Ideas and methods of supersymmetry and supergravity, or, A walk through superspace

Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace Ioseph L Buchbinder and Sergei M Kuzenk...

Author: Ioseph L. Buchbinder and Sergei M. Kuzenko

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Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace Ioseph L Buchbinder and Sergei M Kuzenko Tomsk State University, Russia

Revised Edition

Institute of Physics Publishing Bristol and Philadelphia

Copyright © 1998 IOP Publishing Ltd

@ IOP Publishing Ltd 1995, 1998

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms o f licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. First hardback edition 1995 Revised (paperback) edition 1998

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ISBN 0 7503 0506 1 Library of Congress Cataloging-in-Publication Data are available

Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BSI 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in the UK by P & R Typesetters Ltd Printed in the UK by J W Arrowsmiths Ltd, Bristol Copyright © 1998 IOP Publishing Ltd

Contents Preface to the First Edition Preface to the Revised Edition

1 Mathematical Background I.1

The Poincart group, the Lorentz group 1.1.1 Definitions 1.1.2 Useful decomposition in SO(3, I ) ? 1.1.3 Universal covering group of the Lorentz group 1.1.4 Universal covering group of the PoincarC group

xix 1 1 1 3 4 6

1.2 Finite-dimensional representations of Spin(3, 1) 1.2.1 Connection between representations of SO(3, l ) ? and SL(2, (0 1.2.2 Construction of SL(2, C) irreducible representations 1.2.3 Invariant Lorentz tensors

11

1.3 The Lorentz algebra

13

1.4 Two-component and four-component spinors 1.4.1 Two-component spinors 1.4.2 Dirac spinors 1.4.3 Weyl spinors 1.4.4 Majorana spinors 1.4.5 The reduction rule and the Fierz identity 1.4.6 Two-component and four-component bi-linear combinations 1.5 Representations of the Poincark group 1.5.1 The Poincart algebra 1S . 2 Field representations 1S . 3 Unitary representations 1 S.4 Stability subgroup 1 S.5 Massive irreducible representations 1S . 6 Massless irreducible representations

17 18 19 21 21 22

1.6 Elements of differential geometry and gravity

32

7 7 8

23 23 23 26 26 26 27 30

V


vi

Cori teri ts 1.6.1 1.6.2 1.6.3 1.6.4 1.6.5 1.6.6 1.6.7 1.6.8 1.6.9

Lorentz manifolds Covariant differentiation of world tcnsors Covariant differentiation of the Lorentz tensor Frame deform at i on s The Weyl tensor Four-dimensional topological invariants . gravity and conformal gravity E:Instein Energy-momentum tensor The covariant derivatives algebra in spinor notation

32 35 36 38 39 40 41 43 44

1.7 The conformal group 1.7.1 Conformal Killing vectors 1.7.2 Conformal Killing vectors in Minkowski space I .7.3 The conformal algebra 1.7.4 Conformal transformations 1.7.5 Matrix realization of the conformal group I .7.6 Conformal invariance 1.7.7 Examples of conformally invariant theories 1.7.8 Example of a non-conformal massless theory

45 4.5 46 47 48 50 51 53 55

1.8 The mass-shell field representations 1.8.1 Massive field representations of the Poincar6 group 1.8.2 Real massive field representations 1.8.3 Massless field representations of the PoincarC group 1.8.4 Examples of massless fields 1.8.5 Massless field representations of the conformal group

56 56 59 61 62 66

1.9 Elements of algebra with supernumbers 1.9.1 Grassmann algebras AN and Am 1.9.2 Supervector spaces 1.9.3 Finite-dimensional supervector spaces 1.9.4 Linear operators and supermatrices 1.9.5 Dual supervector spaces, supertransposition 1.9.6 Bi-linear forms

68 70 72 74 77 85 88

1. I O Elements of analysis with supernumbers

1.10.1 1.10.2 1.10.3 1.10.4

Superfunctions Integration over R”q Linear replacements of variables on C-type supermatrices revisited

RPlq

1.1 1 The supergroup of general coordinate transformations on R”lY 1.1 1.1 The exponential form for general coordinate transformations 1.1 1.2 The operators K and K 1.1 1.3 Theorem Copyright © 1998 IOP Publishing Ltd

91 91 96 101 103 106 107 109 110

Contents

I . 1 1.4 The transformation law fhr the volume elcment on JR’’l‘/ I . I I .5 Basic properties of integration theory over IRPI’Y

2 Supersymmetry and Superspace

vii i12 I I3

117

2.0

Introduction: from It/’’‘’ to supersymmetry

I I7

2. I

Superalgebras, Grassmann shells and super Lie groups 2. I . 1 Superalgebras 2 . I .2 Examples of superalgebras 2.1.3 The Grassmann shell of a superalgebra 2.1.4 Examples of Berezin superalgebras and super Lie algebras 2.1S Representations of (Berezin) superalgebras and super Lie algebras 2. I .6 Super Lie groups 2.1.7 Unitary representations of real superalgebras

121 122 124 125

The PoincarC superalgebra 2.2.1 Uniqueness of the N = I Poincare superalgebra 2.2.2 Extended Poincart superalgebras 2.2.3 Matrix realization of the PoincarC superalgebra 2.2.4 Grassmann shell of the PoincarC superalgebra 2.2.5 The super PoincarC group

138 138 141 143 144 145

2.3 Unitary representation of the PoincarC superalgebra 2.3. I Positivity of energy 2.3.2 Casimir operators of the PoincarC superalgebra 2.3.3 Massive irreducible representations 2.3.4 Massless irreducible representations 2.3.5 Superhelicity 2.3.6 Equality of bosonic and fermionic degrees of freedom

146 146 147 149 152 153 154

2.4

155 155 157 160

2.2

2.5

Real superspace R414 and superfields 2.4.1 Minkowski space as the coset space n / S 0 ( 3 , I ) t 2.4.2 Real superspace R414 2.4.3 Supersymmetric interval 2.4.4 Superfields 2.4.5 Superfield representations of the super PoincarC group Complex superspace U?’?, chiral superfields and covariant derivatives 2.5.1 Complex superspace C4I2 2.5.2 Holomorphic superfields 2.5.3 R414 as a surface in C4I2 2.5.4 Chiral superfields 2.5.5 Covariant derivatives 2.5.6 Properties of covariant derivatives


128 132 135 137

160

162

165 166 167 168 169 170 171

I..

Cot1tet1ts

Vlll

2.6 The on-shell massive superfield representations 2.6.1 On-shell massive superfields 2.6.2 Extended super Poincari: algebra 2.6.3 The superspin operator 2.6.4 Decomposition of ?Xi:',B) int.0 irreducible representations 2.6.5 Projection operators 2.6.6 Real representations

172 173 174 175 176 179 179

2.7 The on-shell massless superfield representations 2.7.1 Consistency conditions 2.7.2 On-shell massless superfields 2.7.3 Superhelicity

181

181 182 185

From superfields to component fields 2.8. I Chiral scalar superfield 2.8.2 Chiral tensor superfield of Lorentz type ( n / 2 , 0 ) 2.8.4 Linear real scalar superfield

186 186 188 191

2.9 The superconformal group 2.9.1 Superconformal transformations 2.9.2 The supersymmetric interval and superconformal transformations 2.9.3 The superconformal algebra

191 192

2.8

3 Field Theory in Superspace 3.1

194 195

198 198 198

Supersymmetric field theory 3. I . 1 Quick review of field theory 3.1.2 The space of superfield histories; the action superfunctional 3.1.3 Integration over R"' and superfunctional derivatives 3.1.4 Local supersymmetric field theories 3.1.5 Mass dimensions 3. I .6 Chiral representation

20 1 202 208 21 1 21 1

3.2 Wess-Zumino model 3.2.1 Massive chiral scalar superfield model 3.2.2 Massless chiral scalar superfield model 3.2.3 Wess-Zumino model 3.2.4 Wess-Zumino model in component form 3.2.5 Auxiliary fields 3.2.6 Wess-Zumino model after auxiliary field elimination 3.2.7 Generalization of the model

213 213 215 215 216 2 I7 218 220

3.3 Supersymmetric nonlinear sigma-models 3.3.1 Four-dimensional cr-models 3.3.2 Supersymmetric cr -models 3.3.3 Kahler manifolds

22 1 22 1 222 224


Contents

ix

3.3.4

Kahler geometry and supersymmetric o-models

226

3.4

Vector 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6

multiplet models Massive vector multiplet model Massless vector multiplet model Wess-Zum ino gauge Supersymmetry transformations Super Lorentz gauge Massive vector multiplet model revisited

228 228 230 230 232 233 233

3.5

Supersymmetric Yang-Mills theories 3.5. I Supersymmetric scalar electrodynamics 3.5.2 Supersymmetric spinor electrodynamics 3 3 . 3 Non-Abelian gauge superfield 3.5.4 Infinitesimal gauge transformations 3.5.5 Super Yang-Mills action 3.5.6 Super Yang-Mills models 3.5.7 Real representation

236 236 239 240 24 1 243 245 246

3.6 Geometric approach to super Yang-Mills theories 3.6.1 Complex and c-number shells of compact Lie groups 3.6.2 K-supergroup and A-supergroup 3.6.3 Gauge superfield 3.6.4 Gauge covariant derivatives 3.6.5 Matter equations of motion 3.6.6 Gauge superfield dynamical equations

247 247 249 25 1 25 3 255 257

3.7

259 259 26 1 263

Classically equivalent theories 3.7.1 Massive chiral spinor superfield model 3.7.2 Massless chiral spinor superfield model 3.7.3 Superfield redefinitions

3.8 Non-minimal scalar multiplet 3.8.1 Coinplcx linear scalar superfield 3.8.2 Free non-minimal scalar multiplet 3.8.3 Mass generation I 3.8.4 Mass generation I1 3.8.5 Supersymmetric electrodynamics 3.8.6 Couplings to Yang-Mills superfields 3.8.7 Nonlinear sigma models

I Quantized Superfields 4.1

Picture-change operators 4. I . 1 Functional supermatrices 4.1.2 Superfunctional supermatrices 4. I .3 (Super) functional derivatives 4.1.4 Picture-change operators


264 265 266 268 269 270 27 3 273 275

275 276 278 283 285

X

Corrtetits

4.2 Equivalence o f component field and superfield perturbation theorics 4.2.1 (Super) field Green’s functions 4.2.2 Generating functional 4.2.3 Generating superfunctional 4.2.4 Coincidence of Z [ J ]and Z[Jl

289 289 292 294 296

4.3 Effective action (super) functional 4.3.1 Effective action 4.3.2 Super Poincart invariance of W[J] and r[G] 4.3.3 Short excursion into renormalization theory 4.3.4 Finite pathological supersymmetric theories

29 8 299 302 305 307

4.4 The Wess-Zumino model: perturbative analysis 4.4.1 Preliminary discussion 4.4.2 Feynman superpropagator 4.4.3 Generating superfunctional 4.4.4 Standard Feynman rules 4.4.5 Improved Feynman rules 4.4.6 Example of supergraph calculations 4.4.7 Supersymmetric analytic regularization 4.4.8 Non-renormalization theorem

309 309 31 1 313 316 319 324 325 326

4.5

Note about gauge theories 4.5.1 Gauge theories 4.5.2 Feynman rules for irreducible gauge theories with closed algebras 4.5.3 Supersymmetric gauge theories

328 329

Feynman rules for super Yang-Mills theories 4.6.1 Quantization of the pure super Yang-Mills model 4.6.2 Propagators and vertices 4.6.3 Feynman rules for general super Yang-Mills models 4.6.4 Non-renormalization theorem

340 340 343 346 350

4.6

332 337

4.7 Renormalization 4.7.1 Superficial degree of divergence 4.7.2 Structure of counterterms 4.7.3 Questions of regularization

350 350 353 357

4.8 Examples of counterterm calculations: an alternative technique 4.8.1 One-loop counterterms of matter in an external super Yang-Mills field 4.8.2 One-loop counterterms of the general Wess-Zum ino model

36 1

4.9

36 1 366

Superfield effective potential 37 1 4.9.1 Effective potential in quantum field theory (brief survey) 37 1


Contents 4.9.2 4.9.3 4.9.4 4.9.5

Superfield effective potential Superfield effectivc potential in the Wess-Zumino model Calculation of the one-loop Kahlerian effective potential Calculation of the two-loop efrective chiral superpotential

5 Superspace Geometry of Supergravity 5.1

5.2

Gauge 5.1.1 5.1.2 5 . I .3 5.1.4 5.1.5 5.1.6

group of supergravity and supergravity fields Curved superspace Conformal supergravity Einstein supergravity Einstein supergravity (second formulation) Einstein supergravity multiplet Flat superspace (final definition) and conformally flat superspace

xi 374 376 379 382

386 386 386 390 396 397 398 399

Superspace differential geometry 402 5.2.1 Superfield representations of the general coordinate transformation supergroup 403 5.2.2 The general coordinate transformation supergroup in 405 exponential form 406 5.2.3 Tangent and cotangent supervector spaces 407 5.2.4 Supervierbein 408 5.2.5 Superlocal Lorentz group 410 5.2.6 Superconnection and covariant derivatives 412 5.2.7 Bianchi identities and the Dragon theorem 413 5.2.8 Integration by parts 414 5.2.9 Flat superspace geometry

5.3 Supergeometry with conformal supergravity constraints 5.3.1 Conformal supergravity constraints 5.3.2 The Bianchi identities 5.3.3 Solution to the dim = 1 Bianchi identities 5.3.4 Solution to the dim = 3/2 Bianchi identities 5.3.5 Solution to the dim = 2 Bianchi identities 5.3.6 Algebra of covariant derivatives 5.3.7 Covariantly chiral tensor superfields 5.3.8 Generalized super Weyl transformations

416 416 420 423 424 426 426 427 427

5.4

428 429 430 432 433 435 436 438

Prepotentials 5.4.1 Solution to constraints (5.3.15a) 5.4.2 Useful gauges on the superlocal Lorentz group 5.4.3 The A-supergroup 5.4.4 Expressions for E , T, and R 5.4.5 Gauge fixing for the K - and A-supergroups 5.4.6 Chiral representation 5.4.7 Gravitational superfield


xii

Cot1trrrts

5.4.8

5.5

Gauge fixing on the generalized super Weyl group

Einstein supergravity 5.5.1 Einstein supergravity constraints 5.5.2 Chiral compensator 5.5.3 Minimal algebra of covariant derivatives 5.5.4 Super Weyl transformations 5.5.5 Integration by parts 5.5.6 Chiral integration rule 5.5.7 Matter dynamical systems in a supergravity background

44 1 442 442 442 443 445 446 446 448

5.6 Prepotential deformations 5.6.1 Modified parametrization of prepotentials 5.6.2 Background-quantum splitting 5.6.3 Background-quantum splitting in Einstein supergravity 5.6.4 First-order expressions 5.6.5 Topological invariants

450 450 454 460 46 1 453

5.7

465 465 468 469 47 1 472

Supercurrent and supertrace 5.7.1 Basic construction 5.7.2 The relation with ordinary currents 5.7.3 The supercurrent and the supertrace in flat superspace 5.7.4 Super Weyl invariant models 5.7.5 Example

473 5.8 Supergravity in components 474 5.8.1 Space projection of covariant derivatives 47 8 5.8.2 Space projections of R , and G, 479 5.8.3 Basic construction 483 5.8.4 Algebraic structure of the curvature with torsion -7- 485 , , ~%(aWpys) , 5.8.5 Space projections of 9 - R , 9 ~ ~ 9 c v Gand 5.8.6 Component fields and local supersymmetry 486 transformation laws 488 5.8.7 From superfield action to component action 6 Dynamics in Supergravity

49 1

6.1 Pure supergravity dynamics 6.1. I Einstein supergravity action superfunctional 6.1.2 Supergravity dynamical equations 6.1.3 Einstein supergravity action functional 6.1.4 Supergravity with a cosmological term 6.1.5 Conformal supergravity 6. I .6 Renormalizable supergravity models 6.1.7 Pathological supergravity model

49 1 492 49 3 494 496 498 498 499

6.2 Linearized supergravity

500


Contents 6.2.1 6.2.2 6.2.3 6.3

6.4

Linearized Einstein supergravity action Linearized superfield strengths and dynamical equations Linearized conformal supergravity

Supergravity-matter dynamical equations 6.3.1 Chiral scalar models 6.3.2 Vector multiplet models 6.3.3 Super Yang-Mills models 6.3.4 Chiral spinor model (Conformal) Killing supervectors. Superconformal models 6.4.1 (Conformal) Killing supervector fields 6.4.2 The gravitational superfield and conformal Killing supervec tors 6.4.3 (Conformal) Killing supervectors in flat global superspace 6.4.4 Superconformal models 6.4.5 On-shell massless conformal superfields

xiii 500 503 5 04 505

5 06

509 51 1 5 15 517 518 522 523 5 25 528

6.5 Conformally flat superspaces, anti-de Sitter superspace 6.5.1 Flat superspace 6.5.2 Conformally flat superspace 6.5.3 Physical sense of conformal flatness 6.5.4 Anti-de Sitter superspace 6.5.5 Killing supervectors of anti-de Sitter superspace

53 1 53 1 533 535 536 537

6.6 Non-minimal supergravity 6.6.1 Preliminary discussion 6.6.2 Complex linear compensator 6.6.3 Non-minimal supergeometry 6.6.4 Dynamics in non-minimal supergravity 6.6.5 Prepotentials and field content in non-minimal supergravity 6.6.6 Geometrical approach to non-minimal supergravity 6.6.7 Linearized non-minimal supergravity

539 539 54 1 543 545

6.7 New minimal supergravity 6.7.1 Real linear compensator 6.7.2 Dynamics in new minimal supergravity 6.7.3 Gauge fixing and fieldcontent in new minimal supergravity 6.7.4 Linearized new minimal supergravity

55 1 55 1 554 556 559

6.8

546 547 550

Matter coupling in non-minimal and new minimal supergravities 560 6.8.1 Non-minimal chiral compensator 5 60 6.8.2 Matter dynamical systems in a non-minimal supergravity background 563


xiv

Cor1tents 6.8.3

New minimal supergravity and supersymmetric a-models 564

6.9 Free massless higher superspin theories 6.9.1 Free massless theories of higher 6.9.2 Free massless theories of higher 6.9.3 Free massless theories of higher 6.9.4 Free massless theories of higher 6.9.5 Massless gravitino multiplet 7

integer spins half-integer spins half-integer superspins integer superspins

565 566 570 573 577 5 80

Effective Action in Curved Superspace

584

7.1 The Schwinger-De Witt technique 7. I. 1 When the proper-time technique can be applied 7.1.2 Schwinger’s kernel 7.1.3 One-loop divergences of effective action 7. I .4 Conformal anomaly 7.1.5 The coefficients a l ( x , x ) and u z ( x , x )

5 84 5 84 5 86 590 593 597

7.2 Proper-time representation for covariantly chiral scalar superpropagator 7.2.1 Basic chiral model 7.2.2 Covariantly chiral Feynman superpropagator 7.2.3 The chiral d’Alembertian 7.2.4 Covariantly chiral Schwinger’s superkernel 7.2.5 af(z,z ) and a!j(z,z ) 7.2.6 One-loop divergences 7.2.7 Switching on an external Yang-Mills superfield

