Positive Geometry of the S-Matrix Yuntao Bai A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Nima Arkani-Hamed September 2018 © Copyright by Yuntao Bai, 2018. All rights reserved. Abstract The search for a theory of the S-Matrix has revealed unprecedented structures under- lying amplitudes. In this text, we present a new framework for understanding a class of amplitudes that includes Yang-Mills, Non-linear Sigma Model, the bi-adjoint cubic scalar, planar N = 4 super Yang-Mills, and more. We introduce positive geometries, which are generalizations of convex polytopes to geometries with higher order (i.e. non-linear) boundaries. Our construction provides a unique differential form called the canonical form of the positive geometry, whose pole structure is completely con- trolled by the geometric boundaries. The central claim of this text is that positive geometries play a fundamental role in our class of scattering amplitudes, whereby the corresponding canonical form determines a physical quantity. Our primary examples are (1) the bi-adjoint cubic scalar for which the positive geometry is the famous as- sociahedron polytope whose canonical form gives the color-ordered tree amplitude, and (2) planar N = 4 super Yang-Mills for which the positive geometry is the ampli- tuhedron whose canonical form gives the scattering integrand. One recurrent theme in our text is that physical properties of amplitudes like local poles and factorization are direct consequences of the boundary structure. We are therefore led to the point of view that locality and unitarity are emergent properties of the positive geometry. Furthermore, we discuss unexpected connections between positive geometry and on- shell diagrams, BCFW recursion, scattering equations, color-kinematics duality and the open string. iii Acknowledgements First and foremost, I am deeply indebted to Nima Arkani-Hamed for being my adviser and an amazing collaborator. It is very hard to overstate his intellectual prowess not just in particle physics but more generally in science. His work will undoubtedly continue to shape particle physics for decades to come. It was therefore an absolute privilege to have worked along his side. Nima also taught me the importance of persistence, and I am thankful to him for encouragement during moments of self- doubt. Furthermore, Nima is not just a great physicist, but also an amazingly fun person to talk to. It is rare to find such a fantastic combination of wit, charm and intellect in one person. I am deeply thankful to Thomas Lam for many insightful discussions, for being an outstanding collaborator, and for kindly inviting me to Ann Arbor for two very productive visits. Thomas played the key role in translating our work on the S- Matrix to a more rigorous mathematical framework, which established the bedrock foundations for the subject of positive geometry. I also owe great a debt of gratitude to Song He for many fruitful discussions over the years, and for collaboration on my first paper and subsequent papers. His expertise on the CHY formalism was central to connecting our work on positive geometries to the scattering equations. In real life, he is also a great friend to have. Furthermore, I am grateful to Freddy Cachazo and Jaroslav Trnka for stimulating discussions, and for their hospitality during my travels. I would also like to thank Steven Karp, Hugh Thomas, Lauren Williams, and Gongwang Yan. Moreover, I am grateful to Herman Verlinde and Dan Marlow for serving on my FPO committee, and especially to Dan for letting me help him on an undergraduate lab project. I am grateful to Steven Gubser for kindly providing me with work during the summer. I also want to thank the Department of Physics at Princeton, the Institute for Advanced Study, and NSERC for supporting my academic career. iv I would also like to thank my friends back home for our continued friendship despite the long distances. Finally and most importantly, I would like to thank my parents for supporting me throughout the years. The material presented in this thesis is based on publications [1, 2, 3, 4] by the au- thor and collaborators Nima Arkani-Hamed, Thomas Lam, Song He, and Gongwang Yan. Parts of the material were presented at various physics seminars, conferences and meetings at Harvard University, QMAP UC Davis, Perimeter Institute, Prince- ton University, Bhaumik Institute, Higgs Centre for Theoretical Physics, and National Taiwan University. v For my family vi Contents Abstract . iii Acknowledgements . iv 1 Introduction 1 2 Positive geometries 8 2.1 Positive geometries and their canonical forms . .9 2.2 Triangulations . 13 2.2.1 Triangulations of pseudo-positive geometries . 