GRASSMANNIAN GEOMETRY OF SCATTERING AMPLITUDES

Transcription

1 GRASSMANNIAN GEOMETRY OF SCATTERING AMPLITUDES Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang-Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the broader fields of mathematical physics. Nima Arkani-Hamed is Professor of Physics at the Institute for Advanced Study, Princeton, New Jersey, USA. Jacob Bourjaily is Assistant Professor of Physics at the Niels Bohr International Academy and Discovery Center at the University of Copenhagen, Copenhagen, Denmark. Freddy Cachazo is the Gluskin Sheff Freeman Dyson Chair in Theoretical Physics at the Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada. Alexander Goncharov is Professor of Mathematics at Yale University, New Haven, Connecticut, USA. Alexander Postnikov is Professor of Applied Mathematics and Algebraic Combinatorics at the Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Jaroslav Trnka is Assistant Professor of Physics at the University of California, Davis, California, USA.

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3 GRASSMANNIAN GEOMETRY OF SCATTERING AMPLITUDES NIMA ARKANI-HAMED Institute for Advanced Study JACOB BOURJAILY Niels Bohr Institute FREDDY CACHAZO Perimeter Institute ALEXANDER GONCHAROV Yale University ALEXANDER POSTNIKOV Massachusetts Institute of Technology JAROSLAV TRNKA University of California, Davis

4 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: / c Alexander Postnikov, and Jaroslav Trnka 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Names: Arkani-Hamed, Nima. Title: Grassmannian geometry of scattering amplitudes / Nima Arkani-Hamed (Institute for Advanced Study) [and five others]. Description: New York : Cambridge University Press, Includes bibliographical references and index. Identifiers: LCCN ISBN (hardback) Subjects: LCSH: Quantum field theory. Particles (Nuclear physics) Graph theory. Combinatorial analysis. Geometry, Algebraic. Classification: LCC QC G DDC 516.3/5 dc23 LC record available at ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Contents Acknowledgments page ix 1 Introduction 1 2 Introduction to on-shell functions and diagrams On-shell particles, functions, and kinematical data Scattering amplitudes and their amalgamations On-shell building blocks: the massless three-particle S-matrix On-shell supersymmetry and (maximally) supersymmetric theories Building up diagrams with BCFW-bridges On-shell recursion for all-loop amplitudes Physical equivalences among on-shell diagrams 23 3 Permutations and scattering amplitudes Combinatorial descriptions of scattering processes The BCFW-bridge construction of representative graphs 33 4 From on-shell diagrams to the Grassmannian The Grassmannian of k-planes in n dimensions, G(k,n) Grassmannian description of kinematical data λ and λ Grassmannian representation of on-shell diagrams Amalgamation of on-shell diagrams Boundary measurements and canonical coordinates Coordinate transformations induced by moves and reduction Relation to composite leading singularities 56 5 Configurations of vectors and the positive Grassmannian The geometry and combinatorics of the positroid stratification Canonical coordinates and the equivalence of permutation labeling 66 v

6 vi Contents 5.3 Positroid cells and the positive part of the Grassmannian Canonically positive coordinates for positroids 73 6 Boundary configurations, graphs, and permutations Physical singularities and positroid boundaries Boundary configurations: combinatorics and stratification (Combinatorial) polytopes in the Grassmannian Approaching boundaries in canonical coordinates 81 7 The invariant top-form and the positroid stratification Equivalence with the canonical positroid volume form 85 8 (Super-)conformal and dual conformal invariance The Grassmannian geometry of momentum conservation Twistor space and the superconformality of on-shell forms Momentum-twistors and dual super-conformal invariance 92 9 Positive diffeomorphisms and Yangian invariance The kinematical support of physical on-shell forms Kinematical support of NMHV Yangian-invariants Kinematical support for one-dimensional kinematics A general combinatorial test of kinematical support Homological identities among Yangian-invariants Homological identities in the Grassmannian (Relatively) orienting canonical coordinate charts on positroids configurations Comparing the orientations of canonical coordinate charts Accessing boundary configurations from the canonical atlas Classification of Yangian-invariants and their relations The Yang Baxter relation and ABJM theories The on-shell avatar of the Yang Baxter relation ABJM theories On-shell diagrams for theories with N < 4 supersymmetries Dual graphs and cluster algebras The dual of an on-shell diagram Cluster algebras: seeds, mutations, and cluster coordinates Cluster amalgamation (Brief) overview of cluster structures in physics On-shell representations of scattering amplitudes (Diagrammatic) proof of the BCFW recursion relations The structure of (tree) amplitudes in the Grassmannian 159

7 Contents vii 17.3 Canonical coordinates for loop integrands The transcendentality of loop amplitudes Outlook 178 References 183 Index 192

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9 Acknowledgments This book represents the synthesis of several years of work and was originally made available on the arxiv as reference [1]. There are many people without whom this book would not have been possible, or without whose comments would not be nearly as comprehensive as we hope that it is. Many of the mathematical topics discussed were greatly improved through early discussions with Pierre Deligne, Bob MacPherson, and Mark Goresky at the Institute for Advanced Study; and for the physics topics discussed, we owe an enormous debt of gratitude for discussions with Andrew Hodges and Simon Caron-Huot. For helpful comments regarding early versions of the manuscript, we are grateful to Simon Caron-Huot, Henriette Elvang, Sebastian Franco, Yu-tin Huang, Lionel Mason, Marcus Spradlin, Cristian Vergu, Congkao Wen, and Lauren Williams. This work was made possible through the hospitality of the School of Natural Sciences at the Institute for Advanced Study, and was supported in part by grants from the National Science Foundation DMS (A.P.), DMS (A.G.), and PHY (J.T.), the Harvard Society of Fellows and a grant from the Harvard Milton Fund (J.B.), the NSERC of Canada and MEDT of Ontario (F.C.), and U.S. Department of Energy contracts DE-FG02-91ER40654 (N.A.-H.) and DE-FG02-91ER40654 (J.B.). The authors are also grateful to the Aspen Public Library for their hospitality. ix