600 60 1 604 606 608 610 61 1 612

7.3 Proper-time representation for scalar superpropagators 7.3.1 Quantization of the massless vector multiplet model 7.3.2 Connection between GL’’ and G(’) 7.3.3 Scalar Schwinger’s kernel 7.3.4 Divergences of effective action

613 613 616 617 619

7.4

619 620 623

Super Weyl anomaly 7.4.1 Super Weyl anomaly in a massless chiral scalar model 7.4.2 Anomalous effective action 7.4.3 Solution of effective equations of motion in conformally flat superspace

626

7.5 Quantum equivalence in superspace 7.5.1 Problem of quantum equivalence 7.5.2 Gauge antisymmetric tensor field 7.5.3 Quantization of the chiral spinor model 7.5.4 Analysis of quantum equivalence

627 627 630 633 637

Bibliography

640

Index

644


Preface to the First Edition

The discovery of supersymmetry is one of the most distinguished achievements of theoretical physics in the second half of the twentieth century. The fundamental significance of supersymmetry was realized almost immediately after the pioneering papers by Gol’fand and Likhtman, Volkov and Akulov and Wess and Zumino. Within a short space of time the supersymmetric generalization of the standard model was found, the supersymmetric theory of gravity constructed, and the approaches to supersymmetric string theory developed. It has also emerged that supersymmetry is of interest in certain quantum mechanical problems and even some classical ones. In essence, supersymmetry is the extension of space-time symmetry (Galileo symmetry, Poincare symmetry, conformal symmetry, etc.) by fermionic generators and it serves as the theoretical scheme which naturally unifies bosons and fermions. Therefore, it is natural that supersymmetry should form the basis of most modern approaches to finding a unified theory of all fundamental interactions. At present, the ideas and methods of supersymmetry are widely used by specialists in high-energy theoretical physics. The conceptions of supersymmetry play an important role in quantum field theory, in the theory of elementary particles, in gravity theory and in many aspects of mathematical physics. It is evident that supersymmetry must be an element of the basic education of the modern theoretical physicist. Hence there is a need for a textbook intended as an introduction to the subject, in which the fundamentals of supersymmetry and supergravity are expounded in detail. The present book is just such a textbook and it aims to acquaint readers with the fundamental concepts, ideas and methods of supersymmetry in field theory and gravity. It is written for students specializing in quantum field theory, gravity theory and general mathematical physics, and also for young researchers in these areas. However, we hope that experts will find something of interest too. The main problem which tormented us constantly while working on the book was the choice ofmaterial. We based our decisions on the following principles. xv Copyright © 1998 IOP Publishing Ltd

xvi


(i) The account must be closed and complete. The book should contain everything necessary for the material to be understood. Therefore we included in the book a series of mathematical sections concerning group theory, differential geometry and the foundations of algebra and analysis with anticommuting elements. We also included some material on classical and quantum field theory. (ii) A detailed exposition. Since the book is intended as an introduction to the subject we tried to set forth the material in detail, with all corresponding calculations, and with discussions of the initial motivations and ideas. (iii) The basic content of the subject. We proceeded from the fact that a book that aspires to the role of a textbook should, in the main, contain only completed material, the scientific significance of which will not change in the coming years, which has received recognition, and has already found some applications. These requirements essentially restrict the choice of material. In principle, this means that the basic content of the book should be devoted to four-dimensional N = 1 supersymmetry. It is possible that such a point of view will provoke feelings of disappointment in some readers who would like to study extended supersymmetry, higher-dimensional supersymmetry and two-dimensional supersymmetry. Although realizing the considerable significance of these subjects, we consider, nevertheless, that the corresponding material either does not satisfy the completeness criterion, is too specialized, or, because of its complexity, is far beyond the scope of the present book. As far as two-dimensional supersymmetry is concerned, its detailed exposition, in our view, should be carried out in the context of superstring theory. However, we are sure that the reader, having studied this book, will be well prepared for independent research in any directions of supersymmetry. (iv) The superfield point of view. The natural formulation of fourdimensional N = 1 supersymmetry is realized in a language of superspace and superfields-that is, this formulation is accepted for the material’s account in the book. Initially we did not want to discuss a component formulation at all, considering that it is only of very narrow interest. But then we came to the conclusion that the component language is useful for illustrations, and for analogies and comparisons with conventional field theory; hence we have included it. At present there exists an extensive literature on the problems of supersymmetry and supergravity. Since we give a complete account of material here, we decided that it was unnecessary to provide an exhaustive list of references. Instead we use footnotes which direct the attention of the reader to a few papers and books. Moreover, we give a list of pioneer papers, basic reviews and books, and also the fundamental papers whose results we have used. The book consists of seven chapters. In Chapter 1 the mathematical backgrounds are considered. This material is used widely in the remaining Copyright © 1998 IOP Publishing Ltd


xvii

chapters of the book and is standard for understanding supersymmetry. Chapter 2 is devoted to algebraic aspects of supersymmetry and the concepts of superspace and superfield. In Chapter 3 the classical superfield theory is given. Chapter 4 is devoted to the quantum superfield theory. In Chapters 5 and 6 the superfield formulation of supergravity is studied. Chapter 7 is devoted to the theory of quantized superfields in curved superspace. This chapter can be considered as a synthesis of the results and methods developed in all previous chapters. The material of Chapter 7 allows us to see how the geometrical structure of curved superspace displays itself by studying quantum aspects. The subject of this chapter is more complicated and, to a certain extent, reflects the authors’ interests. The book has pedagogical character and is based on lectures given by the authors at the Department of Quantum Field Theory in Tomsk State University. Certainly, it cannot be considered as an encyclopedia of supersymmetry. As with any book of such extensive volume, this one may not be free of misprints. References to them will be met with gratitude by the authors. We are grateful to Institute of Physics Publishing for its constant support during work on the book and its patience concerning our inability to finish the work on time. We are especially grateful to V N Romanenko for her invaluable help in preparation of the manuscript. One of us (SMK) wishes to express his deep gratitude to J V Yarevskaya, the first reader of this book, and also to the Alexander von Humboldt Foundation for financial support in the final stage of preparing this text. We hope that the book will be useful for the young generation of theoretical physicists and perhaps will influence the formation of their interest in the investigation of supersymmetry. Ioseph L Buchbinder Sergei M Kuzenko Tomsk and Hannover, 1994


Preface to the Revised Edition The first edition of Ideas and Methods of Supersymmetry and Supergravity was published in 1995 and received positive reviews. It demonstrated the necessity for a detailed, closed and complete account of the fundamentals of supersymmetric field theory and its most important applications including discussion of motivations and nuances. We are happy that the first edition proved to be a success and are grateful to Institute of Physics Publishing for publishing this revised edition. The general structure of the book remains without change. N = 1 supersymmetry, which is the main content of the book, is still the basic subject of many modern research papers devoted to supersymmetry. Although field theories possessing extended supersymmetry attract great attention due to their remarkable properties and prospects, their description both at the classical and quantum levels is realized, in many cases, in terms of N = 1 superfields with the help of the methods given in this book. For the revised edition we have corrected a number of misprints and minor errors and supplemented the text with new material that fits naturally into the original content of the book. We have added a new subsection (3.4.6) which is devoted to the component structure of the massive vector multiplet. Of course, this model can be investigated by purely superfield methods given in the book, but as the component approach is more familiar to a great many practitioners of supersymmetry, we decided to present such a consideration for the massive vector multiplet; this point was not given in the first edition. We also included a new section (3.8) which deals with the non-minimal scalar multiplet being a variant realization of the superspin-0 multiplet. The non-minimal scalar multiplet possesses remarkable properties, as a supersymmetric field theory, and is expected to be important for phenomenological applications, in particular, for constructing supersymmetric extensions of the low-energy QCD effective action. Since those supersymmetric theories involving non-minimal scalar multiplets are still under investigation in the research literature, we have restricted ourselves i n section (3.8) to the description of a few relevant models and provided references to recent research papers of interest. xix Copyright © 1998 IOP Publishing Ltd

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Preface to the Revised Edition

One of the difficulties we faced while writing the book was associated with the practical impossibility of composing an adequate bibliography of the works on supersymmetry and supergravity related to our book. A complete list of such works seems to contain more than ten thousand titles and evena shorter list demands an essential increase in the book’s volume. The references given in the book include only the monographs and review papers and also those papers which, in our opinion, have exerted influence on the development of the material under consideration as a whole. Of course, the choice of these papers is rather subjective and reflects our scientific interests. After the first edition was published we received a number of responses from many of our colleagues to whom we are sincerely grateful. While working on the book, we took pleasure in communicating and discussing various aspects of supersymmetry and field theory with N. Dragon, E. S. Fradkin, S. J. Gates, E. A . Ivanov, 0. Lechtenfeld, D.Lust, V. I. Ogievetsky, H. Osborn, B. A. Ovrut, A. A. Tseytlin, I. V . Tyutin, M. A. Vasiliev, P. West and B. M. Zupnik. We express our deepest gratitude to all of them. We thank our editor at Institute of Physics Publishing, Jim Revill, for his friendly support and encouragement on this project. One of us (SMK) is grateful to the Alexander von Humboldt Foundation for financial support and to the Institute for Theoretical Physics at the University of Hannover for kind hospitality during the preparation of this edition. Ioseph L Buchbinder Sergei M Kuzenko Tomsk March 1998


1. Mathematical Background O h my children! my poor children! Listen to the words of wisdom, Listen to the words of warning, From the lips of the Great Spirit, From the Master of Life, who made you! Henry Longfellow: The Soi7g qf' Hiuwuthu

1.1. The Poincare group, the Lorentz group 1.1.1. Definitions Space-time structure in special relativity is determined by the set of general principles: 1 . Space and time are homogeneous. 2. Space is isotropic. 3. In all inertial reference systems the speed of light has the same value c. From the mathematical point of view, these principles mean that the space-time coordinates X" and X" of two arbitrary inertial reference systems are related by a linear non-homogeneous transformation X"

leaving the metric

(1.1.1)

= Am,xn-I-b"

ds2 = qmndx" dxn

( 1.I .2)

invariant. We have used the following notation: x m = ( x 0 ,xl, x2, x3), xo =et, where t is the time coordinate (in what follows, we set c = l), R=(x',x2, x3) are the space coordinates and qmnis the Minkowski metric /-1


0 0

o\

The requirement for invariance of the metric ds2 is equivalent to the equation ( 1.1.3)

ATqA = q.

Here AT is the matrix transpose of A. Equation (1.1.3) is not the only necessary restriction on the parameters A'",l. On physical grounds, two additional conditions are to be taken into account. Using (1.1.3), we find det A = 5 1 (Aoo)2-(A'0)2-(A20)2-(A30)2 = 1.

It is now easy to show that in order to preserve the direction of time one must demand Aoo2 1

(1.1.4~)

and to preserve parity (spatial orientation), one has to choose the branch det A = 1.

(1.1.4b)

The transformations (1.1.1) with parameters A", constrained by the relations (1.1.3, 1.1.4) are called the 'Poincare transformations'. In the homogeneous case, when b" =0, they are called the 'Lorentz transformations'. We shall denote the Poincare transformations symbolically as (A, b) and the Lorentz transformations simply as A, The union of all PoincarC transformations forms a real Lie group under the multiplication law (A,,bz)x(A,,b,)=(A,A,, b2+&bi).

(1.1.5)

This is the 'Poincare group', denoted below the symbol n. Analogously, the union of all Lorentz transformations forms a real (semisimple) Lie group. This is called the (proper orthochroneous) 'Lorentz group'. We denote it by SO(3, 1)f.

Note that the set of all homogeneous transformations (1.1.1) constrained by the relation (1.1.3) forms a real Lie group denoted by O(3, 1) which consists of four disconnected pieces O(3, 1)=(SO(3,

where

Ap =

[i -' 0

0

0

0

- 1O

0

ApSO(3,

0 - 1

ATsO(3,

'1

ApTSO(3,

- 1 0 0 0

AT=[

O0 0 O1 0

Api-=ApAT

0 0 0 1

hence it includes the transformations of spatial reflection Ap, time reversal A T and total reflection APT. The set of all homogeneous transformations Copyright © 1998 IOP Publishing Ltd

Muthemutical Background

3

( 1.1.1 ) constrained

by the equations (1.1.3) and (1.1.4h)also forms a Lie group, SO(3, l ) , which consists of two disconnected pieces SO(3, I ) = ISO(3,

Ap~S0(3,

1.1.2. Ust$ul clwonzposition in SO(3, 1 ) T Now we shall give a deeper insight into the structure of the Lorentz group. To begin with, note that an arbitrary element A E SO(3, 1)f may be represented as

(1.1.6)

A = RAx($)R

where R and

R' are space rotations: 0

0

detR=l

RTR=U

0 R 2 , R2, R2,

(1.1.7)

0 R3, R3, R3, and Ax is a standard Lorentz boost in the xo, x1 plane

A,(*)=

(cosh$

sinh$

0 0

sinh$

cosh$

0 0

0

1 0

0

0 1/

I

\ o

\ ( 1.1.8)

There is a strict mathematical proof of equation (1.1.6),but it can be most easily seen by considering the following physical argument. The transformation (1.1.8) corresponds to the situation where the x2- and x3-directions of two inertial systems K and K' coincide, and the system K' moves along the x'-axis of the system K. If A#&($), we can rotate the spatial axes of systems K and K'at t=Osoas toobtainjustsuchasituation.Thisgivesequation(1.1.6). Let us also recall a well-known fact about the rotation group SO(3).Namely that an arbitrary element R e S O ( 3 ) may be represented as a product of rotations around the coordinate axes:

'

=R x ( ~ 1 ) R y ( ~ 2 ) R z ( ( p 3 )

[: I

Rx(cp)=

O

0 0

0

cos cp sin cp -sin cp

cos cp


coAcp p -':') 0

R,,(cp)=[i

0

0 sincp 0

0

coscp

/I

0

o\

0

(1.1.9)

I . I .3. Unicersul covering group of the Lorentz group From equations (1.1.4, 8, 9) it follows that the Lorentz group is connected. But SO(3, l ) ? is not simply connected. Let us recall that a connected Lie group G is termed simply connected if any closed path in G can be shrunk down to a single point. The main property of simply connected groups is a one-to-one correspondence between representations of the group and the corresponding Lie algebra. Namely, any representation of the Lie algebra 9 of a simply connected Lie group G is the differential of some representation of G. But this is not true for non-simply connected Lie groups. However, to any connected Lie group G one can relate a (unique up to isomorphism) 'universal covering group' G with the following properties: 1. G is simply connected. 2. There exists an analytic homomorphism p: G G such that G z c/Ker p, where Ker p is a discrete subgroup of the centre of G.Since the homomorphism p is locally one-to-one, the group G and its universal covering group G have isomorphic Lie algebras. In quantum field theory, one needs to know representations of the Lie algebra so(3, 1) associated with S0(3,1)? rather than representations of SO(3, l)t itself. As is seen from the discussion above, to construct representations of the Lorentz algebra so(3, l), it is sufficient to find a universal covering group for SO(3, l)f,denoted by Spin(3, l),and to determine its representations. The main property of Spin(3, 1) is given by the following theorem. --f

Theorem. Spin(3,l) SL(2, C), where SL(2,C) is the Lie group of 2 x 2 complex unimodular matrices. Proof. Introduce in the linear space of complex 2 x 2 matrices the basis om,m=O, 1, 2, 3,

0

1 0

-i

0

-1

where ;are the Pauli matrices. It is convenient to define the one-to-one map of the Minkowski space on the set of 2 x 2 Hermitian matrices xm +

x = xmom=

x c =x Copyright © 1998 IOP Publishing Ltd

x0+x3

x'-ix2

x'+ix2

xo-x3

det x = - qm,,xmx".

(1.1.10)

Mathematical Background

5

Here qm,! is the Minkowski metric. Let us now consider a transformation of the form X+

xI

fm

= X

crm = NxN'

N E SL(2,C).

(1.1.11)

Since det N = 1 , this transformation preserves the interval qmnxmxn = q,,,x"~'~. Thus x" =(A(N))",x",where A(N)EO(3, 1). From relation (1.1.1 1 ) we see that the map x: SL(2, @)+ O(3, 1)

defined by the rule

N

+ A(N)

is an analytic homomorphism. In fact, we shall show that n(SL(2,e))= S 0 ( 3 , l)T. First, note that

).

Kern=*( 1 0 0 1

(1.1.12)

Indeed, N E Ker TC if and only if NxN' = x for any 2 x 2 matrix x. In particular, the choice x = 0 gives NN' = 0, hence N + = N - ' . So, our condition takes the form N x N - = x for any x. This is possible if and only if N 0. As the next step, we reconstruct elements of SL(2, e),which are mapped into rotations R,(cp), R,,(cp),R,(cp) defined in expressions (1.1.9). It is a simple exercise to check that

-

1; 1= 1.; 1=

n-'(Rx(cp))=+exp i-o,

+N,(cp)

n-'(R,(cp))= +exp i-oz

+N,(cp)

(1.1.13)

Analogously, the Lorentz boost Ax($) defined in equation (1.1.8) and the Lorentz boosts Ay($),A,($) (i.e. Lorentz transformations in the planes xo,x2 and xo, x3, correspondingly) are generated by x-l(A,($))=

TC-

+exp

' ( A y ( $ )=) iexp


(1.1.14)

The expressions ( I . 1.13, 14) show that the one-parameter subgroups N,(cp), Mi($).i = I , 2. 3. in SL(2,C) are mapped into SO(3,l)T. But these subgroups generate SL(2, C).Hence, all elements of SL(2, @) are mapped into SO(3, l)T, so n(SL(2,C ) ) c S O ( 3 , On the other hand, the identities (1.1.6,9) mean

). R,(cp) and the boost A,($) generate the Lorentz that the rotations R , ( ~ J R,(cp), group. Then equations ( I . I . 13, 14) tell us that n(SL(2,C))= SO(3, 1)t. From this and equation (1.1 .I 2), we see that SO(3, 1jT2 SL(2,@)/Z2

(1.1.15)

So. SL(2,C)is a double-covering group of SO(3, 1)f. Finally, we briefly prove that SL(2,C)is simply connected. Every element h' E SL(2, C)can be represented uniquely in the form (1.1.16)

N=,p

where g is a unimodular unitary matrix and z is a unimodular Hermitian matrix with positive trace: g E S U ( 2 ) z+ = z

detz = 1

Trz > 0.

(1.1.17)

The group S U ( 2 )is simply connected. Indeed, anyg E S U ( 2 )can be written as q=(-;*

;*)

M2+I4l2=1.

+

Thus topologically, S U ( 2 ) is a three-sphere S 3 ( ( u ' ) ~ ( u ~+ )( ~u ~ +) ~( u ~ = ) ~1, where p = u1 + iu2, q = u3 + iu4), which is a simply connected manifold. Now consider the manifold of Hermitian 2 x 2 matrices z constrained by (1.1.17). If we parameterize z as z=zmam,(zm)*=zm, then the constraints (1.1.17) imply that (20)2 -(z')2 -( 2 j 2

- (z3j2 = 1

zo > 0.

This manifold is evidently simply connected. So, we may represent S L ( 2 , C ) as a product of two simply connected manifolds, and hence it is also simply connected. Due to relation (1.1.15), SL(2,C) is the universal covering group of the Lorenz group. This completes the proof of the theorem. I . I .4. Universul cooering group of the Poincari group Our goal now is to construct a universal covering group of the Poincare group. For this purpose, we return once again to the space of 2 x 2 Hermitian matrices (1.1.10) and consider a new class of linear transformations over it Copyright © 1998 IOP Publishing Ltd

Muthmuticul Buckground

7

by adding a non-homogeneous term in the right-hand side ofequation (1.1.1 1): x + X ' = ximom = NxN

+

+b

b = b+ = b"a,.

