP for adjacent MHV amplitude for SYM. Each colored arrow represents a collinear pole that contributed to the bubble coefficient. Pairs of dashed arrows in the same color cancel. Only those represented by the solid arrows one on the two ends remain; they are the only non-trivial contribution to the overall bubble coefficient..................... 67 3.12 A schematic representation of the cancellation of CCP for adjacent six-point MHV amplitude. The dashed lines are common collinear poles which cancel pairwise....................... 69 3.13 The two terminal cuts for a given helicity configuration for the loop legs. For the choice of reference spinor jαi = jai only diagram (a) is non vanishing. If one instead choose jαi = jbi, then it is diagram (b) that gives the non-trivial contribution................
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