13 2.2.2 Physical vs. spurious boundaries . 15 2.3 Maps between positive geometries . 16 2.4 Generalized simplices . 17 2.4.1 The standard simplex . 18 2.4.2 Projective simplices . 20 2.4.3 Generalized simplices on the projective plane . 22 2.4.4 An example of a geometry with self-intersections . 27 2.4.5 Generalized simplices in higher-dimensional projective spaces . 29 2.4.6 Grassmannians and positivity . 32 2.5 Generalized polytopes . 34 2.5.1 Projective polytopes . 34 2.5.2 Cyclic polytopes . 36 vii 2.5.3 Dual polytopes . 37 2.5.4 Generalized polytopes on the projective plane . 39 2.5.5 Loop Grassmannians . 41 3 Canonical forms 46 3.1 Direct construction from poles and zeros . 47 3.1.1 Cyclic polytopes . 49 3.1.2 Generalized polytopes on the projective plane . 52 3.2 Triangulations . 55 3.2.1 Projective polytopes . 55 3.2.2 Generalized polytopes on the projective plane . 57 3.3 Pushforwards . 58 3.3.1 Projective simplices . 59 3.3.2 Projective polytopes from Newton polytopes . 64 3.3.3 Recursive properties of the Newton polytope map . 69 3.3.4 Newton polytopes from constraints . 72 3.3.5 Generalized polytopes on the projective plane . 77 3.4 Integral representations . 80 3.4.1 Dual polytopes . 80 3.4.2 Laplace transforms . 85 3.4.3 Projective space contours . 89 3.4.4 Projective space contours part II . 99 4 The associahedron 102 4.1 The planar scattering form on kinematic space . 103 4.1.1 Kinematic space . 103 4.1.2 Planar kinematic variables . 104 4.1.3 The planar scattering form . 105 viii 4.2 The kinematic associahedron . 109 4.2.1 The associahedron from planar cubic diagrams . 109 4.2.2 The kinematic associahedron . 112 4.2.3 Bi-adjoint cubic scalar amplitudes . 115 4.2.4 All ordering pairs of bi-adjoint cubic amplitudes . 119 4.2.5 The associahedron as the amplituhedron for bi-adjoint cubic theory . 125 4.3 Factorization and\soft" limit . 127 4.3.1 Factorization . 128 4.3.2 \Soft" limit . 132 4.4 Triangulations and recursion relations . 133 4.4.1 The dual associahedron and its volume as the bi-adjoint amplitude134 4.4.2 Feynman diagrams as a triangulation of the dual associahedron volume . 135 4.4.3 More triangulations of the dual associahedron . 138 4.4.4 Direct triangulations of the kinematic associahedron . 139 4.5 Vertex coordinates of the kinematic associahedron . 141 5 The worldsheet 143 5.1 Associahedron from the open string moduli space . 144 5.2 Scattering equations as a diffeomorphism between associahedra . 151 6 Color and kinematics 157 6.1 The big kinematic space . 157 6.2 Scattering forms and projectivity . 160 6.3 Duality between color and form . 165 6.4 Trace decomposition from scattering form . 169 6.5 BCJ relations . 174 ix 6.6 Scattering forms for gluons and pions . 175 6.6.1 Gauge invariance, Adler zero, and uniqueness of scattering forms for gluons and pions . 175 6.6.2 Scattering forms from the worldsheet . 177 7 The amplituhedron 181 7.1 Properties of the amplituhedron . 181 7.2 The tree amplituhedron for m = 1; 2 .................. 182 7.3 Grassmannian contours . 189 7.4 Wilson loops and surfaces . 191 7.5 Pushforwards . 197 7.6 Dual amplituhedra . 201 8 Planar N = 4 super Yang-Mills 203 8.1 Supersymmetric momentum twistors . 203 8.2 Scattering amplitudes from the amplituhedron . 205 8.3 BCFW recursion and positive geometry . 206 9 Momentum twistor diagrams 211 9.1 On-shell diagrams in momentum twistor space . 211 9.1.1 The Grassmannian representation of momentum twistor diagrams213 9.1.2 Examples and operations on the diagrams . 215 9.2 Boundary diagrams . 220 9.3 Factorization diagrams . 221 9.4 Forward limit diagrams . 223 9.5 Tree diagrams . 226 9.5.1 NMHV tree . 226 9.5.2 N2MHV trees . 227 9.6 One loop diagrams . 229 x 9.6.1 One loop diagrams for n-point MHV . 229 9.6.2 One loop Kermit expansion . 232 9.6.3 One loop 5 point NMHV . 233 9.7 Two loop diagrams . 237 9.7.1 Two loop diagrams for 4 point MHV . 239 10 One-loop amplitudes of planar N = 4 super Yang-Mills 245 10.1 The one-loop k = 1 Grassmannian . 246 10.1.1 0-dimensional cells . 247 10.1.2 Top cells . 249 10.1.3 Shifts . 251 10.1.4 5 point . 253 10.1.5 6 point . 257 10.2 The one-loop Grassmannian measure . 262 10.2.1 The general setup . 262 10.2.2 The measure . 264 10.2.3 Top cells and BCFW terms . 268 10.2.4 Triangulation of the one-loop Grassmannian with top cells . 270 10.2.5 Geometric factor . 271 11 Conclusion 273 xi Chapter 1 Introduction In recent years, we have witnessed tremendous progress in the theory of scattering amplitudes, including Witten's twistor string [5], on-shell recursion relations [6, 7], hidden symmetries such as dual conformal symmetry [8], unprecedented simplifica- tions (e.g.