N E SL(2, @)

(1.1.18)

Such a transformation associated with a pair (N, b) looks in components like .$"I= (A(N))",x"+ b", i.e. it coincides with the Poincare transformation ( A ( N ) ,h). The set fi of all pairs (N,b), N and b being as in relations (1.1.18), forms a ten-dimensional real Lie group with respect to the multiplication law (N2, b 2 ) x ( N l , b1)=(N2N,, N2blN: + b d

(1.1.19)

Evidently, this group is simply connected. The above correspondence (1.1.18) constitutes the covering mapping cp:

if+n

cp((N,b)) = (A"

b)

( 1* 1.20)

of the simply connected group if on to the Poincare group. Since the correspondence N + A ( N ) is a group homomorphism, and by virtue of equations ( 1 . 1 3 and {1.1.19), the mapping (1.1.20) is an analytic holomorphism of fi on to the Poincare group. Its kernel consists of two elements, Ker cp = {( & U, O)}. The above arguments show that fi is the universal covering group of the Poincare group. 1.2. Finite-dimensional representations of Spin(3,l) 1.2.1. Connection between representations of SO(3, 1)t and SL(2, C) A linear represenation T of a Lie group G in an n-dimensional vector space V, is defined as a homomorphism of G into the Lie group of non-singular linear transformations acting on this vector space,

T : g-T(g) T(gJT(g,)=T(g,g,)

gEG 91, 92 E G.

Let T: A + T(A)be a representation of the Lorentz group SO(3, l)t. Then we automatically obtain a representation ;f of its universal covering group SL(2, C ) by the rule T: N + T(n(N)) (1.2.1) where n is the covering mapping constructed in subsection 1.1.3. For example, the vector representation (1.2.2) Copyright © 1998 IOP Publishing Ltd

or the covector representation Tcv:A -+(AT)Vm-+VL=A,"V,

(1.2.3)

Amn= qmkAklql" qmkqk"= 6," of the Lorentz group immediately generate representations of SL(2,C). In equations ( 1.2.2,3), V" and V , are the components of some Lorentz vector and covector correspondingly. Any representation ?. of SL(2, C),associated with a representation T of SO(3, 1 ) f according to expression (1.2.1),satisfies the property

T(N ) = T(- N )

N E SL(2, C).

(1.2.4)

But there exist representations of SL(2, C)for which this property is not true. As we shall see, one can construct irreducible representations (irreps) of SL(2, C)for which T(N)= -T(-N). (1.2.5) Any such SL(2, C)irrep is not a representation of SO(3, l)T. However, it may be treated as a double-valued representation of the Lorentz group. 1.2.2. Construction of SL(2,C) irreducible representations In what follows, we denote SL(2,C) indices by small Greek letters. In particular, components of a matrix N E S L ( ~ , Care ) N!, a, /?=1, 2. Components of the complex conjugate matrix N * are denoted by dotted indices, N * / . Define the fundamental representation of SL(2, C) ( 1.2.6)

An object $ %transforming according to this representation is called a 'two-component left-handed Weyl spinor'. The representation T, is called the (left-handed) Weyl spinor representation of the Lorentz group. It is denoted by symbol ( 3 . 0 ) .

Taking an n-fold tensor product of T,, n=2, 3, . . ., T,OTs,O...@I;, we obtain new representations of SL(2, C) of the form * 3 , 3 ~ . . . ~ , , ~ ~ ~ , ~ ~ . . . 2 , , = ~ 2 ~ 1 N(1.2.7) ~ ~ ~ . . . N ~

The representation contragradient to the representation T, (1.2.6)is given Tcs:N - + ( N T ) - l $2

-+ $'"=$P(N

-

y.

(1.2.8)

This representation is equivalent to T,. Indeed, the condition of unimodularity Copyright © 1998 IOP Publishing Ltd

Mathemaricul Buckground

9

for NeSL(2, C)can be written in the form or

(1.2.9) E21j=

where

E,/{

and

~"~(N-')~~(N-l)~~fi

are antisymmetric tensors defined by ( 1.2.10)

Equation (1.2.9)means that ( N - ' ) p z = ~ 2 y N ~ ~and n l j hence , the representations T, and T,, are equivalent. and cXB are invariant tensors of the The identities (1.2.9) imply that Lorentz group. Hence we can use them for lowering or raising spinor indices, which will be done in this text according to the rules

*"

= &,4+b,,

*,

= &*p*P.

(1.2.11)

Now consider the complex conjugate representation of the representation

T,: (1.2.12) An object transforming according to this representation is said to be a 'two-component right-handed Weyl spinor'. The representation Ts is called the (right-handed) Weyl spinor representation of the Lorentz group. It is denoted by the symbol (0,i). Taking the tensor product of Ts with itself m times, T s Q . .. @ Ts, m=2, 3, . .., one obtains new representations of SL(2, C)of the form

--

We can also consider the more general situation

T @ . . . @ T@ G @ . . . @ G tI

m

obtaining, as a result, Lorentz spin-tensors with dotted and undotted spinor indices


10

Ideus and Methods of Supersymmetry and Supergravity

The representation contragradient to the representation Ts (1.2.12) is

Tcs: N + ( N + ) - ' ( 1.2.15 )

$"$\I/'"$p(N-')*/f This representation is equivalent to Ts since antisymmetric tensors ck/', where

E+B

and

(1 $2.16) are invariant Lorentz tensors and can be used for lowering or raising dotted spinor indices by the rules =

$k

$k

( 1.2.17)

= &+/j@

Representations of the form (1.2.14) with unconstrained tensors z,r/jIp?,,p,,, are reducible when n > 1 or m > 1. For example, an arbitrary second-rank tensor with undotted indices $ x p may be decomposed in a Lorentz invariant way as $xlx?,

,,

,

$ap

5I

= ( h p + $fix)

1

1

+ j ($28 - $ p z ) =

-5 & x p ( & ' V y s )

$(1/?,

where ( a p .. .) denotes symmetrization in indices CL, p, . . .. In general, a tensor of the form (1.2.14) turns out to be irreducible if t,hzILZ1... .+,jl.,,bni is totally symmetric in its undotted indices and independently in its dotted indices. Thus irreducible representations are realized on tensors $xlaL...z,,plp?..

( 1.2.18)

. B , , ~ = $. ZC, , ) C~P ~ , P ~.~ ../ j .~ ,~ . .

The corresponding irrep is denoted as (n/2,m/2). Its dimension is

( n + l)(m+ 1). Note that (n/2,m/2) is a single-valued representation of the Lorentz group if ( n+ m) is even, otherwise, it is double-valued.

J,,, be a (n/2, m/2)-type tensor. Taking the complex Let $ x l z 2 , , .xf,plbz., , , ~ ~ ~ ~ * as conjugate of equation (1.2.14), we find that ( $ z l l L , ~ , x , , ~ , ~ L ,transforms a (m/2, n/2)-type tensor. Therefore, the following mapping

*: VWZ"2) +V"Z."i2) $ x 1 x 2 . . . x , , p , / j 2 . ..R,,,

+

4

$plp2... / ~ , , ,.+ . k1 , , =i (2 $ z. I Z 2 . .

.I,,,)* (1.2.19)

.~,,B,B?..

is defined and is said to be the complex conjugate spin-tensor of $. Evidently, its square coincides with the identity operator. If n # m, the (n/2, m/2)-representation is complex. But with every (n/2, m/2) representation, n # m,one may associate a real representation of the Lorentz group in the following way. Taking the direct sum representation (n/2, m/2) @(n/2, m / 2 ) we consider within the space of this representation V ( n , 2 , m ; 2V1(@ m , Z . n(which 12, Copyright © 1998 IOP Publishing Ltd

Mathemuticul Buckground

11

is mapped on to itself by *) a Lorentz invariant subspace denoted by V&2,m12, and selected by the condition that * coincides with the identity operator on this subspace. Arbitrary pairs

-

($Zl,. '

$XI..

. X , , / I I . . .p,,,3 $ Z l . .

.?,,,/h.../L)

.2.p1 . . .A,,= $ ( X I . . . Z,,,tPI

.I

.),,,I

span V & z . m / z l . In the case n=m, we can define real tensors. By definition, they satisfy the equation

1.2.3. Invariant Lorentz tensors Invariant tensors of the Lorentz group are useful for lowering, raising or covariant contraction of indices. Up until now, we have found the following invariant tensors: the Minkowski metric qms and its inverse q"", the spinor metrics and their inverse E@, E'P, and the Levi-Civita totally E = -~ 1). Now ~ ~we find ~ one more invariant antisymmetric tensor caficd( tensor carrying Lorentz as well as spinor indices. Let us rewrite the relation ( 1 . 1 . 1 1 ) in the form

(~l(N))~,x"o, =x"Na,N+ or om=Na,N+(A(N)-')",.

(1.2.21)

Here A(N) is the Lorentz transformation corresponding to an element N ES y 2 , C).The identify (1.2.21)shows that, denoting components of u, as (0m)zri

(1.2.22)

we obtain a Lorentz invariant tensor with one space-time index m, one undotted spinor index a and one dotted spinor index it. We may also introduce o-matrices with upper spinor indices (1.2.23) (1.2.24) (1.2.25) (1.2.26) (1.2.27) Copyright © 1998 IOP Publishing Ltd

I d ~ ~ (ind i s Methotis of' Supersymmetry

12

oa8bac=(qacob-~bcaa-

tind

qab'c)+

8 a o b 8 c =( q a c 8 b - qbC8a - q a b 8 , ) -

Superyrmity

iEabcdod

(1.2.28)

i&abcd8'.

(1.2.29)

It is seen from these relations that the Minkowski metric and the Levi-Civita tensor are expressible in terms of the o-matrices. The a-matrices are usually used to convert space-time indices into spinor ones and vice versa according to the general rule: one vector index is equivalent to a pair of spinor indices, dotted and undotted,

va= - 1 (8,p v

v,,= (aa),&va

-

2

11' '

(1.2.30)

In many cases, the conversion of vector indices into spinor ones leads to some technical advantages. Moreover, conditions of irreducibility are much simpler when working with two-component objects. For example, let us consider a second-rank tensor x , b . Convert the space-time indices a, 6 into spinor ones according to relation (1.2.30): xab

* x,B&.B

=(a%i(ab)pjXab.

The spin-tensor Xzp+ can be decomposed into irreducible components as follows

X m a = XIzBlcib, + X('P,[@l+ XIlBlr&.al+X i X P N i P , =&&@I),

+ & & j X ( ' P , + &'/+U@

+ x,,p,(&p,

(1.2.31)

where Cap] denotes antisymmetrization in indices a, p. The irreducible components of xUPhb are x(,pXk,p,, x =~ s % i b ~ , ~ xg ,( ~=--) j ~ i P ~ ( , ~ ) ~ , j , x ( i B ) - - - - E rBX,P,e,c), Here we have used the identities E' E , ~ - E ' ~ E ~ ~-2. = If X a b is a symmetric tensor, X a b = X b a , then the first and second terms in equation (1.2.31)vanish. If, in addition, X , , is traceless, X: =0, then the third term in equation (1.2.31) is also zero. In the case when x , b = - X b a , only the first and the second terms in equation (1.2.31) are non-zero. Hence, an arbitrary antisymmetric tensor x a b is equivalent to a pair of symmetric bi-spinors. To write down explicitly this correspondence, introduce the matrices ,

1

(gab)/=

--(oa8b-ob8a)z

4

(1.2.32) Then we obtain


13


Xab

= -xba

x,p = X p x

X & #= X#?.

If xu,is a real tensor, then X,, and X&#are complex conjugates of each other, X , g = X i b . Now consider another example. Let Cabcd be a tensor subjected to the constraints Cabcd

= - Cbacd = - Cabdc=

(1.2.34a)

erdab

Caboc= 0

(1.2.34b)

EabcdC/bcd= 0.

(1.2.344

The corresponding spin-tensor is seen to be

+

Cz/j$@>d = (aa)z&(ob)ss(o‘)~(od)Bs Cabcd = E?t#Ei,dC,p$ EzpEy6 ck#$

where tensors C,..,s and C,B?Jare symmetric in all their indices. Note that equations (1.2.34)are the algebraic constraints on the Weyl tensor in general relativity. Remark. In what follows, by representations of the Lorentz group we mean both the single- and double-valued representations, i.e. arbitrary SL(2, C)representations. Analogously, by representations of the Poincare group we mean its single- and double-valued representations (both being infinitedimensional, in general), i.e. arbitrary fi-representations. 1.3. The Lorentz algebra

The Lorentz group and its universal covering group SL(2,C) are locally isomorphic. So, the Lie algebra so(3, 1 ) of the Lorentz group (the ‘Lorentz algebra’) and the Lie algebra sl(2, C)of the Lie group SL(2, C )are isomorphic. Nearly all elements N ESL(2, C)can be expressed in the exponential form N = exp(i6)= exp(z)

z E 5742, C)

(1.3.1)

where i = ( z l r z 2 ,z 3 ) is a complex three-vector, and 6 are the Pauli matrices. The exponential from 4 2 , C)covers SL(2,C)apart from only a (complex) two-dimensional surface in SL(2, C)consisting of elements

- 1 +ab

0

-1

where (a,b) is a non-vanishing complex two-vector. The union of all SL(2, C) Copyright © 1998 IOP Publishing Ltd

14

Idiws

trnti

Metliotls o f Supiwymmetry and Superyravitj

points, given in the form (1.3.1), is an everywhere dense set in SL(2,C). On these grounds, we shall later write SL(2, C )elements in the form (1.3.1)without comment. Note that if N E SL(2, C), then at least one of the matrices N and ( - N ) admits the exponential form (1.3.1). Considering SL(2,C) as a complex Lie group, the parameters 2 from equation (1.3.1) play the role of local complex coordinates, and the Pauli matrices form a basis in the corresponding Lie algebra. Since we treat S 4 2 , C) as the universal covering group of SO(3, l)?, a real Lie group, we must understand SL(2, C) as a real six-dimensional Lie group. Introduce real local coordinates in SL(2, C)using the following parametrization:

(Kab)*= Kab

KQb= -Kb".

(1.3.2)

The matrices a'" were defined in equation (1.2.32).The complex ( 2 ) and real ( K a b )coordinates in SL42, C)are related by the rule 1 z,=~(K0'+iKZ3)

1 1 z , = - ( K " + ~ K ~ ~ ) z3=-(KO3+iK12), 2 2

By virtue of equation ( 1 . 1 . 1 l), the infinitesimal Lorentz transformation corresponding to an infinitesimal SL(2, C)transformation 1 N = 0 +- Kabaab 2

(1.3.3)

is given by the expression x m + X" = x m

+ Km,,x".

(1.3.4)

When deriving equation (1.3.4), we have used the identities (1.2.28, 29). Recall one important result from group theory. If f G + H is a homomorphism of Lie groups, df Q + % is the corresponding homomorphism ' and exp X is the of Lie algebras, then f(exp X) =exp df(X), where X E 3 exponential from Y into G. In accordance with equation (1.3.2), the matrices c a b form a basis of the real Lie algebra of SL(2, C). Let T be a representation of S t ( 2 , C). Then, using equation (1.3.2)and the above result, we deduce that T(N)=exp(

KabMa,)

N E SL42, C).

(1.3.5)

Here the 'generators' Mab= dT(oab)define a representation of the Lorentz algebra. Note that Ma, = - Mba. The commutation relations for the generators can be easily obtained by noting that d T is a Lie algebra homomorphism, so that Mcdl

= [dT(oab), dT(acd)l


=dT([aab,

acd]h

Muthemutical Buckyround

15

Recalling the explicit form (1.2.32) for the a-matrices, one finds [ c a b , Ocdl

= qadbhc

- qacahd

f VhcOad

- qhd'ac.

(1.3.6)

This gives CMah. M c d l = V a d M h c - ~ a c M h d + V b c M a d - V h d M a c .

(1.3.7)

These commutation relations define the Lorentz algebra and its representations. In the cases of the representations (+,O), (0,i) and (f,*), the Lorentz generators are 5Mr2b(ll/%)= (Oub)!$\j

(1.3.8~)

sMab(p)=(8ab)i,j$'

(1.3.8b)

VMub(1/C)=6~l/b-6~I/a.

(1.3.8,)

The matrices ' M a b form a basis of the Lie algebra so(& 1) of the Lorentz group SO(3, l)r. Recall that any element AcSO(3, 1)T can be written as A = A ( N )= A( - N )

for some N E SL(2, C), with respect to the covering map 71: SL(2, C) -,SO(3, l)T. As was pointed out above, N or (- N ) or both can be represented in the exponential form (1.3.2), therefore we deduce that

for every Lorentz matrix AeSO(3, l)T. Let us introduce a new basis for the Lorentz algebra, replacing the generators Mu,with vector indices by operators Mu, and Akfiwith spinor indices defined as follows

(1.3.9)

*

1 R' 4 -- --(dub)$fiMab.

After this redefinition, equation (1.3.5) takes the form T ( N )= exp(K",M,p

+ PfiA,fi)

(1.3.10)

Using the commutation relations (1.3.7), it is not difficult to obtain the Copyright © 1998 IOP Publishing Ltd

16

Idetis

utid

Metliocls

of' Supersymmetry

and Superyravit.v

commutation relations for generators M T aand

Mi,j. These are:

1 [ M , p M,iil=- (c,;,Mpa + E , ~ M P ; . + & ~ ; . M , ~ + E / ~ ~ M , ; . ~

2

[ M r p , Mi,jI = 0

(1.3.11 )

1

-

[A+,A??

(1.5.23)

(3) for a given family of Lorentz transformations i 2 [ p ] , which parameterizes the coset space l l / H q , we have Ip, a; m2>= u(fl[p], 0) 14, a;m2>

(1.5.24)

(4)for every Lorentz transformation (A, 0), we have WA, 0 )I P ,

0;m2> =

WCbl, o)u(Q-' C ~ P l A w - p l 0) , 14, a;m2> (1.5.25)

where (Ap)"=Aabpb.Note that f2-'[Ap]An[p] EH,, so the action of the operator U(l2-'[Ap]AsZ[p], 0) on the subspace V, is given by requirement (1). It is a technical matter to check that we have indeed obtained some unitary representation of the Poincare group. Evidently, this representation is irreducible. What we have described above is simply Wigner's method of induced representations as applied to the Poincare group. I S.5. Massive irreducible representations We proceed by finding massive irreducible representations of the Poincare group. As explained above, it is sufficient to construct all (unitary) irreducible finite-dimensional representations of the stability subgroup H, corresponding to an arbitrary four-momentum on the 'mass-shell' p 2 = - m 2 , Copyright © 1998 IOP Publishing Ltd

30

Ideus und Methods of' Supersymmetry und Superyruvity

p o O, a basis in the space of representation consists of states IF)

IWIIF)=r,IF)

R2/F)=r21F)

labelled by points F=(rl, r 2 ) of the one-sphere

(r,)’ +(r2)’ = p 2 . Thus, if p2 > 0, the operators R, and R 2 have a continuous spectrum, and the representation is infinite-dimensional. On the other hand, the subspace V, should have a finite dimension. Therefore, no other possibility is available, but p2 = O and RI, R, are trivial on V,, hence

w,= w2=o.

