COU R A N T V. S . V A R A D A R A J A N LECTURE NOTES Supersymmetry for Mathematicians: An Introduction American Mathematical Society AMS Courant Institute of Mathematical Sciences A...,. A i N., I I. I M A. 1.11.... 11 11 Supersymmetry for Mathematicians: An Introduction Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Assistant Editor Reeva Goldsmith Copy Editor Will Klump V. S. Varadarajan University of California, Los Angeles 11 Supersymmetry for Mathematicians: An Introduction Courant Institute of Mathematical Sciences New York University New York, New York American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 58A50, 58C50, 17B70, 22E99; Secondary 14A22, 14M30, 32C11. For additional information and updates on this book, visit www.ams.org/bookpages/cln-11 Library of Congress Cataloging-in-Publication Data Varadarajan, V. S. Supersymmetry for mathematicians: an introduction/V. S. Varadarajan. p. cm. (Courant lecture notes; 11) Includes bibliographical references. ISBN 0-8218-3574-2 (alk. paper) 1. Supersymmetry.I. Title.II. Series. QA174.17.S9V372004 539.7'25-dc22 2004052349 Copying and reprinting.Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint -permission®ams.org. © 2004 by the author. 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Visit the AMS home page at http://www.ams.org/ 10987654321 090807060504 Contents Preface vii Chapter 1.Introduction 1 1.1.Introductory Remarks on Supersymmetry 1 1.2.Classical Mechanics and the Electromagnetic and Gravitational Fields 2 1.3.Principles of Quantum Mechanics 9 1.4.Symmetries and Projective Unitary Representations 17 1.5.Poincare Symmetry and Particle Classification 23 1.6.Vector Bundles and Wave Equations: The Maxwell, Dirac, and Weyl Equations 37 1.7.Bosons and Fermions 48 1.8.Suipersymmetry as the Symmetry of a Z2-Graded Geometry 51 1.9.References 52 Chapter 2.The Concept of a Supermanifold 59 2.1.Geometry of Physical Space 59 2.2.Riemann's Inaugural Talk 63 2.3.Einstein and the Geometry of Spacetime 66 2.4.Mathematical Evolution of the Concept of Space and Its Symmetries 67 2.5.Geometry and Algebra 72 2.6.A Brief Look Ahead 76 2.7.References 79 Chapter 3.Super Linear Algebra 83 3.1.The Category of Super Vector Spaces 83 3.2.The Super Poincare Algebra of Gol'fand-Likhtman and Volkov- Akulov 90 3.3.Conformal Spacetime 95 3.4.The Superconformal Algebra of Wess and Zumino 108 3.5.Modules over a Supercommutative Superalgebra 113 3.6.The Berezinian (Superdeterminant) 116 3.7.The Categorical Point of View 119 3.8.References 124 Chapter 4.Elementary Theory of Supermanifolds 127 4.1.The Category of Ringed Spaces 127 V Vi CONTENTS 4.2.Supermanifolds 130 4.3.Morphisms 138 4.4.Differential Calculus 143 4.5.Functor of Points 150 4.6.Integration on Supermanifolds 152 4.7.Submanifolds: Theorem of Frobenius 157 4.8.References 167 Chapter 5.Clifford Algebras, Spin Groups, and Spin Representations 169 5.1.Prologue 169 5.2.Cartan's Theorem on Reflections in Orthogonal Groups 174 5.3.Clifford Algebras and Their Representations 178 5.4.Spin Groups and Spin Representations 192 5.5.Spin Representations as Clifford Modules 203 5.6.References 208 Chapter 6.Fine Structure of Spin Modules 211 6.1.Introduction 211 6.2.The Central Simple Superalgebras 212 6.3.The Super Brauer Group of a Field 222 6.4.Real Clifford Modules 227 6.5.Invariant Forms 236 6.6.Morphisms from Spin Modules to Vectors and Exterior Tensors 250 6.7.The Minkowski Signature and Extended Supersymmetry 256 6.8.Image of the Real Spin Group in the Complex Spin Module 262 6.9.References 272 Chapter 7.Superspacetimes and Super Poincare Groups 273 7.1.Super Lie Groups and Their Super Lie Algebras 273 7.2.The Poincare-Birkhoff-Wilt Theorem 279 7.3.The Classical Series of Super Lie Algebras and Groups 289 7.4.Superspacetimes 294 7.5.Super Poincare Groups 299 7.6.References 299 Preface These notes are essentially the contents of a minicourse I gave at the Courant Institute in the fall of 2002. I have expanded the lectures by discussing spinors at greater length and by including treatments of integration theory and the local Frobenius theorem, but otherwise have not altered the plan of the course. My aim was and is to give an introduction to some of the mathematical aspects of supersymmetry with occasional physical motivation. I do not discuss supergravity. Not