Supersymmetry fo r 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 Supersymmetr y for Mathematicians : An Introduction

Courant Institute of Mathematical Sciences New York University New York, New York

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/cln/011

2000 Mathematics Subject Classification. Primar y 58A50 , 58C50 , 17B70 , 22E99 ; Secondary 14A22 , 14M30 , 32C11 .

For additiona l informatio n an d updates o n this book , visi t www.ams.org/bookpages/cln-ll

Library o f Congres s Cataloging-in-Publicatio n D a t a

Varadarajan, V . S. Supersymmetry fo r mathematicians : a n introduction/V . S . Varadarajan .

p. cm . (Couran t lectur e notes ; 11) Includes bibliographica l references .

ISBN 0-8218-3574- 2 (alk . paper ) 1. Supersymmetry . I . Title . II . Series .

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Contents

Preface vi i

Chapter 1 . Introductio n 1 1.1. Introductor y Remarks on Supersymmetry 1 1.2. Classica l Mechanics an d the Electromagnetic an d Gravitationa l

Fields 2 1.3. Principle s of Quantum Mechanics 9 1.4. Symmetrie s and Projective Unitary Representations 1 7 1.5. Poincar e Symmetry and Particle Classification 2 3 1.6. Vecto r Bundles and Wave Equations:

The Maxwell, Dirac, and Weyl Equations 3 7 1.7. Boson s and Fermions 4 8 1.8. Supersymmetr y as the Symmetry of a Z2-Graded Geometry 5 1 1.9. Reference s 5 2

Chapter 2. Th e Concept of a Supermanifold 5 9 2.1. Geometr y of Physical Space 5 9 2.2. Riemann' s Inaugural Talk 6 3 2.3. Einstei n and the Geometry of Space time 6 6 2.4. Mathematica l Evolution of the Concept of Space and Its Symmetries 6 7 2.5. Geometr y and Algebra 7 2 2.6. A Brief Look Ahead 7 6 2.7. Reference s 7 9

Chapter 3. Supe r Linear Algebra 8 3 3.1. Th e Category of Super Vector Spaces 8 3 3.2. Th e Super Poincare Algebra of Gol'fand-Likhtman an d Volkov-

Akulov 9 0 3.3. Conforma l Spacetim e 9 5 3.4. Th e Superconformal Algebr a of Wess and Zumino 10 8 3.5. Module s over a Supercommutative Superalgebra 11 3 3.6. Th e Berezinian (Superdeterminant) 11 6 3.7. Th e Categorical Point of View 11 9 3.8. Reference s 12 4

Chapter 4. Elementar y Theory of Supermanifolds 12 7 4.1. Th e Category of Ringed Spaces 12 7

vi CONTENT S

4.2. Supermanifold s 13 0 4.3. Morphism s 13 8 4.4. Differentia l Calculu s 14 3 4.5. Functo r of Points 15 0 4.6. Integratio n on Supermanifolds 15 2 4.7. Submanifolds : Theore m of Frobenius 15 7 4.8. Reference s 16 7

Chapter 5. Cliffor d Algebras , Spin Groups, and Spin Representations 16 9 5.1. Prologu e 16 9 5.2. Cartan' s Theorem on Reflections in Orthogonal Groups 17 4 5.3. Cliffor d Algebra s and Their Representations 17 8 5.4. Spi n Groups and Spin Representations 19 2 5.5. Spi n Representations as Clifford Module s 20 3 5.6. Reference s 20 8

Chapter 6. Fin e Structure of Spin Modules 21 1 6.1. Introductio n 21 1 6.2. Th e Central Simple Superalgebras 21 2 6.3. Th e Super Brauer Group of a Field 22 2 6.4. Rea l Clifford Module s 22 7 6.5. Invarian t Forms 23 6 6.6. Morphism s from Spin Modules to Vectors and Exterior Tensors 25 0 6.7. Th e Minkowski Signature and Extended Supersymmetry 25 6 6.8. Imag e of the Real Spin Group in the Complex Spin Module 26 2 6.9. Reference s 27 2

Chapter 7. Superspacetime s and Super Poincare Groups 27 3 7.1. Supe r Lie Groups and Their Super Lie Algebras 27 3 7.2. Th e Poincare-Birkhoff-Witt Theore m 27 9 7.3. Th e Classical Series of Super Lie Algebras and Groups 28 9 7.4. Superspacetime s 29 4 7.5. Supe r Poincare Groups 29 9 7.6. Reference s 29 9

Preface

These notes are essentially the contents of a minicourse I gave at the Courant Institute i n the fal l o f 2002 . I have expanded th e lectures b y discussin g spinor s at greate r lengt h an d by including treatment s o f integratio n theor y an d th e loca l Frobenius theorem , bu t otherwis e hav e no t altere d th e pla n o f th e course . M y aim wa s an d i s t o giv e a n introductio n t o som e o f th e mathematica l aspect s o f supersymmetry with occasional physical motivation. I do not discuss supergravity.

Not much is original in these notes. I have drawn freely an d heavily from th e beautiful exposition of P. Deligne and J. Morgan, which is part of the AMS volumes on quantum field theory and strings for mathematicians , an d from th e 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 wel l a s fro m man y extende d conversation s wit h him , bot h a t UCL A an d a t CERN, wher e I spen t a month i n 2001 . H e introduced m e to this part o f math -ematical physics and was a guide and participant on a seminar on supersymmetry that I ran in UCLA in 2000 with Rita Fioresi. I am deeply grateful t o him for hi s unfailing patienc e an d courtesy. I also gave a course in UCLA an d 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 manuscrip t and furnished m e with a list of errors and misprints i n the original version o f the notes, and whose infectious enthusias m lifted m y spirit s in the last stages of thi s work. I am very grateful t o 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 whos e tremendous effor t i n preparing an d editing th e manuscrip t ha s made this book enormously better than what it was when I sent it to them.

The cours e i n th e Couran t Institut e wa s give n a t th e suggestio n o f Professor S . R. S . Varadhan. M y visi t cam e a t a time o f mournin g an d traged y for him in the aftermath o f the 9/11 catastrophe, and I do not know how he found the time and energy to take care of all of us. I t 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 o n my part to alleviate the pain and grief by thinking abou t some beautifu l things that are far bigger than ourselves.

V. S. Varadarajan Pacific Palisades March 2004

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Titles i n Thi s Serie s

Volume

11 V . S . Varadaraja n

Supersymmetry fo r mathematicians : A n introductio n

2004

10 Thierr y Cazenav e

Semilinear Schrodinge r equation s

2003

9 Andre w Majd a

Introduction t o PDE s an d wave s fo r th e atmospher e an d ocea n

2003

8 Fedo r Bogomolo v an d Tihomi r Petro v

Algebraic curve s an d one-dimensiona l fields

2003

7 S . R . S . Varadha n

Probability theor y

2001

6 Loui s Nirenber g

Topics i n nonlinea r functiona l analysi s

2001

5 Emmanue l Hebe y

Nonlinear analysi s o n manifolds : Sobole v space s an d inequalitie s

2000

3 Perc y Deif t

Orthogonal polynomial s an d rando m matrices : A Riemann-Hilber t approac h

2000

2 Jala l Shata h an d Michae l Struw e

Geometric wav e equation s

2000

1 Qin g Ha n an d Fanghu a Li n

Elliptic partia l differentia l equation s

2000