( 1.5.35)

Recalling equation (1.5,19), we see now that H , acts on V, as the product of a U(1) phase group and an Abelian group generated by the only operator J 1 2 . Since the action of Hq on V, should be irreducible, this space includes only one non-trivial state,

J12~L)=4A)

(1.5.36)

i = O , &1/2, f l , *3/2 , . . .

(1.5.37)

where E. can takes values The quantum number 3. is called ‘helicity’. Sometimes, the quantity 1E.J is called the ‘spin of a massless particle’. The restriction ( 1 -5.37) is quite understandable. Indeed, operators exp(icp9 2) describe space rotations along the direction of particle motion. Copyright © 1998 IOP Publishing Ltd

Making the rotation on cp=2n, we obtain e 2 n i J ~ ' ?j.) / =e2nill

j")

The resulting phase factor must be equal to k 1 depending on whether our representation of the PoincarC group is single- or two-valued. The set of relations (1.5.32, 35, 36) means that, in the reference system (1.531) the equality WO= jPo

(1 5 3 8 )

is fulfilled. Since WOand Potransform as Lorentz vectors, equation (1.5.38) is satisfied in any coordinate system. Hence, the helicity is a Poincare invariant characteristic of massless particles. Our conclusion is that massless irreducible representations of the Poincare group are classified by helicity.

1.6. Elements of differential geometry and gravity 1.6.1. Lorent: )nanifolis In general relativity, space-time is a four-dimensional connected smooth manifold, i.e. a connected topological space M covered by a set of open , each of which a homeomorphism charts { U i j i E Afor xi: Ui-+R4

of U , onto an open subset of R4is defined, such that the transition functions fij

= xi 0 x,:

1:

x j ui n U j )+ X i ( U ,n U j )

are smooth whenever U i n U j # D.The functions x l ( p ) = {xy, x!, x;, x:), P E Ui, are called the local coordinates of p in the chart Ui. Orientedness of the manifold M means that local coordinates may be chosen in such a way that the transition functions satisfy the condition det(dxy/dxr) > O for any two charts U , and U j with non-empty overlap. The orientedness is needed to define an integration over the manifold. We anticipate that the reader is familiar with the concept of tensors and tensor fields on manifolds. In particular, a vector field is defined in any chart U by smooth functions um(x),such that the operator u ( p ) = u m a / a x m ( , , pE M , does not depend on the choice of chart. The set of vectors {d/d.xml,} forms a basis (holonomic basis) in the tangent space T,(M) at EM. A covector field is given in any coordinate chart U by smooth functions w,(x), such that the one-form w(p)= w,(x) dxmlpdoes not depend on the choice of chart. The set of one-forms (dx"1,) is a basis in the cotangent space T,*(M)at point p E M. Any manifold always admits a globally well-defined smooth Riemannian metric but not, in general, a pseudo-Riemannian metric. It can be shown Copyright © 1998 IOP Publishing Ltd

Muthemutical Buckground

33

that a manifold admits a metric of Lorentzian signature (-, +, +, + ) if and only if there exists a global vector field, non-vanishing at each point of the manifold (global line field). In fact, any non-compact manifold admits a global line field and, hence, a Lorentzian metric. For a compact orientable manifold, existence of a Lorentzian metric is equivalent to the fact that the manifold has zero Euler-Poincare characteristic. For definiteness, we shall assume through this text that our space-time manifold is a non-compact, topologically Euclidean manifold. This means that the manifold M , considered as a topological space, is homeomorphic to Euclidean space R4. In particular, M may be covered by a single chart with local coordinates xm, m=O, 1, 2, 3. The choice of local coordinates on M is usually a matter of convenience. So, the group of (invertible) general coordinate transformations x m-+ X" =fm(x)

det(dfm/dx,) # O

(1 -6.la)

or, in the infinitesimal form,

(1.6.1b)

x m -+ X" = x"' - Km(x)

naturally acts on M . Every general coordinate transformation changes components of tensor fields according to tensorial laws. For example, given a vector field U = um(x)i?,, dm = d/c?xm,its components in new local coordinates are U"(X')

ax"

= -U " ( X ) .

ax"

In the infinitesimal case, we have 6um(x)= ~'"'(x)- um(x)= K" anum- U" d,K"

(1-6.2)

or, in terms of a Lie bracket, dumam

= [Kna,,,

(1.6.3)

~a,].

Analogously, the transformation law W(x') = @(x)

of a scalar field @(x) can be written in the infinitesimal case as ~@(x)=W(X)--X)=[K"~,,

@I.

(1.6.4)

When a space-time metric ds2 =gmn(x)dx" dx" has been introduced, one can define (in addition to the general coordinate transformation group) an action on M of the so called local Lorentz group by the following rule. In the tangent space T,(M) at some point p E M , we consider an orthonormal frame [e,}, e, =e,"'?, ,I where a = 0, 1, 2, 3, <ea> eb> =gmneamebn=


qab.

Then, an infinitesimal transformation e,--+eh=e,+ Kabeb

Kab=K:qcb= -Kba

transfers the frame { e , ) to another orthonormal frame { e l } , and therefore represents some infinitesimal Lorentz transformation in T,(M). Now let { e n ( p ) } ,e,(p)=e,"(x)?,, be a set of smooth vector fields on M , forming an orthonormal frame at each point p E M , gmn(x)eam(x)eb"(x)

= qab.

(16 5 )

The set { e , ( p ) ) is called a 'vierbein'. Then, the relation eam(x) e: "(x) = e,"(x) -+

Kab=

+ K,b(x)ebm(x)

(1.6.6)

-Kbn

defines an infinitesimal local Lorentz transformation. Here K,, are scalar fields on M . Exponentiating equation (1.6.6), we obtain finite local Lorentz transformations. In contrast to the holonomic basis {d/axm}, the vierbein forms, as usual, an anholonomic basis in the sense that a commutator of basis fields does not vanish

(1.6.7) where gnbcare the 'anholonomy coefficients' and emc(x)is the inverse vierbein, eamemb= 62.

emaenn = 6,"

(1 -6.8)

By virtue of equations (1.6.6,8), the local Lorentz transformations act on the inverse vierbein as follows

+

ema(x)-+ e , ( x )= ema(x) Kab(X)emb(X). 1 1 1

(1.6.9)

Owing to (1.6.5), the inverse vierbein satisfies the relation gmn(x)= qabema(X)enb(X)*

(1.6.10)

It is clear that the local Lorentz transformations do not change the metric. So any two vierbeins connected by some local Lorentz transformation are physically equivalent. Note that the smooth one-forms (8=emadx"} represent a basis in the cotangent space T,*(M)of any point P E M. Let u ( p ) be a vector field. In the tangent spaces T,(M) there are two natural frames: one with curved-space indices {a/ax"> and another with flat-space indices { e , } . Decomposing the vector field with respect to these two frames, one obtains U( p ) = V"

d / a x m= u'e,

vu = emavm Copyright © 1998 IOP Publishing Ltd

vm= eamva

35

Mutlzemuticul Buckground

The components U" with curved-space indices transform by the vector law (1.6.2) with respect to the general coordinate group and stay invariant with respect to the local Lorentz group. On the other hand, the components v" with flat-space indices are scalar fields with respect to the general coordinate group and transform by the vector law with respect to the local Lorentz group, u"(.x)

t""(X)

+ K"b(X)vb(X).

= t'"(x)

This example illustrates the general situation. Namely, starting from a world tensor (a tensor with curved-space indices only) and using the vierbein and its inverse, one can convert all curved-space indices into flat-space ones, obtaining as a result an object which is a world scalar field and a Lorentz tensor field. Of course, we can also consider tensor fields which carry curved-space indices at the same time. The frame eamgives such an example. Note that in this text we use the following notational conventions. Small letters from the beginning of the Latin alphabet are used for flat-space vector indices and small letters from the middle are used for curved-space indices. In principle, there is no great advantage in working with Lorentz tensors instead of world tensors. The main importance of the local Lorentz group structure on a space-time manifold lies in the fact that spinor fields, which are used for describing half-integer spin particles, can be defined only as (linear) representations of the Lorentz group. There is no way to realize spinors as linear representations of the general coordinate group. In conclusion, we write down the transformation law of a spin-tensor field @ ) 2 , , , , 1 4 h l , , , i B with A undotted indices and B dotted indices with respect to infinitesimal general coordinate (1.6.1b) and local Lorentz (1.6.6) transformations:

a@%,. . . ) 2 4 h l . ..&,(XI = WZ, . .

.Z,il.,

,h,(X)--@ZI

... Z , h l . .

.&AX)

where Ma, are the Lorentz generators. 1.6.2. Covariant diyerentiation of world tensors The prescription of how to covariantly differentiate world tensors is well known. One simply has to introduce the Christoffel symbols

and to replace operators d, by 'covariant derivatives' V, defined as follows: (1 -6.13)


and so on. Given some world tensor, its covariant derivative is also a world tensor. The metric gmnis covariantly constant VpYnrn = 0.

The covariant derivatives commute only when acting on a scalar field. In the tensor cases, we have

[V,,

= &kpmnl'P

V,]Ck

[vm, Vn]Wk=

- d pk m n W p

and so on, where ./Akpmn is the 'curvature tensor': .dkpmn

= 2 m r k n p - d,rk,,

+ rkmrrrnp -rkn,rrmp.

(1-6.14)

and has the following algebraic properties: ( 1.6.15 )

where

.'Akpmn

=qk@pm,,

as well as satisfying the Bianchi identities

+

+

Vr.%kpm,, VmBkpnrVnWkprm = 0.

(1.6.16)

Extracting from the Ricci tensor W,, = Wkmkn,W,, = Wnmrand the scalar curvature .R = ym"Bm,, one obtains the Weyl tensor 1 1 Ckpmn = akpmn - (wmpS: - anPS: ymp9; -gflpWk) - (s:gflp - G;gmp)W. 2 6

+

+

+

(1.6.17) The Weyl tensor is traceless in any pair of its indices and has the same algebraic properties (1.6.15) as the curvature. I .6.3 Couuriunt diflerentiation of the Lorentz tensor T o obtain a covariant differentiation rule moving some Lorentz tensor to another one, it is sufficient to introduce a spin connection W,b(x), C?),,b = - W m b l l ,taking its values in the Lorentz algebra and transforming by the law (swm&= KnanWm&

+ (8mKn)Wn,h-

8mKab

+

KkWmcb

+

KbCWmQc

(1.6.18)

with respect to the general coordinate and local Lorentz transformations. Then the operators 1

6,= eamzm+ -2 wObcMbCenme, w o bc


= eamWmbc

(1.6.19)

Muthemuricul Buckground have the following transformation law

1

31

(1.6.20)

when acting on tensors with flat-space indices only. Recalling equation (1.6.1 l ) , we see that the operators 6, transfer any Lorentz tensor to another one (with an additional vector index). So, 6, are 'Lorentz covariant derivatives'. The covariant derivatives satisfy the algebra 1

[6,, 6 b ] = y a b c Q c + - g a i Mcd d 2

F a t= qat+ a , b c

-W b t

(1.6.21)

where the anholonomy coefficients e,; were defined in equation (1.6.7).The field strengths Fa; and .%'atd are called the 'torsion tensor' and the 'curvature tensor', respectively. In general, the vierbein and the spin connection are completely independent fields. This is clear from their physical interpretation: the vierbein is a gauge field for the general coordinate group, while the spin connection is a gauge field for the local Lorentz group. However, one can consider a geometry in which the vierbein and the spin connection are related to each other in a covariant way, due to some constraints on the torsion. For example, the torsion-free condition

Yat=O

(1.6.22)

determines, by virtue of (1.6.211,the spin connection in terms of vierbein as follows

(1.6.23) In this case, the curvatures (1.6.14)and (1.6.21)appear to be the curved-space form and the flat-space form of the same tensor, Cgabcd

= ekaebPecmedn9kpmn.

(1.6.24)

Sometimes, it seems reasonable to consider tensors with both flat-space and cuved-space indices. Then one has to modify the definition of covariant derivatives (1.6.19) by including terms with the Christoffel symbols acting on curved-space indices. For example, if YY",is a Lorentz vector and world covector field, then

Qmy = amyl;+ wmabyf:-r p m n y . Copyright © 1998 IOP Publishing Ltd

In the torsion-free case (1.6.22), the vierbein is covariantly constant,

Vmene= 0 as a consequence of equations (1.6.12,23). Now, the derivatives V , (1.6.13) and 6, (1.6.19) are consistent in the sense that they represent the curved-space form and the flat-space form of the same operator. For example, for a vector field va = emaC",we have vmiln = emaebnG0vb.

Equation (1.6.24) is a consequence of the last assertion. From now on, we consider only torsion-free covariant derivatives and denote operators V and 9 by the same symbol V . 1.6.4. Frame deformations We are going to discuss a rather technical question-how to transform geometrical objects (the covariant derivatives and curvature) with respect to an arbitrary variation of the vierbein

+

eam.+eom de,"

6e,"(.x) = H,b(x)eb"(x)

(1.6.25)

where H is a second rank Lorentz tensor field. For this purpose, it is useful to decompose H into its symmetric and antisymmetric parts: (1.6.26) The frame deformation heam= K,bebm corresponds to a local Lorentz transformation, under which the covariant derivatives change as in equation (1.6.20), or, expanding the commutator, as 1 6 v , = KnbVb- - (V,Kbc)Mbc 2 and the curvature changes as a fourth-rank Lorentz tensor. We must now study the case of a symmetric H . Let us consider the frame deformation

de," = A,bebm

(1.6.27)

fi being a symmetric tensor field. In accordance with equation (1.6.19), we can represent the corresponding changing of covariant derivatives in the form (1.6.28) where habc is a 'connection deformation'. To determine it, one must impose the torsion-free condition on the derivatives V i . This leads to

&abc= V b E 3 a c Copyright © 1998 IOP Publishing Ltd

Vcfiab.

(1.6.29)

Muthemuticul Buckground

39

Commuting the derivatives Vb, we find the change in the curvature, the Ricci tensor and the scalar curvature: (5.@,,cd

vu(V,fihd- v d f i h , ) - vh(v,fi,d- vdfi,,)+ f i a k c 2 k b c d - f i b k g k a c d * 6 ’ 8 a h = VcVcfiah- VoVCfi~c-VbVCfiac + v,v,fif + 2f?cdgC4db(1.6.30)

=

6 d = 2v‘vcfi:

+ 2Pb9?ab.

- 2vaVbfiab

Note that the vierbein deformation (1.6.27) induces the following (in fact, arbitrary) metric variation 6gmn= - 2Amn= - 2emaenbA,,.

(1.6.31)

A particular transformation of the type (1.6.27)

(1.6.32) is known as a ‘Weyl transformation’. Making the specialization of equations (1.6.28-30) to the case f i n , = oqab, we find how the ~ e transformations y ~ change all geometrical objects:

v, +v, + OV, -(Vbo)M,b

69abcd

= qbdvuvcg-

qbcVoVda + qacVhVda-

qadVbVcc

+ 2oBabcd

(1*6.33)

69ab = qabVcVco+ 2Vavbo f 209ab

6.3 = 6VcVco+ 203. The Weyl tensor (1.6.17) is seen to transform homogeneously: 8Cabcd=

2oCabcd.

(1.6.34)

1.6.5. The Weyl tensor The Weyl tensor is an important characteristic of space-time. Namely the Weyl tensor measures whether our space-time is conformally flat or not. Recall that a space-time M is called ‘conformally flat’ if there exists a coordinate system on M in which the metric has the form g m n ( X ) = V(X)Vmn

(1.6.35)

for some positive-definite scalar function cp on M . It can be shown that a space-time is conformally flat if and only if the Weyl tensor vanishes, i.e. Cabcd

= O.

(1.6.36)

Now we give a deeper insight into the structure of the Weyl tensor. Recall that it is traceless in any pair of its indices and has all the algebraic properties ( 1 . 2 . 3 4 ~of) the curvature. Let us decompose Cabcd into its self-dual and Copyright © 1998 IOP Publishing Ltd

antiself-dual components using the Levi-Civita tensor:

'

-

2

p h

rf

c( I t ) c d = k i c ( i ) ( l h c d ef

(1.6.37)

+ c(-)abed.

= c(+ )abed

Using equations (1.2.34h,c),one finds

c( )ahad = 0

Cabcd C ( i ) / h c d = o

(1.6.38)

and therefore (1.6.39)

c(i )abcd = c(i)cdab.

is (anti) self-dual in the first and the second pairs of its We see that C(i)abcd indices. Further, making use of equation (1.6.37) and the properties of the Levi-Civita tensor, one can prove the identities C(+)ahcdC(-)(lbef

c(i)abcdC( k abcf )

=O

(1.6.40)

1 = - c&) 8,' 4

where c:t)

c(*)abcdC(k) abcd .

Algebraically, the Weyl tensor and the Ricci curvature are independent. But they are connected by some differential relations. Indeed, based on the identities Vdgdahc

= v b 9 a c -v c g a b

(1.6.41)

1

V?Bab= - va2 2 which are consequences of the Bianchi identities (1,6.16), one obtains

I .6.6. Four-dimensional topoloyical invurianls In four dimensions, there exist two functionals, quadratic in curvature, with purely topological origin: the Pontrjagin invariant P=

s

d4x e-'(C?+)--C:-))


e=det(eam)

(1 6 4 3 )


41

and the Euler invariant

S (

%= dxe-l

)

C : + , + C : - , - 2 . ~ a h ~ a , + - . ~.2 3

Being explicitly constructed from a metric gmn,P and its arbitrary variations

6

6

-P=O

---~XO.

Jgmn

6gmn

(1.6.44)

x do not change under (1.6.45)

Therefore, the Pontrjagin invariant and the Euler invariant depend only on the topological structure of space-time. To prove relations (1.6.45),we employ the results of two previous subsections. As usual, we represent the metric in the form (1.6.10) and consider an arbitrary vierbein variation (1.6.25).The functionals P and x are scalars with respect to the general coordinate and local Lorentz transformations. Thus, they are evidently invariant under the deformations (1.6.25) with any antisymmetric Hab. It is convenient to start with the Weyl transformations (1.6.32) which leave invariant the functionals

s

I(*)= d4xe-’ C:*) by virtue of the transformation laws (1.6.34) and 6e = 4ae. Analogously, the functional

is Weyl invariant as a consequence of equation (1.6.33). We see that the Pontrjagin invariant and the Euler invariant do not react to the Weyl deformations. On these grounds, it is sufficient to consider only the deformations (1.6.25)with a traceless symmetric H o b . This is a tedious exercise, involving employment of the relations (1.6.30,37-42), to show that

,

61,+ = 61, - 1 = 6 J

s

= 2 d4x e-’

i

Pbv

1 3

2 3

c v ~ ~ ~ a b - - v ~ v ~ ~ - - ~ ~ a b + 2 ~ c d ~ c a ~ b

Aab=Aba

R=o.

This completes the proof. 1.6.7. Einstein gravity and conformal gravity We now recall two gravity models based on different gauge groups. The first one is Einstein gravity describing (as a field theory) propagation of a spin-two massless particle (the gravition). The theory is characterized by the Copyright © 1998 IOP Publishing Ltd

42

I(1etr.y

tmd

Method7 o f Supersymmetry und Superyruvity

action

(1.6.47) where K is a gravitational coupling constant. The gravitational field can be treated in terms of the vierbein or the metric. In the vierbein approach, the symmetry group of Einstein gravity is a product of the general coordinate group and the local Lorentz group. The vierbein transformation law is

6eam= Kna,,eam- eaKm+Kabebm.