much is original in these notes. I have drawn freely and heavily from the beautiful exposition of P. Deligne and J. Morgan, which is part of the AMS volumes on quantum field theory and strings for mathematicians, and from the books and articles of D. S. Freed and D. A. Leites, all of which and more are referred to in the introduction. I have profited greatly from the lectures that Professor S. Ferrara gave at UCLA as well as from many extended conversations with him, both at UCLA and at CERN, where I spent a month in 2001. He introduced me to this part of math- ematical physics and was a guide and participant on a semintr on supersymmetry that I ran in UCLA in 2000 with Rita Fioresi. I am deeply grateful to him for his unfailing patience and courtesy. I also gave a course in UCLA and a miniwork- shop on supersymmetry in 2000 in Genoa, Italy, in the Istituto Nazionale di Fisica Nucleare. I am very grateful to Professors E. Beltrametti and G. Cassinelli, who ar- ranged that visit; to Paolo Aniello, who made notes of my UCLA course; to Ernesto De Vito and Alberto Levrero, whose enthusiasm and energy made the Genoa work- shop so memorable; and finally to Lauren Caston, who participated in the Courant course with great energy and enthusiasm. I also wish to thank Alessandro Toigo and Claudio Carmeli of INFN, Genoa, who worked through the entire manuscript and furnished me with a list of errors and misprints in the original version of the notes, and whose infectious enthusiasm lifted my spirits in the last stages of this work. I am very grateful to Julie Honig for her help during all stages of this work. Last, but not least, I wish to record my special thanks to Paul Monsour and Reeva Goldsmith whose tremendous effort in preparing and editing the manuscript has made this book enormously better than what it was when I sent it to them. The coursein the Courant Institute was givenatthe suggestion of Professor S. R. S. Varadhan. My visit came at a time of mourning and tragedy for him in the aftermath of the 9/11 catastrophe, and I do not know how he found the time and energy to take care of all of us. It was a very special time for us and for him, and in my mind this course and these notes will always appear as a small effort on my part to alleviate the pain and grief by thinking about some beautiful things that are far bigger than ourselves. V. S. Varadarajan Pacific Palisades March 2004 vii CHAPTER 1 Introduction 1.1. Introductory Remarks on Supersymmetry The subject of supersymmetry (SUSY) is a part of the theory of elementary particles and their interactions and the still unfinished quest of obtaining a unified view of all the elementary forces in a manner compatible with quantum theory and general relativity. Supersymmetry was discovered in the early 1970s and in the intervening years has become a major component of theoretical physics. Its novel mathematical features have led to a deeper understanding of the geometrical structure of spacetime, a theme to which great thinkers like Riemann, Poincare, Einstein, Weyl, and many others have contributed. Symmetry has always played a fundamental role in quantum theory: rotational symmetry in the theory of spin, Poincare symmetry in the classification of ele- mentary particles, and permutation symmetry in the treatment of systems of iden- tical particles. Supersymmetry is a new kind of symmetry that was discovered by physicists in the early 1970s. However, it is different from all other discoveries in physics in the sense that there has been no experimental evidence supporting it so far. Nevertheless, an enormous effort has been expended by many physicists in developing it because of its many unique features and also because of its beauty and coherence.' Here are some of its salient features:2 It gives rise to symmetries between bosons and fermions at a fundamental level. Supersymmetric quantum field theories have "softer" divergences. Supersymmetric string theory (superstrings) offers the best context known so far for constructing unified field theories. The development of supersymmetry has led to a number of remarkable pre- dictions. One of the most striking of these is that every elementary particle has a SUSY partner of opposite spin parity, i.e., if the particle is a boson (resp., fermion), its partner is a fermion (resp., boson).