(1.6.48)

It is instructive to rewrite this deformation in the form (1.6.25).Using the torsion-free condition, one finds

6eOm= H,beb" + Rabebm Hob = - V(,,Kb) Rab=

Ka = Kmema

(1.6.49)

Kab - KCWc,b-v[,Kb].

In the metric approach, the symmetry group of Einstein gravity is reduced to the general coordinate group. The metric transformation law can be easily obtained from equations (1.6.31,49):

6gm,=VmK,+V,Km.

(1.6.50)

The equations of motion for S , are 1 2

gab--vabg=o,

*

gab=().

(1 -6.51)

To derive them, one must use equation (1.6.30). Recalling equation (1.6.42), we see that the Weyl tensor satisfies the on-shell equations

vdCdabc= 0.

(1.6.52)

The second gravity model we would like to discuss is conformal gravity. It is characterized by a larger gauge group with respect to Einstein gravity since the corresponding action ( 1.6.53)

is invariant, as has been shown above, under the Weyl transformations. The price for this additional symmetry is that S , is a higher-derivative model. In the vierbein approach, S , is invariant under the general coordinate, local Lorentz and Weyl transformations:

+

+

Beom= Kna,eam- eaKm Kabebm tream. Copyright © 1998 IOP Publishing Ltd

(1.6.54)

43

Mathematical Background In the metric approach, the transformation law is given in the form:

+

6g,, = V,K, V,K, - 20g,,. The equations of motion for S , are 1 v'vce%ab--

3

vavbw--

2 ggab 3

+

(1.6.55)

29?cdgcfadb

(1.6.56)

1 6

-- qa),(vcvcd - :'A2 + 3& + (?ab)&$(?d)dp

(1.6+66)

+ (pb)&a(8cd)8';} c(

+ )abed.

To derive the last two relations, we have used the observation that the matrices gab and Pb are (anti) self-dual: (1.6.67)

(1.6.68)

1.7. The conformal group 1.7.1. Conformal Killing vectors Let M be a space-time manifold with local coordinates X" and metric ds2=gmn(x)dx" dx"(of Lorentzian signature). Given a vector field t = tm(x)dm we can define an infinitesimal general coordinate transformation X m + X"

=x m

+ yyx)

(1.7.1)

which changes the metric as follows 6gmn(x)=g6n(x)-gg,n(x)= - Vmtn- V n t m .

(1.7.2)

A vector field =f~/?Pa+ ;f~a Jab ab*

(2.2.4a)

and

(2.2.4b) and 1 2

I q x ,q d = f,?~,+ - f a t h j a b f,&h

= - fa?.

(2.2.44

Recall that spinor indices are raised and lowered with the help of the spinor (see equations (1.2.1 1, 17)). In accordance with the results metric cxP and of subsection 2.1.1, the set of structure constants { fx//', fa/?', f,$, f,/f',fa$', fa&'} should form an invariant tensor of the Poincare algebra, i.e. i t must Copyright © 1998 IOP Publishing Ltd

satisfy equation (2.1.18).The indices i, j and 1 from (2.1.18)are now Poincare indices, and the indices E , p and y from (2.1.18) are now spinor indices. The non-vanishing structure constants of the Poincare algebra are fa,,dh = -&, h and Lrh.,.d' = -L.d..d', and their explicit values can be readily found from (1.5.5).Then, choosing in equation (2.1.18)i = a and 1 = b and using equation (2.2.3),one obtains =

f$h

,,p

= 0,

= f,fh

The relations (2.1.18)tell us that the other structure constants fa,;, f.$' and f3; are invariant tensors of the Lorentz algebra. But the Lorentz group has no invariant tensors like fa,; or f+,f.Further, the only candidate for the role of f,f is the invariant tensor (GO),&. As a result, the anticommutation relations (2.2.4) are simplified drastically: (2.2.5) 19x9 q h j = 2k(a,hPa

with k being some constant. It is a simple exercise to check that the second equation (2.1.19) for the structure constants is satisfied now identically. So, we have obtained a superalgebra. Finally, it would be desirable to demand that in any unitary representation T of the obtained superalgebra (from the physical point of view, such representations are of the greatest importance) the generators Qa = T(q,) and U& = T(qh)were Hermitian conjugate to each other,

Q, = (QJ+.

(2.2.6)

In other words, we wish to treat the pair (q,, qi) as a Majorana spinor. Then, the constant k in (2.2.5) should be real and positive. Indeed, equation (2.2.5) leads to 1 , ' 4

- kP, = -(~J,)'"{Q,,

1 4

Ob} = -(8,)"{Q

35

(Q,j)t}

(2.2.7)

+

where P, = T(p,)= ( - E, P) is the (Hermitian) energy-momentum operator, hence k is real. Choosing here a = 0 gives

Since physically acceptable unitary Poincare representations are characterized by condition (1.5.16) (positivity of energy) and due to positive definiteness of the operator in the right-hand side of equation (2.2.8), we must choose k > 0. Hence one can set k = 1 by making a simple rescaling of q, and q+. Let us now write down the complete list of (anti)commutation relations of the superalgebra: Copyright © 1998 IOP Publishing Ltd

Supersymmetry and Superspuce

141

(2.2.9)

{ 41, q?} = 2(‘a)Z&pQ This real superalgebra is known as the ‘Poincare superalgebra’. It will be denoted by SP. Its a-type generators q* and qjl are called ‘supersymmetry generators’. Every element X of the Poincare superalgebra can be represented, in agreement with subsection 2.1.7, as follows:

+ ,,h(K”q, + i;*q&) b“, Kab =

[w

K”, E’ = (K”)*

(2.2.10)

For later use, we rewrite the Poincare superalgebra in spinor notation, converting every vector index into a pair of dotted and undotted indices. When applied to the Poincare generators, this operation leads to the change {pa,jQb} I P , ~ i,/j,Ls}. , Then one finds -+

ti,g,

i P;J = - E;,P/C

i + -E;pP*j. 2

2

i Cjlp,j,d = -(E;&&

+ ~ ; , r j , d + ~ s & p + ~sPj;,,)

2

tiqh

s;l

i

= -E y Q p 2

{QZ,

zi’>

i

+ -2 E;’&

(2.2.1 1)

= 2P.,

Other (anti)commutators vanish or may be found by Hermitian conjugation. 2.2.2. Extended PoincurP superulgebrus It was shown above that the Poincare superalgebra is the only possible superalgebra such that its even part coincides with 9 and the odd part transforms in the real (Majorana) representation (i,0) 0 (0, $) of the Lorentz group. However, one can consider a more general problem: to find all possible superalgebras A with ‘4being of the form (2.2.2)and ‘Abeing generated by a set of elements, each of which carries at least one spinor index. Then, ‘A!} c ‘A, it follows from the Coleman-Mandula theorem that since {‘A, every generator of ’AV should carry exactly one spinor index (otherwise, O C k’ Copyright © 1998 IOP Publishing Ltd

142

Iu‘eas and Metimis of Supersymmetry and Supergravity

will contain a Lorentz tensor generator different from the Poincare generators). Of course, we wish to have a real superalgebra A’. Therefore, ‘A’ is a direct sum of several real (& 0) 0 (0, $) representations, & ’e.i! is generated by elements qlA and qhB,where A , B = 1,2,. . . , N, with qCtAbeing Hermitian conjugate of q / . Further, one must satisfy equations (2.1.18)and (2.1.19). These equations give very strong restrictions on the structure constants. An analysis similar to that of subsection 2.2.1 leads to the following (anti)commutation relations: &b,

q,Al = i(crab),Pq/jA

uab, q’A1

[Pm q / l =

= i(aab)k@”A

k-hpB3=

{qiA, qflB)

{q/,

qkB)

[CAB, q z D ] LEAB,

[ti,

[pay q ’ A ]

= &hPAB

= 2(a”)diPa

=

[CAB, q & D ]

=0

=0

(2.2.12)

q?] = [ C A B , q k D 1 =

q/l

=

-(Si)ABqa

B

[ti, q k A 1 = ( s i ) B A q k B I t . t . 1 = if..kt 1’ J V k together with the Poincare commutation relations. Here {ti} are generators {c“” = -cBA, CAB = -eBA) are generators of the semi-simple Lie algebra of the Abelian algebra g2,see (2.2.2), with CAB the Hermitian conjugate of cAB. The matrices ( S i ) A B are Hermitian, S: = Si, and they form a representation of the algebra Q,, [Si, S j ] = if,kS,. Finally, the generators cAB and CAB of g 2 should be ‘invariant tensors’ of %,, i.e. one has (2.2.13) As may be seen, the generators cAB and CAB commute with every element of the superalgebra. On these grounds, they are called ‘central charges’. Note that central charges may exist when N 2 2. For a more detailed derivation of equations (2.2.12)and (2.2.13),see the book by J. Wess and J. Bagger. The superalgebra (2.2.12)is known as an “-extended Poincare superalgebra’, depending on the number of spinor generators. The Poincare superalgebra (2.2.9)is called the N = 1 (or ‘simple’) Poincare superalgebra. It follows from the above consideration that the superalgebra (2.2.12)is the most general (finite-dimensional) extension of the Poincare algebra, consistent with axioms of quantum field theory. This assertion is known as the Haag, Lopuszanski and Sohnius theorem. Copyright © 1998 IOP Publishing Ltd

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143

From now on, we shall study the N = 1 Poincare superalgebra only; this case, being simple, is worked out in detail and contains the main ingredients of all supersymmetric theories. 2.2.3. Matrix realization of the Poincari superalgebra The Poincare superalgebra has been introduced above as an abstract superalgebra. Now we give its realization in terms of matrices. Let us consider in Mat(4,l I C) matrices {pa,jab,qrs,q&}defined as follows

- iCab

1:

0

- ioab

02

- Gab

\ o

0

0

0

0 0 0 0

I o (2.2.14)


144

Ideus und Methods of Supersjtmmetrj and Superyrat.itj

where 0, means the zero n x n matrix. The matrices ya, x a b and y s were introduced in Section 1.4. One can readily check that the matrices (2.2.14) satisfy the (anti)commutation relations (2.2.9). Note that the superalgebra, generated by the matrices (2.2.14), is a subalgebra of 4 4 , 1 IC).

2.2.4. Grussnzunn shell of the Poincarh superalgebru As we know, with every complex superalgebra 9(@) one can relate a Berezin superalgebra %(Ax) (the Grassmann shell of $(@)) and a super Lie algebra OS(A,) (the even part of %(Az)). We now construct these objects for the complex shell SP(@) of the Poincare superalgebra. It is a complex superalgebra of dimension (10 4) with a general element X E S 9 ( @ )of the form

+

X = xap,

1 + -xabjab + x'q, + xiqq 2

xa, xab=

-Pa,x,

(2.2.15)

X i E @.

Recall that elements of the Poincare superalgebra have the form (2.2.10). Now, introduce the Grassmann shell S9P(Am) of SP(@).It is a Berezin superalgebra of dimension (10,4) with a pure basis {pa,jab,q,, qi) such that (2.2.16) for any pure supernumber Z E A , . Every element i c M ' ( A = ) can be represented as follows 1 2

2 = tapa+ - 2, vanishes. Then, projectors on the Copyright © 1998 IOP Publishing Ltd

subspaces X

(

+

,- X , -, and

X ( * ,turn out to be

(2.3.16) P(i)P,j)= dijP(i)

where the indices i, j take values

+ , -, 0. Further, one can prove the relation

z u f l , z ,= w a

1 2

f -pa

Then, due to equation (2.3.13), we have (2.3.19) Therefore, each of the subspaces X ( + and ) X ( - )carries the spin Y . I t is worth pointing out one more important observation. Namely, any state 1") E X may be obtained by acting with the supersymmetry generators For example, on states from one of the subspaces 2 , +&)( - ), and Sf(,,,. , be represented as starting from #(+), every state / " ) E H ( -can

every state I Y ) E . X ( is~ )represented as

Together with equations (2.3.19),this observation means that a representation Copyright © 1998 IOP Publishing Ltd

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Supersynimetry und Superspace

of the Poincare superalgebra in a Hilbert space H is irreducible if and only if the corresponding Poincare representations in the subspaces A?(+) and .H, - ) are irreducible. The only remaining task is to decompose X,,, into irreducible Poincare representations. For this purpose, let us once more consider in H the subspace V,, where, as usual, qa is the momentum of a particle at rest. The Pauli-Lubanski vector reduces on V , to the form (1.5.27),where the operators S/satisfy the commutation relations (1.5.28)of the algebra 4 2 ) . The equation (2.3.4) reduces on V , to the form (2.3.20) where (a,),", I = 1,2,3, are the ordinary Pauli matrices. So, the supersymmetry generators Q, form an SU(2)-spinor on the subspace V,. The subspace V , is decomposed into the direct sum V, =

Vq(+ 1

(2.3.21)

0 V,( - ,eV q o ,

where Vq(+ = V , n .X,+ ) and so on. Since X (+ ) describes the spin- Y Poincare representation, one can choose a basis { lY,l,2,,, ,,,)} in V q ( + ,to be totally symmetric SU(2)-tensor of rank n = 2Y, l y % l z 2 . . .,,,>

=

l~(,,,~...,,,)>

V4(+).

E

Then, the states Q,

I y x , , > . . . x , , > E Vq(0)

generate V,,,,. In accordance with equation (2.3.20), these states represent an SU(2)-tensor. It contains two irreducible (totally symmetric) SU(Z)-tensors: 1 n+l I'yY,12...z,,+,) = Q 2 1 1 % . . . x , / _ I > - - & , * A Q ~ l y % 2 .A ... .. e,,,+,> n 1&=2

+ 1

and

I y,,x2,. . x,,

~

,> = Q'I

y;'l,?z.. . z,, ,>. ~

Evidently, we have Wy,,,? . . . 1, , _ I > = ms'(s'+ w % l * 2 . . . x , ,

,> =

,S"(S"

+

l)l~,,m2.,.X"+,>

,>

~ ) l ~ X , m ~ . . ~ x "

S I =

S" =

y + 1/2 y - 1/2.

As a result, Sc0, contains two irreducible Poincare representations: of spin ( Y $) and ( Y - $), respectively, when Y >O. In the case Y = 0, H c 0 , describes only one Poincare representation, of spin f-. To summarize, the unitary representations of the Poincare superalgebra are classified by mass and superspin. In the case Y # 0, the corresponding

+


irreducible representation describes four particles of spin Y - i, Y , Y , Y + with the same mass 171 # 0. When Y = 0, the representation describes two scalar particles and a spin-$ particle with the same mass. 2.3.4. M r r . s s l c . s , s ir.rcriuc,ihlc wprvseniutions N o w we are going to study massless unitary representations of the Poincare superalgebra. They are characterized by the massless equation P"P, = 0.

(2.3.22)

This equation gives very strong constraints on the supersymmetry generators. Using the supersymmetry algebra (2.2.9), one can prove the identity jP,p0'$, P@") = -2P,,P.

In accordance with equation (2.3.22), this operator should vanish. Then, for every state 1") from a Hilbert space Y? of particle states, we have .Po, i. being the helicity, one finds that the operators [I, and P, must be proportional to each other, L, K P, in every irreducible massless representation of the Poincare Copyright © 1998 IOP Publishing Ltd

154

Idem unci Methods of’ Supersymmetry und Supergruvity

superalgebra. So, we introduce a new quantum number K defined by

La = (IC

+ ;)Pa.

(2.3.33)

This quantum number is called ‘superhelicity’. Superhelicity characterizes massless representations of the Poincare superalgebra. Given some representation of superhelicity K, let us calculate helicity values on the subspaces 2 ( + and ) defined above. For any state IY) E Ha(+), we have

This gives

Analogously, one finds WZ&JY)= K f f x i l Y )

VIY)€Y?(-).

(2.3.35)

Therefore, the helicities of the S, + )- and X (- ,-Poincare representations are equal to ( K + +j and K, respectively. To summarize, the massless unitary representations of the Poincare i 1, i;,. . . . For a superalgebra are classified by superhelicity K, K = 0, given superhelicity K, the corresponding representation describes two massless f). particles of helicities K and (ICi-

++,

2.3.6. Equality of bosonic and fermionic degrees of freedom Our discussion of unitary representations of the Poincare superalgebra would be incomplete without pointing out one important consequence of the above results - that is, each S9-representation describes an equal number of bosonic and fermionic degrees of freedom. It is worth recalling what is usually understood by the notion ‘number of degrees of freedom’. Given a massive spin-s particle, its number of degrees of freedom, denoted by N , , is defined to be equal to the number of different spin polarizations of the particle at rest. In other words, N , coincides with the dimension of every subspace V , in a Hilbert space of one-particle spin-s states. In the massive case we have N , = dim V , = (2s 1). A massless helicity-i. particle has only one degree of freedom. Since the relation W, = EP, is Poincare covariant, every massless particle of definite helicity possesses one and only one spin polarization.

+


Supersymmetry and Superspace

155

We have seen above that the massive SP-representation of superspin Y describes four particles with spins Y - $, Y , Y , Y + i, i.e. two bosonic and two fermionic particles. Therefore, the complete number of bosonic (fermionic) degrees of freedom is equal to 2(2Y 1). For a given superhelicity ti, the corresponding massless S9-representation describes two massless particles having helicities ti and (ti + i),i.e. one bosonic and one fermionic particle.

+

2.4. Real superspace R4I4and superfields As is known, the unitary Poincare representations can be realized in terms of fields on Minkowski space (see Section 1.8.) Undoubtedly, it would be interesting to obtain analogous realizations for the unitary representations of the Poincare superalgebra. How is this done? Clearly, the existence of field Poincare representations was possible merely due to the fact that from the very beginning the Poincare group was introduced as the group of transformations in Minkowski space. As for the Poincare superalgebra, one cannot relate it to some ordinary Lie group but only to a super Lie group -the super Poincare group. A super Lie group cannot be realized as a group of transformations acting in some ordinary space R", only in some superspace W"7. So, to achieve the above aim, one may choose some reasonable superspace, then define on it some reasonable action of the super Poincare group and so on, and so forth. But what principle do we have to be guided by? Evidently, we cannot make use of the principle leading to the Poincare transformations. Recall, they are those transformations of R4 which preserve the Minkowski metric. In our case, we know nothing about either superspace or its metric. We have only the super Poincare group at our disposal. Fortunately, there exists a purely algebraic way to introduce Minkowski space and Poincare transformations starting from the Poincare group. It is this approach which may be generalized to the super case.

2.4.1. Minkowski space as the coset space l l j S O ( 3 , Let us consider the left coset space ll/S0(3, l)t, where ll is the Poincare group and SO(3,l)T is the Lorentz group. We are going to show that this coset space can be identified with Minkowski space. Points of the coset space SO(3, l ) T are equivalence classes. For any group element g E IT, its equivalence class Q is defined to be the following set of group elements:

J = (yh, hESO(3, l)T]..

(2.4.1)

Recall, every equivalence class Q is uniquely determined by its arbitrary Copyright © 1998 IOP Publishing Ltd

1 56

Idetrs trnd Methods of Supersymmetrp and Supergrwiry

element, i.e. 4 = 8,'d'h~SO(3,1)t. How can one parametrize points of the coset space? Elements of the Poincare group are parametrized by ten real variables ha, K O h = - Kha, via the exponential mapping,

In particular, translations and Lorentz transformations look like g(b, 0) and g(0, K ) , respectively. Since E, p ] p, making use of the Baker-Hausdorff formula gives (2.4.2~) y(h, K ) = g(x3 O M O , K )

-

d b , K ) = do, K M Y , 0 )

(2.4.2b)

where xa and y" are functions of b" and K". One can take the variables ( x a , K a b }or { y a , K a b }in the role of local coordinates on lI. The variables jx", K a b } prove to be best adapted to the coset space lIlSO(3, l)t, since for any elements g(bl, K , ) and g(b2,K , ) from the same equivalence class BE lI/S0(3,1)T we have g(b1,K 1) = g ( x , O)g(O,K 1)

g(b2, K,) = g(.-G O)g(O, K2).

As a result, every equivalence class g ( b , K ) is uniquely determined by the Since translations g(x,0) = translation g(x, 0) defined by equation (2.4.2~). exp (-ix"p,) are uniquely determined by four real numbers xa, we obtain a one-to-one correspondence betwen njSO(3,l)fand R4.Thus, we can identify ll/S0(3, l)f and R4 by the rule:

- g(b, K ) = g(x, 0)

exp [ - ix"p,].

(2.4.3)

Let us consider the left action of the Poincare group on the coset space Namely, with every group element g o ~ lwel relate a mapping &: lIlSO(3, lI/S0(3, l)f -+ lIjSO(3, 1)f of the coset space to itself, defined as follows: J

+ J'

= bo(#)= 909

VJ E lIlSO(3, l)t.

(2.4.4)

Clearly, we have d 1 o O 2 = gG2.Further, every transformation (2.4.4)induces some mapping of R, to itself, due to the identification (2.4.3).Let us analyse these transformations. It is sufficient to study two particular cases: the translations go = g(b, 0), and the Lorentz transformations g o = g(0, K ) . Translations In accordance with equation (2.4.4),one has g(x, 0)

-+

g(b, O)g(x,0) = g(x

+ b, 0).

Therefore, the corresponding transformation in R4 is x a + x'" = xa + b". Copyright © 1998 IOP Publishing Ltd

(2.4.5)


157

Lorenrz transformations In accordance with equation (2.4.4), one has

so

+

do, K M x , 0).

Since &j= 3, V h E SO(3, 1)T, we can write g(0, K M x , 0) = do, K)g(x,O)g(O, - K). The expression under the bar can be rewritten as g(0, K)g(x,O)g(O, - K ) = e j K h ' j h , e - i X U p " e - j K h ' j h ,= exp ( - iX"e:Kh'Lp,e-fKh~jh1). Making use of the Poincare algebra gives ,:Kh'jh,

Finally, since

K,b

-fKh'jh,

Pae = - K b a , we obtain

= (e -K)abpb,

g(0, K)g(x,0) = d x ' , 0)

where

x" = (eK)obxb.

(2.4.6)

The expressions(2.4.5)and (2.4.6)reproduce ordinary Poincare transformations. The above discussion shows that Minkowski space can be identified with the coset space lIlSO(3, l)t.

2.4.2. Real superspace R4I4 Let us try to generalize the left coset construction described above to the super case. Consider the coset space SIT/SO(3,1IR,)t, where SIT is the super Poincare group and SO(3,ll RJt is the Lorentz group over R,. Its elements are equivalence classes

S

= (gh, h E SO(3, 1 I RJr}

for any g E SlI. Elements of the Poincare supergroup are parametrized as in equation (2.2.20).Then, since U, p] p and G, q] q, the Baker-Hausdorff formula gives

-

-

Ab, E, 5 K ) = g(x, 6, 8,O)g(O,O, 0, K )

(2.4.7)

where real c-numbers X" and complex a-numbers 8" and 8' (conjugate to each other) are functions of the supergroup coordinates. Evidently, equation (2.4.7)generalizes equation (2.4.2a).The variables {x', e", 8', K O b } may be used to parametrize SIT, instead of the original coordinates. Now, it is easy to prove that points of the coset space Sn/SO(3,1I R,)? can be identified with points of real superspace R4I4 parametrized by the rule

W 4= { ( z A ) = (xa, e", 8&),8'


=

(e,)*,X a E R,,e, E e,}.

(2.4.8)

The identification is as follows g(b, E, F, K) =

exp [i( - xapa

+ eq + &)I.

(2.4.9)

Furthermore, one may define the left action of the super Poincare group

on the coset space SnjSO(3, 1 I RJr in the same fashion as in (2.4.4). This induces some action of the super Poincare group on the superspace R4I4due

to the identification (2.4.9). It is sufficient to find transformations of R4I4 corresponding to the supergroup elements (2.2.21-23). Using the Poincare superalgebra (2.2.9), one obtains: Translat ions .y'"

0'" = 92

+ b"

=

8;

= gi.

(2.4.10)

Lorenrz trun:forniations

x'" = (eK)abxb (2.4.11 )

We see that odd superspace coordinates 8" and 8' transform as (un)dotted spinors. T o derive equations (2.4.10, 1 l), one has to perform the same steps as in deriving equations (2.4.5,6). Supersymmetry trunsformations (2.4.12) Let us comment on equation (2.4.12). In accordance with prescription (2.4.4), the element g(E,E) (2.2.23) of the super Poincare group acts on the coset space by the rule g(z, 0 ) g(E, 0). One can write +

w,

+ Eq)] exp [i( - x B p a+ eq + Gj)] i.x"p,) exp [i(Eq + Eq)] exp [i(eq + @)I

g(c, ?) y(z, 0 ) = exp [i(Eq = exp ( -

because [p,,q,] = 0. Then equation (2.2.25) leads to equation (2.4.12). Combining equations (2.4.10-12) leads to the most general form of super Poincare transformations on ~ ~ 1 ~ : x'" = (eK)abxb+ i(Ba"C - &I)+ b" 8'" = (e- )

+ E" o e; = (eK),QBp+ ex

gi = (e")Jjap + E ~ * ~ B ' * Copyright © 1998 IOP Publishing Ltd

=

(e-")'*,@ + ?'*

(2.4.13)


159

So, the super Poincare group acts on the superspace R4I4as a group of linear inhomogeneous transformations. Using expressions (2.4.13),it is not difficult to obtain a supermatrix realization of Sll analogous to the matrix realization of ll given in subsection 1.5.1. Looking at equation (2.4.12),we see that the supersymmetry transformations represent z-independent shifts of the odd superspace coordinates together with &dependent shifts of the even superspace coordinates. More precisely, decomposing each even coordinate x", a = 0 , 1 , 2 , 3 , into its body and soul,

+

x" = (xQ), (x")s

(2.4.14)

the supersymmetry transformations change the soul leaving the body invariant, ( x ~+ ) ~( x " ) ~ + i ( h a E - EO"@ (x")~ + (xa), Even if all x" were soulless before making a supersymmetry transformation, they acquire some soul afterwards. Supersymmetry requires soul. Recall that the transformation (2.4.12) corresponds to the supergroup element g(E, E) (2.2.23). Let us consider an element 9 = (g(E1,?l)l-1(9(E29E d - 1g(~1J1)g(E29 E*)

E:

(2.4.15)

E:

where and are arbitrary undotted spinors. Due to equation (2.2.25),we have (g(E,Z))- = g( - E, - Z). O n the same grounds, one finds = e - ih'lp,,

b" = 2i(~~o"E, - E,G"E,). (2.4.16) Therefore, the sequence (2.4.15) of supersymmetry transformations on R4I4 presents a bodiless translation of the even superspace coordinates. In fact, every bodiless translation (2.4.10) may be represented as a sequence of supersymmetry transformations. It is useful to treat the superspace R4I4 as a trivial fibre bundle over Minkowski space such that the projection n: R4I4+ R4 from the fibre bundle into the base manifold-Minkowski space-is given by

e", 8,)) = ((x"),)

n((xU,

for any superspace point (x", e", 6,). Every Poincare transformation (2.4.5) or (2.4.6) on the base space can be extended to a transformation (2.4.10) or (2.4.1l), respectively, on the fibre bundle. Every super Poincare transformation (2.4.13) on the fibre bundle is projected into a Poincare transformation (X'a)B

= ((eKyb)R(Xh)R

+ (ba)B

on the base manifold, where (K"b)Band (ba)Bare the bodies of the parameters Kab and b". Copyright © 1998 IOP Publishing Ltd

160

Idmy

tind

Methods of Supersymmetry unci Supergruuitj-

2.4.3. Supersyrtiinetric intertial As is known, the Poincare transformations leave invariant the interval (1.1.2). It is not difficult to find its superspace generalization invariant under all super Poincare transformations. Consider a superspace two-point function w"(z,,z2), where z t and z: are arbitrary points of R4I4, defined by

w ~ zz 2~) =, (x2 - xly

+ i0,0"(8,

- 8,) - i(02 -

B,)a"B,. (2.4.17)

This two-point function proves to be invariant under supersymmetry transformations (2.4.12).Clearly, it is also invariant under translations (2.4.10). Hence every super Poincare transformation (2.4.13) leaves invariant the two-point function ds2 = wawa.

(2.4.18)

This is called a 'supersymmetric interval'. Let us recall also that every space-time transformation X" + x'" = f " ( x ) preserving the interval (1.1.2)is a Poincare transformation. For this reason, flat space-time admits a preferable class of reference systems - inertial systems, in which the space-time metric has flat form (1.1.2).Below we shall show that every superspace transformation z A zIA = f " ( z ) preserving the supersymmetric interval (2.4.17)presents some super Poincare transformation. So, by analogy with Minkowski space, one can speak about super-inertial systems - that is, reference systems (zA)on R4I4in which the supersymmetric interval has the form (2.4.17). Given two super-inertial systems, their coordinates are related by a super Poincare transformation. --f

2.4.4. Superfields A supersmooth function I/: R4I4+ Am on real superspace R4I4is said to be a 'superfield' (recall, supersmooth functions are defined to be smooth with respect to even superspace coordinates and analytic with respect to odd superspace coordinates). Since superspace is parametrized by rule (2.4.8)and spinor indices ci or 5 take only two values, the odd superspace coordinates satisfy the identities

e,e,e,

=

o

B , B ~= ~0.~

(2.4.19)

Due to the reduction rules (1.4.6), we also have 1 6I,gp - - E Ipe 2 2

eqp =

- -1& @ e 2

2 (2.4.20)


Supersj"ry

161

unci Superspuce

Therefore, an expansion of a superfield V ( z )= V ( x ,8, Q) in a power series in 0" and 8, reads V ( . Y , 0,O)

+ 02$z(.Y) + O h $ ( X ) + 02F(x) + P G ( x ) + ~ W , ( +X82OZE.,(x) ) + e2e,ijh(x)+ e 2 P q X )(2.4.21) .

= A(x)

The coefficients in this expansion are said to be 'component fields of the superfield'. From now on we restrict ourselves to the consideration of 'bosonic superfields'

v:R4I4+ @, and 'fermionic superfields'

v:R4I4+ c, only. In these cases component fields are ordinary bosonic and fermionic fields over Minkowski space. This should be understood as follows. Let V ( z ) be, for example, a bosonic superfield. Then, component fields A(x), F(x), G(x), C,(x) and D ( x ) are bosonic supersmooth functions on the c-number space R,: the other component fields are fermionic supersmooth functions on R.: Their restrictions from R,4 to R4 represent smooth bosonic and fermionic fields on R4 identified with Minkowski space. Given a superfield V ( z ) ,we define its complex conjugate superfield V*(z) by the rule

V * ( z )E (V(z))*

vz E

w4.

(2.4.22)

Since conjugation rules for the superspace coordinates have the form = x~

(ex)* = eh

(B4)* = ea

(e,e,{)*= ggeh =, (eZ)*= 8 2 (B,~B,)* = e,e,{ a (PI*= e 2 = o,~B, (eaag)* = w e

(2.4.23)

the component fields of V*(z)are related to the component fields of V ( z )in the following way v * ( ~ , = A*(x)+ ( - lp"ec"cp,(x) + ( - I ) " ( ~ ) B , $ ~ ( X ) eZG*(x)

+

e, e)

+ B ~ F * (+~ eau8c,*(x) ) + (- i p Y " t & ) + ( - i)1:(v)e2BJh(x) + e2e2D*(x) where cpx(x) = ( ( P h ( X ) ) *

9"(4 = ( @ W *

q x ( x )= (&(X))*

P(x)= (i"(x))*

and E ( V )is the Grassmann parity of V(z). Copyright © 1998 IOP Publishing Ltd

(2.4.24)

Introduce partial derivatives of the superspace coordinates (2.4.25) F a x h = dah

i,4zB

=S

, B o

[@ ,;

= 6;j

H p = 6",{

?,Q" =

?,e, = 0

s,ej = 0

?,xb

=

;+xb

= pep = 0.

We also define partial derivatives d A corresponding to the variables ZA

= (Xa,

o,, e"),

(2.4.26)

Recall that partial derivatives of even superspace coordinates are always left ones. Basic properties of the partial derivatives are: 1.

[a, 2B) =

- ( - 1 ) c 4 c ~ a g 2 A = 0;

(2.4.28) 4. (?,V)* = ZaV*, (?,V)*

=

- ( - l ) ~ ( V + V * (W)* = - (- 1)"'")av*.

Their derivation duplicates the general analysis of Section 1.10 with one modification: in Section 1.10 we parametrized superspaces R P l 4 by real even and odd variables, R4I4 is parametrized by four real c-number variables xu and four complex a-number variables 8" and 8' conjugate to each other. 2.4.5. SuperJield representations of the super Poincare group The notion of tensor fields is easily generalized to the superspace. For example, a tensor superfield of Lorentz type (n/2,m/2) is defined by two requirements: ( 1 ) in every super-inertial system, it is determined by a set of (n + l)(m + 1) superfields V,,,?. . . ,,,d,h2.. . *,,, (2) (component superfields), totally symmetric in n undotted indices and m dotted indices; (2) a super Poincare transformation (2.4.13)changes the component superfields according to the Copyright © 1998 IOP Publishing Ltd

163

Siipersynirtzeiry and Superspuce

law where the Lorentz generators M,, act on the external superfield indices only. , f , , l (totally z) symmetric in its Removing the restriction of ~ , l z ~ . , , ~ , , ~ l f , , ,being undotted indices and dotted indices, the transformation law (2.4.29) defines an arbitrary Lorentz tensor superfield. The operation of complex conjugation maps every tensor superfield & ? ? . . .,,p,/j,. . . p,,,(=) into ~ / ~~ ~, . ./ . / ~ , , , ~( , ~~ ~) ,~. .( ~ ,~.,.,,,/jl/~~.../j,,,(z))* ~, ,, ,

(2.4.30)

which is also a tensor superfield, in accordance with equation (2.4.29). Given a tensor superfield of Lorentz type (n/2,m/2), its complex conjugate tensor superfield has Lorentz type (m/2,n/2). In the case n = m, we can consider real tensor superfields defined by the equation As a rule, we will assume every tensor superfield carrying an even (odd) total number of indices to be bosonic (fermionic),

4&x,...

z , , f , i , .. . &))

=n

(mod 2).

+m

(2.4.32)

The notion of tensor superfields given above will play a most important role in subsequent chapters for two basic reasons. First, in the case of Poincare transformations (2.4.10, 1 l), equation (2.4.29) means nothing more than the fact that the component fields of V,,,?, z , , k l h 2 , ,f,#lz)are ordinary bosonic and fermionic tensor fields. For example, let V ( z ) be a scalar superfield transforming according to the law ,,

,

(2.4.33)

V'(z')= V ( z )

with respect to the super Poincare group. Choosing a space-time translation (2.4.10) gives

+ B'f?j'"X') + . =A ' ( ~ + ' ) e%$',(x') + & p ( x ' ) + = ~ ( x+)e%*%(.x)+ B,@ + .. . . +

A'(x') O'5yX(.Y')

* *

In the case of Lorentz transformations (2.4.11) equation (2.4.31) leads to

+ B l h ? j / i ( ~ '+) . . .

+

A ' ( ~ ' ) efZ$'&') - A ! ( ~ ! ) -

=A(x)

eye-+Kt"ot,i. ),P +

+

$'/{(x')+ Bi(e-tK'"'~,")'~?j~p(x') ,,.

+ e%$%(x)+ B&X) + . . .

Therefore, in expansion (2.4.2l), component fields A ( x ) , F(x), G(x) and D ( x ) are scalar fields, C,(x) is a vector field, cLl(x) and E&) are undotted spinor Copyright © 1998 IOP Publishing Ltd

fields and 4:,*(.~)and V'(x) are dotted spinor fields. The second reason why tensor superfields are of primary importance is that the transformation law (2.4.29) automatically provides us with a realization of supersymmetry transformations on component tensor fields. Therefore, if we work out a technique to handle superfields, in particular to construct super Poincare invariant functionals of superfields, we shall arrive at a supersymmetric field theory. Supersymmetry transformation laws of component fields will be analysed below. Tensor superfields provide us with representations of the super Poincare of tensor superfields of Lorentz group. For example, consider the space 2'(,,,,) type (n/2.m/2).With every element g =y(b, E, E, K ) of the super Poincare group we relate a one-to-one mapping

T(g):p ( H . , ? l ) --* p(ifn,m) which is given by rewriting the transformation law (2.4.29)in the form: ~ ( 8 )V: ( Z )-, ~ ' ( = z )e+K"h.wc,l'V(g.z )

v V ( Z E) Y ( ~ , ~ (2.4.34) )

where ~ 7 ehave suppressed indices. Here zIA= g - l . z A is the super Poincare transformation corresponding to the element g - ' . Evidently, the correspondence g -, T(g)determines a representation of the super Poincare group. Therefore, every representation operator T(g) can be expressed as T(g(b,&,E,K)) = exp[i( - b"P, + +KnbJan EQ+%)I

+

= exp[i($b"4P,h

+ K"/'J,,'+

R%,fi

+ E Q+a)](2.4.35)

where P,=dT(p,)

Jab=dTQub)

Q,=dT(q,)

Q4=dT(qi)

are generators of the representation. One can easily find the generators by recalling the explicit form for the super Poincare transformations. The result is

P, = - lC, . ?

Jab=i(XbZa-X,Llb

Q, = ia, Qh

+

(o,b)"'~B,d/j-(~,bP'Bh~fi--,b)

(2.4.36)

+(oU),$Vu

= - i?*-

dz(8),h2,

or, in spinor notation,

(2.4.37)


165

Supersymmetry und Superspuce where

As an exercise, one can check that the operators (2.4.36) really form a

representation of the Poincare superalgebra. The reader should always keep in mind that the Lorentz generators M u , in equations (2.4.34) and (2.4.36) act on external superfield indices. Infinitesimal supersymmetry transformations act on any tensor superfields by the law (2.4.39) As for the superspace supersymmetry transformations (2.4.12), they can also be written in terms of supersymmetry generators Q, and Qdras follows z '= ~ z A- i(cQ

+a ) z A .

(2.4.40)

2.4.6. Muss dimensions In conclusion, let us discuss a technical question regarding dimensions (in units of mass) of the superspace coordinates. Evidently, for the even superspace coordinates we have [xu]= - 1

To determine the dimension of 8" and (2.4.12), which forces us to demand

[e"] =[PI= - 1/2

[a,]

= 1.

(2.4.41)

Bi', it is necessary to recall equation

[a,]

= [&I = 1/2

(2.4.42)

After this, having a superfield of some fixed dimension, one can easily determine dimensions of its component fields. The concept of superspace and superfields was first introduced by A. Salam and J. Strathdee.

2.5. Complex superspace C4I2,chiral superfields and covariant derivatives We are now going to describe one more realization of the super Poincare group as a group of transformations in superspace. Using this realization will turn out to be very helpful in two respects, In the present section it will facilitate an introduction of chiral superfields-important representations of supersymmetry. Later, in Chapter 5, this realization will serve as a starting point for constructing supergravity-that is, a gauge theory of supersymmetry. The point of departure of our discussion is the observation that the set { xp+l, 8') of complex variables xp+ xa + i8aa8and 8" is closed with respect to the super Poincare group, because the supersymmetry transformations Copyright © 1998 IOP Publishing Ltd

166

1ticti.s (inti Methods of Supersjw"ry and Supergracitji

act on these variables as follows xp+) + x;:

+

= .XI+) 2iWG

+i W C

8" +

= 8" + E "

(2.5.1)

All the other results of this present section are in a sense consequences of this observation.

2.5.I. CotnpIe.u superspuce c4I2 Consider a complex superspace C4I2 spanned by four complex c-number coordinates ya and two complex a-number coordinates 8". We introduce super Poincare transformations on C4I2 by associating with every element q E SFI a one-to-one mapping ya -, 4''"= g yo, 8" + 8'" = g .8" of c4l2 to itself, defined as follows:

-

Translations

Lorentz transformations g(K).ya =(eK)abyb

9 8, =(exp(iKaboa,)),B8p

(2.5.26)

Supersymmetry transformations g ( c , E).4,a=ya+2i8aa~+i€aoC g(E, E ) . 8 " = 8 z + ~ a

(2.5.2~)

The most general form of super PoincarC transformations on C4I2reads (2.5.3) Evidently, expressions (2.5.2) define a group of transformations on C4I2, i.e. g 1 ( g 2 ~ y a ) = ( g l g 2 ) ~and y " g 1 ( g 2 ~ 8 " ) = ( g , g , ) ~ 8 afor , any elements g1 and g2 of the super Poincare group. Beautifully, since C4I2is complex, one can define an action on C4I2 of a complex super Lie group corresponding to the complex shell (2.2.17) of the super Poincare algebra. Elements of this super Lie group are parametrized by complex c-number variables p, K Z B =KB" and t'b=zLPu and complex a-number variables E" and

rz,

rx#(~,)*.

where, in contrast to the super Poincare group, E @ # ( K Z S ) * and Transformations of C4I2absent in expressions (2.5.2) have the form elKi' j, . yxk = (e - K ) l p y B ~ etK"i f 1 , ,g" = (e - K)z,ll@ (2.5.4~) e~Lf'jjj .y m = (e - E ) k p y x / j elLiJ ia . 8% = 8" (2.5.4b) '


Supersymmetry und Superspuce e". y" = y"

e't9.8" = 8" + E"

-

eicq,4'" = y o + 2ie0"r

-

ei:q,

8" = 8"

167 (2.5.4~) (2.5.4d)

The reader may forget, for a time, transformations (2.5.4) because they will be used only in Section 2.9 when studying the superconformal group.

2.5.2. Holomorphic superfields Complex superspace C4I2can equivalently be considered as a real superspace Rgi4with coordinates y", p=(y")*, 0" and P=(P)*. In general, a superfield on C4I2 is a supersmooth function U : C4I2+ A m of all these variables, U = U(y,J,f?,@. However, objects best adapted to the complex structure on C4I2are 'holomorphic superfields' depending on the variables y" and 8" only, Q =q y , e )

a@/aya= aqagi = o

(2.5.5)

and antiholomorphic superfields defined by \V

=\v ( g , ~ ) ay = a q a e i = 0.

(2.5.6)

Clearly, an antiholomorphic superfield is complex conjugate to a holomorphic superfield. Every holomorphic superfield can be expanded in a power series in 8": @(Y,@= 4 y ) + W , ( y )+ 02F(Y).

(2.5.7)

In accordance with equation (2.5.3), the super Poincare transformations represent holomorphic mappings of C412 to itself. Therefore, it is possible to introduce into consideration tensor holomorphic superfields like @z,z2. , , z , , + , + 2 , , . i , ~defined { y , 0 ) by the transformation law @ : , " 2 . . . z , , i l i 2..i,,$ . (",el)

= eiK"hMoh @w2 . . ",,i(&>. . . .i,,,(Y8)

(2.5.8)

with respect to the super Poincare transformations, where the Lorentz generators Mab act, as usual, on external superfield indices only. Similarly to ordinary holomorphic functions 011 C",a holomorphic superfield @(y,O) proves to be an analytic function of complex c-number variables y". This means that @(y,O) possesses a convergent Taylor expansion in y" near each point of the superspace. In particular, holomorphic superfields are not localizable in y", i.e. one cannot make a holomorphic superfield on C4I2non-vanishing in a small neighbourhood of some point y$ only. Evidently, this is a very severe restriction from the point of view of field theory. Nevertheless, there is a way to obtain localizable holomorphic superfields. One should simply restrict the region of the superfield definition from the whole superspace C412 to a surface in C4I2 such that, for every point (y",8") on the surface, the imaginary part of y" is bodiless, (y" - ya)B = 0. Copyright © 1998 IOP Publishing Ltd

(2.5.9)

This surface will be denoted and termed the 'complex truncated superspace'. The surface C?l2 is interesting for two reasons. First, due to equation (2.5.3), every super Poincare transformation maps C:12 on to itself. Therefore, the notion of tensor holomorphic superfields can be transferred from C4I2 to @: '. Secondly, every holomorphic superfield on may be represented in the form O,cJ,O)=

1-1

?'qX,O)

PI! i.Y"'

I

. . . ?XU" , ~\= + \

(Y-Y)"' . . . (4'- W"

-

2

2

(2.5.10)

i,

where O(.~,fl) is a supersmooth function of four real c-number variables xu and two complex a-number variables 8". Conversely, for every such superfunction O(x,O), equation (2.5.10) defines a holomorphic superfield on C:I2 (but not, in general, on C4I2).To confirm these assertions, the reader may recall the discussion in Section I . 10 concerning supersmooth functions. We adopt the convention that every holomorphic superfield appearing below is defined on 2.5.3. W 4us n surfiice in C41' Let us introduce a family of surfaces in C4I2determined by

4'" - pa = 2iH""(+j+ +j, 8, B)

(2.5.11)

Here X u ,a =0, 1, 2, 3, are real superfields on [W4I4. Every surface (2.5.1 1) can be parametrized by four real c-number variables xu, given by

xu =+(y + y)"

(2.5.12)

and four complex a-number variables 8" and conjugate to each other. Therefore, we may look on this surface as a real superspace R4I4equipped with a set of four real superfields Ha(x,8,B)on R4I4. So, it seems natural to use the notation R4I4(2) for such a surface. Now, let us try to find a super Poincare invariant surface lW414(#)-that is, a surface which moves into itselfwhen applying an arbitrary super Poincare transformation. First, invariance with respect to the translations (2.5.2~) imposes the restriction

+

S U ( x b,O,8) = #'(x,@,B)

because the combination (y- j ) is translationally invariant. We see that 2"" is a function of 8 and B only. Secondly, the invariance with respect to the Lorentz transformations (2.5.2b) leads to

S""(O',iJ') = (eK)",,Sb(8,B,. = k80'B, where The most general solution of this equation proves to be ,Ha



169

k is a constant real c-number. Finally, the invariance with respect to the supersymmetry transformations (2.5.2~)fixes this constant resulting in ,%a

=

wo.

(2.5.13)

As a result, we arrive at a beautiful conclusion. Namely, the family (R414(Af)> of surfaces in C4I2 includes a unique super Poincare invariant

~urface-Ft~~~(8afl. But this is not the whole story. Restricting transformations (2.5.2) from C4I2 to R414(&78)one finds they coincide with the super-Poincare transformations on R4I4(see equation (2.4.10-12)). Therefore, one can identify the superspace R4I4 with the surface R4i4(Ba@in C4I2. The identification works as follows (xa,

e", 8,)

C)

+

(xa ieaaB, 8%)

(2.5.14)

2.5.4. Chiral superfields Let z,,dr,, &,,Jy,O) be a tensor holomorphic superfield. Restricting it to the surface R414(8a8)and applying the prescription (2.5.14)leads to the tensor superfield ,

,,

(2.5.15) on real superspace [w4l4. Evidently, it is a quite trivial fact that we have really obtained a tensor superfield on R4I4.However, due to its importance, let us comment on it in detail. Under a super Poincare transformation, OorI,,,z,8,1 ,,,hnJy,O)changes according to the law (2.5.8). For every point (ya,eor) from the surface R414(0a8),the transformed point (y'",@') will also lie on R414(80@. Therefore, we have @;,,

,

+

. a , h , , , , z n ~ ~ ie'a8',8') '

= e+KYhM"h@orl,. , z n h l . ., ,,JX

+i e d e ) .

This gives which represents a tensor transformation law. Note that the transformed superfield depends on the superspace coordinates in the same fashion as in superfield (2.5.15),

@k, ...or,$

. . .,,,(Xkm = %,, , . z,,h, . , . h , j X + i e o m .

(2.5.16)

A tensor superfield on R4I4 defined by equation (2.5.15) is said to be a 'tensor chiral superfield'. Its conjugate tensor superfield

%,.. . l " , b , , . z,(x,m = @ a , . , . z , , , h , . .z,(X - i e a m ,

,

(2.5.17)

is said to be a 'tensor antichiral superfield'. As may be seen, chiral superfields depend on x and 8 only through the combination (x+iea8), antichiral superfields depend on x and 0 only through the combination (x-iea8). Obviously, the set of tensor chiral (or antichiral) superfields of Lorentz Copyright © 1998 IOP Publishing Ltd

type (n/2,m/2) forms a super Poincare invariant subspace in the space of tensor superfields of Lorentz type (n/2,m/2). So it would be desirable to obtain a covariant constraint selecting this subspace. This is an easy problem upon taking into account the observation that equation (2.5.15) can be rewritten in the form

Analogously, every tensor antichiral superfield can be represented in the form

2.5 .S. Covariant derivatives The differential operators D, and Dh introduced in the previous subsection anticommute with the supersymmetry generators

(2.5.22) These identities can be checked explicitly. However, there are two different ways to understand relations (2.5.22). First, consider a tensor chiral superfield @(z) (indices are suppressed). After applying an infinitesimal supersymmetry transformation, it takes the form

@'(z)= @(z)+ 6@(z)

6@(z)= i(eQ + EQ)@(z).

Superfields @(z)and W(z) are chiral D,@ = DZ@'= 0. Therefore, we must have [Db,

+ EQ] = 0

EQ

at least when acting on chiral superfields. Secondly, let us recall how the coset space SII/S0(3,1liw,)? has been identified above with real superspace iw4l4. Namely, points of the coset space are in one-to-one correspondence with elements g(z) = ei(- X ' P ~+ fh+ of the super Poincare group. The supersymmetry transformations act on these Copyright © 1998 IOP Publishing Ltd


elements by left shifts ly(,:,E)

17 1

-

g ( z ) -, g(z’)= lycEag(z)= ei(cqj- “)g(z)

and lead to the superspace transformation zA+ f

A

= z A - i(EQ

+a ) z A

(2.3.23)

Now, consider right shifts ry(,,q)

+a

g(2) + g(2’) = vycslT,g(z)=g(2)e”qq which induce the superspace transformation zA+

=

+ (vD + Q)zA

(2.5.24)

Since the left and right shifts commute, lg(c,E)ry(q,q) = rg(q,e+g(c,z) and from their explicit expressions (2.5.23) and (2.5.24),one obtains expression (2.5.22). Derivatives

D,

= (aa,D,,

D’)

(2.5.25)

are seen to form a cor,plete set of first-order differential operators commuting with the supersymmetry transformations,

(2.5.26) [D,, EQ+ $1 = 0 This is the first reason why D, are called ‘covariant derivatives’. The second reason is that for every tensor superfield Vx, ,,&, , +,,jz), the superfield ,, ,

U,,, . . . l,,‘,. . . &&I = D,

K, .

,,

, ,

z,,’,

.. .

turns out also to be a tensor superfield, with the transformation law U% . . . ,,,&I . . . &’) = eiKh‘Mh‘ U,,,

, ,,

a,,’,

,,,

&,,jZ)

with respect to the super Poincare group. Here the Lorentz generators Mbc act on all external indices including index A. In other words, a covariant derivative moves every tensor superfieid into a tensor superfield. The operators D, and Dh are said to be ‘spinor covariant derivatives’.

2.5.6. Properties of couariant deriuritives The covariant derivatives satisfy the algebra

We see that the spinor covariant derivatives and the supersymmetry generators satisfy similar anticommutation relations. Furthermore, the dotted and undotted covariant derivatives are related by complex conjugation as Copyright © 1998 IOP Publishing Ltd

1 72

Ideas and Methods of' Supersyrnmetrj und Superyruuity

follows

(D,I/)*=(- l)':(')Dkl/*

(D21/)*=D2V*

(2.5.28)

for an arbitrary superfield V ( z ) (bosonic or fermionic), where we have introduced the notation D2= D"D, (2.5.29) D' = D * D k Let us list the identities involving the covariant derivatives which are known to be most relevant for practical superfield calculations:

D,D,j = $&,pD' D,D,jD;, = 0

- -

DkD) = - + E . 3P.D2

(2.5.304

D,DpD, = 0

(2.5.30b)

[D', D,]

(2.5.304

[D', Dk] = - 4id,%D2

= 4i?,+Dk

D X D ~ D=,D p D k

(2.5.30d)

D2D2+ D2D2- 2D"D2D,= 1 6 0

(2.5.304

D2DkD2= 0 D2D2D2= 16D2n

D2D,D2 = 0

(2.5.30f )

D2D2DZ= 16D20

(2.5.309)

All the identities can be readily proven with the help of relations (2.5.27). On the same grounds, one can see that a product of n 2 5 spinor covariant derivatives may be reduced to an expression containing terms with at most four D and D factors. It is worth pointing out two simple applications of the above identities. Given a tensor superfield ,, ,,,,&), the object ,

(2.5.31 ) is a tensor chiral superfield, and the object (2.5.32) D2v,,. . . %,+, . . .+,,,(4 is a tensor antichiral superfield. The reader may check that every tensor chiral superfield can be represented in the form (2.5.31). Furthermore, for every chiral superfield @(z),D+@= 0, we have

D2D,@=0

(2.5.33)

as a consequence of equation (2.5.30~). 2.6. The on-shell massive superfield representations

The main goal of the present section is to give a realization in terms of superfields for the massive super Poincare representations described in Section 2.3. Copyright © 1998 IOP Publishing Ltd


173

2.6. I . On-shell massive superfields To begin with, we must formulate what is to be understood by the notion ‘on-shell massive superfield’. By analogy with the Poincare case, every tensor superfield of Lorentz type (A/2,B/2),

Kl... x , , h l I . .

= v x l . . .z,,Hfl , . h,,)(z)

(2.6.1)

satisfying the mass-shell equation

(U-m2)v,I . . .

x,&

/.,.

&)=O

m2>0

(2.6.2)

Pz&=-id,&

(2.6.3)

and the supplementary condition

pz%,

I...

x,, ] h U l , . . * ,

,(z)=O

imposed when A # O and BfO, is said to be an on-shell massive superfield of Lorentz type (A/2,B/2). For arbitrary non-negative integers A and B, A + B = 2 X the space of all on-shell massive (A/2,B/2)-type superfields, denoted by &?(A,B), forms a representation of the super Poincare group. Given a non-negative (half-)integer spaces &?(2LO), %(2y-1,1), . . ., %(0,2y) describe equivalent representations of the super Poincare group. This assertion can be proved in the same fashion as was done in the Poincare case. Namely, the operator Au& (1.8.6) is invertible under the fulfilment of equation (2.6.2) and provides us with a one-to-one mapping of &?(2z0) on X,A,B), where B # 0 and A + B = 2 I: defined as follows K 1 . . . z A j l l . . . ~ g = A ’ l.**A’lBh,K, ttl . . . aAyl . . . y B

(2.6.4)

where Vzl..,zA+iz)is an arbitrary element of jiEo(2xo). In contrast to the Poincare case, every space %(A,B) constitutes a reducible representation of the super Poincare group, because &?(A,B) contains at least three super Poincare invariant subspaces:

{ v(z) E * ( A , B )

D&v(z) = O> (2.6.5)

D2v(Z)= D2v(Z)= o} { v(Z)E &(A,B) where superfield indices have been suppressed. It is seen that X{A+,)B) consists of chiral on-shell superfields and includes antichiral on-shell superfields only. Every element of is said to be a ‘linear superfield’. One can easily find projection operators for the subspaces XI&, %{,&, and &?[:!Bp Making use of identities (2.5.30) gives i%f[:!~):

&?I;,$

1 D2D2- 1 D 2 D 2 g?(+)=16 0 16m2 1 D2D2- 1 D2D2 --16 0 16m2

g?(-)=-


(2.6.6)

1 74

I d e m and Methods of Supersymmetry und Supergruvit,s 1 1 D,D*D, = - _ _ D'D'D, B(,)= - 8 0 8m2

= -- 1

8m2

D'D

2D'

9@(,) = bijY(,) where 9(i) = (?(+

j,

2,-), 9(,,). The equation (2.5.30e) leads to

9( + j + Y(-) + P C O=,0

(2.6.7)

therefore we have the decomposition yi"A.Bj=

s[,&)@2~A.)B)@2{~)B).

(2.6.8)

It is not difficult to see that the super Poincare representations on the are irreducible. Now, we have to decompose spaces 2[,$h) (but not on 2[i)B)) * [ i ) B ) into a direct sum of invariant subspaces and then determine superspin values corresponding to each of the irreducible representations under consideration. But before doing this, it is worth discussing the question of where the difference between the Poincare and super Poincare cases lies. Why were restrictions (2.6.1-3) sufficient in the Poincare case to select out irreducible representations and yet they have proved to be incomplete in order to play the same role in the super Poincare case? The point is that in superspace, in contrast to Minkowski space, we have at our disposal not only the super Poincare generators acting on superfields but also the spinor covariant derivatives which possess the property of preserving tensor is endowed structure when acting on superfields. As a result, each space with the action of a super Lie algebra including the super Poincare algebra as a subalgebra. 2.6.2. Extended super-Poincurt. algebra Following E. Sokatchev, let us extend the set of super Poincare generators (2.4.37) by adding the spinor covariant derivatives and consider the linear differential operators on

i(tPPP,h + K ~ P J , +~ R ' B J ~ ~ + EQ+ EQ) + VD+

(2.6.9)

where the super Poincare parameter are defined in the standard way and (yl",ijh) are a-numbers forming a Majorana spinor. The set of all operators (2.6.9) is seen to form a super Lie algebra with respect to the ordinary Lie bracket, and the generators of the algebra satisfy the following (anti)commutation relations: [J,,, Pyj,]= f ~ . , , ~ P f)~?pP,j, i [J,,,

+ Jy6]= ) ( g y a J p 6 +

gj.pJa6

+ ~d,J;,p+ E ~ P J ~ , )

CJap3 Q;'l= ~ E ~ ~+igrpQ1 Q P

[J,,, D?] = ) E ~ , D + i~~ ~ , j D ,

IQ,, 0')= {D,, DtJ = 2P,h.


(2.6.10)


175

The remaining (anti)commutators vanish or may be found by Hermitian conjugation (using the rule (P?,)' =PE,, (Jup!+=Jkb, (Q,)' =Q, and (D,)+ = D,). The algebra (2.6.10) presents nothing more than the N = 2 Poincare superalgebra without central charges (see Section 2.2). It is a simple exercise to check that every space jlE4!,~) has no non-trivial subspaces invariant under all of the operators (2.6.9). It IS the spinor covariant derivatives which mix superfields from X[A+,)B), X & B )and X$,!Bp 2.6.3. The superspin operator As the next step, we express the superspin operator (see subsection 2.3.2),

corresponding to superfield representations,

c =(Z, P)2 - Z2P2

(2.6.1 1)

where (2.6.12) in an explicitly supersymmetricaliy invariant form. The point is that superspace is endowed with not only the supersymmetry generators but the covariant derivatives also. The covariant derivatives, being supersymmetric invariant objects, have a structure similar to the structure of the supersymmetry generators. On these grounds, it is worth expecting that the Casimir operator C can be re-expressed in terms of D, and Dk. The basic observation is that the operator Z,can be rewritten, after some algebra with the super Poincare generators (2.4.36), using the rule Z, = 2, - A . P,

Z, = -$&abcdMbCpd +$(C,)k"[Da, D&] A =)(@ap - 888').

(2.6.13)

Here A is the generator of 7,-rotations v(x,B,@ + v'(x,e,@=V(x,eW,e-+q@.

It commutes with P,, so we have Z[aPb]=z[RPb]

and the superspin operator takes the form

c = (Z, P)2 - Z2P2

(2.6.14)

Obviously, this expression is explicitly supersymmetrically invariant. The operator 2, consists of two terms. The first one,

w q =-)&,bCdMbcpd Copyright © 1998 IOP Publishing Ltd

qz2= M,/jPflk- M,lP,'i

(2.6.15)

is the ‘space-time’ Pauli-Lubanski vector. It is sensitive to superfield tensor types and x-dependence but not to &dependence. The second one, iL?:‘[D,, D&],is sensitive to superfield x- and 6-dependence and is inert with respect to superfield tensor type. After using the identity *Pa = 0 and the mass-shell equation P2= - m2, one obtains

m2

1

+-m2 WX4[Dx,D’]. 4

+

C = - (P’”[D,, D4])2-- [D’, D‘][D,, Dk] m 2 q 2 64 32

(2.6.16) As for the third term here, we can profit from our old result (1.8.12),which leads to

*I.,’

I i.ni

=m2Y(Y+1)0

Y =(A+B)/2

(2.6.17)

The first and second terms in expression (2.6.16)can be simplified with the help of equation (2.5.27).The final expression for the superspin operator is (2.6.18)

where P(,,)is the projector onto the subspace of linear superfields in X((A,B). The operator B turns out to have the following interesting properties: (2.6.19) where B is assumed to act on X(A,B). So, we can rewrite equation (2.6.18)in the form CIfY‘”.”’=m4{ Y ( Y + l)O+(++B)P(,,}

(2.6.20)

Recalling equation (2.6.5), this immediately gives C I . ~= CI ~ H~. l A ~81= , m4 y(y + 1~

(2.6.21)

Therefore, each of the spaces A?[& of (anti)chiral on-shell superfields of Lorentz type (A/2,B/2) provides us with the superspin- Y representation of the Poincare superalgebra. 2.6.4. Decomposition of ~ F f iinto ),~ irreducible ) representations We are going to show that every space A B # 0, describes two irreducible super PoincarC representations of superspins ( Y ki),and the space &‘{& describes the super Poincare representation of superspin Since all of the spaces %‘&o), %IS:- 1 , 1 ) , . . . ,&‘[g,\lzy) realize equivalent

+

4.



I77

representations of the super Poincare group (see subsection 2.6.1), it is sufficient to restrict our consideration to the case B = 0. Let V,, 2 , E . Y T { ~be ! ~ an , arbitrary linear on-shell superfield, ,

When A Z O , we have the identity

(2.6.23) where parentheses (. . .) denote, as usual, the total symmetrization of indices, for example, i

A

and symbol !ik means that index ak is omitted. In accordance with equation ~ , exist chiral on-shell superfields (2.6.23), for every V,, ,, ~ & f [ s !there ( A + 1 . 0 ) and P / X , , , , Z ~ l E ~ ~ A+~l,O), %2 , . . .z , . , ,

,,

-

Dd?,

=(U-m2)Xal...lA+, =o

h,, .. , = (0 - m2)4a1.. . ,

z4

CtA

I

=0

such that the following representation

Kl.. . ZA= DYX./2,, . . a A+ D(a4a2. ..l a ) takes

(2.6.24)

Conversely, for arbitrary chiral on-shell superfields 1 , ~ ) the , superfield V,, x A constructed ) Val a A ~I E x,, . . ,4+ I E + 1 , ~and by the rule (2.6.24) belongs to %[i!o). Moreover, the correspondence ,

place.

(+I

,,,

,,,

( X a l . . . z A + l4 ' ai...aA&l)+

v,

I... 2 4

is one-to-one, because equation (2.6.24) can be resolved as follows: (2.6.25)

We come to the conclusion that the super Poincare representation on the space X$!o) is equivalent to the super Poincare representation on the direct l,o). For every positive (half-)integer K the space sum space X[Afil,o)@X[a+l&f[:ho), provides us with two irreducible super Poincare representations of superspins ( Y It is not difficult to find a suppplementary condition selecting out each superspin. Namely, the highest superspin, Y + 3, is extracted Copyright © 1998 IOP Publishing Ltd

178

Ideas wid Methods of' Supersymmetry und Supergravity

by the condition

D7y'21...?n I =o

(2.6.26)

in accordance with equation (2.6.24).The lowest superspin, Y -& is extracted by the condition D(Z,V,> ,A-l)=o.

(2.6.27)

Now, i t is worth recalling the decomposition (2.6.8). In accordance with ) equation (2.6.24), we can write every on-shell superfield U , , ,A E & ' ( ~ , ~in the form U P I

21

=@?,

I4

+Tal

Iq

+ wq"

Pq

+ D(cc,Va2

In)

(2.6.28)

X?, . . . l q c lE X

(+)

( A + 1,O)

rll, . . . X A

I

E

&'I2I,O)

which represents a decomposition into irreducible superfields. ~ ) be The case Y = O is treated similarly. Every superfield V E ~ Y ) ! ~can represented as 1 V = -- DaD2D,V 8m2

and hence

I/=D"X~

xa=

1 -

--

8m2

D2DzV

Xr E

x{:,b,.

(2.6.29)

These relations establish the equivalence of the super Poincare representations on H$!o,and i@{:,b), therefore the space 2{E!realizes o) the superspin-3 representation. By analogy with equation (2.6.28),every scalar on-shell superfield U E 3Y(o.o, can be written in the form (2.6.30) We summarize the results. If A # O or B Z O , the representation is the direct sum of four irreducible super Poincare representations with X X Y + i , where 2 Y = A + B . When A = B = O , the superspins Y-$, representation of 3Y(o,ol is the direct sum of three super Poincare representations with superspin 0, 0, 4. Every massive super Poincare representation can be realized in terms of (anti)chiral superfields. This is the reason why (anti)chiral superfields are objects of primary importance. Copyright © 1998 IOP Publishing Ltd

Supersymmerry und Superspuce

179

2.6.5. Projection operutors To complete the above consideration, it is worth finding projection operators extracting from H$!B) the subspaces of superspin Y - i and Y + i , respectively. To do this, one can use the following simple observation. If a linear operator F on a vector space 9takes eigenvalues f l , f 2 , . . . , f , such that 9 = Y 1 @ 9 * @ . . . @ 9FI,=Jf;:O ,

then projection operators on eigenspaces Piare given in the form

ninj=sijni

r I l + ... +n,=o.

In our case, the superspin operator acts on m 4 (~

Yf(,,B,

+ 1)

+

M~Y(Y

1 / 2 ) ( ~ 1/21

and its eigenvalues are

m4(y

+1

/ 2 +3/2) ~ ~

Then equations (2.6.19, 20) and the prescription just given lead to (2.6.31)

2.6.6. Real representutions The mapping of superfield complex conjugation defined by equation (2.4.30) converts a mass-shell space to 3?(B,A). Using this mapping and the operator Ax+which changes tensor type, one can define real massive superfield representations in the same fashion as was done in Section 1.8 for the massive field representations. In particular, in the case A = B one can impose the reality condition

-

T/11...lqhl...h4(4=

Kl...xAhl.,,&4(4

(2.6.32)

which defines a real tensor superfield. Evidently, the set of real on-shell superfields of Lorentz type (A/2,A/2) represents a super Poincare invariant subspace in To decompose a real on-shell superfield onto irreducible superfields, one can, as a first step, represent it in the form

v,,... ,”&,

-

. . . h” - U , , , . . , , h , . . .h4 + U , , . . , x 4 * , , , , h ,

(2.6.33)

with U , , , A h , , &, being some complex on-shell superfield; after this, it is sufficient to apply to U , , . a decomposition onto irreducible superfields as adopted in For example, given a real scalar superfield ,,,

,,

, ,,


,

1 80

Idem ~ 1 7 Methods d ?f Sirpersynzntetrjs und Supergrucity

V ( z ) ,we write it as

v=u+u and make use of equation (2.6.30). This gives

v = Q, + % + ~ “ q+?D$ Di@= D4q, = O

@

+ G + V,

D2vL= D2v, = 0.

(2.6.34)

I t is possible to subject superfields to extraordinary reality conditions for the simple reason that we have at our disposal the spinor covariant derivatives in addition to the space-time derivatives. In particular, when A = B + 1, one can demand the equation

D Y U y a ,...a w l ..ah = D y & , . . , a # a ,

..a,y

.

(2.6.35)

Let us analyse this equation in the simplest case A = 1 and B=O. Consider an arbitrary on-shell superfield U,(z). In accordance with equation (2.6.28), it can be represented in the form

Imposing the equation

D”U, = Djt(7? leads to the equality

(2.6.37)

- -.

DW, - DjtW = D2q- D2q. Since 0,is a chiral superfield, the left-hand side is a linear superfield. The expression on the right is the sum of (anti)chiral superfields. Therefore, we have q =0

-

-.

DW,, = DkW.

(2.6.38)

Taking D, and D4 on both sides gives -

D2Q,,= P l i a k

a D2G5=p,k@?z.

(2.6.39)

These relations provide us with one more example of possible reality conditions. A natural generalization of the final conditions to the scalar case reads -iD2Q,=pa

--41 D2G = pQ,

where p is a complex constant. Since here @ ( z ) is a chiral superfield, equation (2.6.40) leads to the mass-shell equations

(0 - lpl2)Q, = 0

(0 - Ip12)d,=O.

(2.6.40) = 0, (L.b.41 )

So, one can look on the reality condition (2.6.40) as an equation of motion for a chiral scalar superfield. When imposing the mass-shell equations (2.6.41j Copyright © 1998 IOP Publishing Ltd


18 1

only, (anti)chiral superfields (5 and Q, are independent and describe two irreducible representations of superspin Y = 0. However, choosing stronger equations (2.6.40)leads to dependence between (5 and Q,, therefore the system is reduced to describing a single superspin Y =O. In conclusion, let us point out that the constant p in equation (2.6.40) can be made real after making a redefinition @(z)+ e”Q,(z).

2.7. The on-shell massless superfield representations In this section, we give a realization in terms of superfields for the massless super Poincare representations described in Section 2.3. 2.7.I . Consistency conditions It has been shown in Section 2.3 that every massless unitary representation of the Poincare superalgebra is characterized by the operatorial constraints p,@* = P X h l i y= Q Z = IQ2 = 0.

Recall that these constraints were necessary to make the supersymmetry algebra consistent with the unitarity and the on-shell equation Pz=O. Since we intend to realize in superspace the massless unitary representations, we subject massless superfields to the operatorial constraints PxbQx= Px@ = 0

(2.7.1)

Qz=Qz=o

(2.7.2)

and

in addition to the on-shell equation P2=0.

(2.7.3)

Note that the supersymmetry generators in superspace can be written as Q, = i(8,

+ B’P,,)

+ OcxPadr)

Qh = - i(ah

which leads to POLdrQ, = ipzkaa, -i p p 2 .

Then, equations (2.7.1) and (2.7.3) lead to P,&P = P,&P= 0.

(2.7.4)

Analogously, equations (2.7.2-4) lead to

aua, = akah=0.


(2.7.5)

Furthermore, the covariant derivatives can be written in the form

D, = ?,

- PP?j,

Dj, = - ?j,

+ PP,j,.

Then, equations (2.7.3, 4) give

P,kD' = P,$D' = 0.

(2.7.6)

Analogously, equations (2.7.3, 5 , 6) give

D2= D2=O.

(2.7.7)

The above considerations show that the set of equations (2.7.1-3) is equivalent to the set of equations (2.7.3, 6, 7). However, the second set seems preferable to the first one, because the corresponding equations are explicitly supersymmetrically invariant. 2.7.2. On-shell massless superfields A tensor superfield of Lorentz type (A/2,B/2), V21...aahl,,,hB(~), is said to be an 'on-shell massless superfield' if it satisfies equations (2.7.3, 6, 7) and the supplementary conditions P 9 K ; z , , , , z 4 ,jlI . . .

(2.7.84

&)=O

pi+va i . . .C X A S ' P I . . . h&

,(4= 0

(2.7.8b)

as well. In fact, when A Z O or B#O, the on-shell equation (2.7.3) is a consequence of the supplementary conditions. We are going to classify on-shell massless superfields. First, consider the case A #O, B #O. Given a (A/Z,B/2)-type superfield under the supplementary conditions, we impose the first equation (2.7.6), p y p VI. . . Z A h i . . . h B ( 4 = 0.

Then, making use of equation (2.7.8a), this leads to p a l p K ) a 2 . .. a 4 4 , .. . h&)

= 0.

Analogously, imposing the second equation (2.7.6)and making use of equation (2.7.8b) gives

P 7 & , ~. .~. a Ab$ l- h 2 . . . h S ( Z ) = 0 . Therefore, we have

D'K,a,

] h ,. . . h " ( z ) = O

(2.7.9~1)

(2.7.9b) D'V,, . . . z,'ih, . . . hB ,(4= 0 since the superfields in the left-hand sides of equations (2.7.9) carry zero momentum. Now, both equations (2.7.7) are satisfied identically,



183

It is seen that the set of equations (2.7.6-8) is equivalent to equations (2.7.8,9). Furthermore, we are to consider three possible superfield types to which V,l .,4+ , , . h B may belong: (a) chiral; (b) antichiral; (c) neither chiral nor antichiral. In the first case, the total set of massless constraints (equivalent to equations (2.7.8, 9)) is (2.7.10) Similarly, in the second case we have

Dy% .. . XAhf .. . h , ( 4 = 0 D.@UI,.

P’qT,

.cc,+bl., , , ,ZA

,

h,

(2.7.11)

,(z)= 0

f h ,. . . &)

= 0.

Equations (2.7.10, 1 1) determine massless (anti)chiral superfields. Finally, let us consider the last case. Now, we can construct, starting from V,, he, two secondary superfields: ,, ,

, ,,

(2.7.12) and ‘el

,..*A+]hf . . . h B ( Z ) = D c r l ~ l . . , , A + l d r l

(2.7.13)

...hB(z)‘

The first superfield is chiral and symmetric (due to equation (2.7.9b)) in its dotted indices, hence it belongs to Lorentz type (A/2,(B+ 1)/2).What is more, it satisfies all the constraints (2.7.10). Similarly, Ga,,. a , , l h , . kB(z)is an antichiral superfield of Lorentz type ( ( A+ 1)/2,B/2) under constraints (2.7.11). Therefore, the (anti)chiral secondary superfields are also on-shell massless superfields. One can look on a general massless superfield as a superposition of massless (anti)chiral superfields. As the next step, let us treat the case A # O , B=O. Now, one can readily see that the total set of massless constraints is given by the equations ,

,,

Dj’y,l,..3L”,(z)=O (2.7.14)

P ~ ~ y , a l ,, ~ ( z,) ,=A O

D*V,]

,,,

JZ)

=o.

= P;,,B?,l,, ,aA(z)

When taking the massless superfield to be chiral, equations (2.7.14) are . J Z ) under simplified drastically. Namely, each (A/2,0)-type superfield the constraints (2.7.15)


proves to be massless. One more solution of equations (2.7.14) reads (2.7.16)

which defines an antichiral massless superfield of Lorentz type (A/2,0). Finally, in the case of a general massless superfield, neither chiral nor antichiral, one can construct two secondary superfields: . . . 1 ,&) =

0 2 ,

and a x l ...2

9

~

D J , , .. . 2,(4

(2.7.17)

,(4= D%,V2?.. . x j ,(z) +

(2.7.18)

which are chiral and antichiral massless superfields, respectively. The case A = 0 and B # 0 is treated in complete analogy with the previous one. So, we investigate the last possibility, A = B = 0. Now, the total set of massless constraints can be represented in the form D’V(z)

=

D’D,V(z)

=0

(2.7.19)

D2V(z)= D’D,V(z) = 0 due to the identities

[D’, Di] = 4P,iD“

[D’, D,]

=

-4P,kD‘.

There is no need to impose the on-shell equation

OV(z) = 0 since it follows by virtue of equation (2.5.30e), from equations (2.7.19). In accordance with constraints (2.7.19), a massless chiral superfield is defined by

D,@(z)= 0

D2@(z)= 0

(2.7.20)

and a massless antichiral scalar superfield is defined by

D,G(z) = 0

D’G(z) = 0.

(2.7.2 1)

In the case of a general massless scalar superfield V(z), one can construct two secondary massless superfields: chiral

0&) = D,V(z)

(2.7.22)

@,(z) = D%V(z).

(2.7.23)

and antichiral In the following chapters of our book it will be shown how massless superfields, described in this section, arise in supersymmetric field theories. Copyright © 1998 IOP Publishing Ltd

Supersymmetry und Superspuce

185

2.7.3.Superlielic~iiy

As is known, massless super Poincare representations are classified by

superhelicity K (see subsection 2.3.5).We are going to determine superhelicity values corresponding to all of the massless superfields considered earlier. For this purpose, we rewrite the superhelicity operator 1 16

- -((a,)"lCQ,,

Lo = W,

Qf]

w, = -1 &,bcdJbCpd 2

in an explicitly supersymmetric invariant form. Recalling expressions for the super-Poincare generators (2.4.36) and taking into account massless constraints (2.7.4),one finds 1

L,, = W,f - - [D,, DJ 8

(2.7.24)

where qzfis the 'space-time' Pauli-Lubanski vector (2.6.15). It acts on a superfield of Lorentz type ( 4 2 , B/2) subject to the supplementary conditions (2.7.8) as follows @;,:Vx1

..,,Acil

.,,&)

1 = - ( A - B)P,,S.V,,, , , , 4 f l .,.&)

2

(2.7.25)

(see subsection 1.8.3). We say that a massless superfield has a superhelicity ti if it satisfies the equation

(2.7.26) Only (anti)chiral massless superfields have definite superhelicities. Every chiral massless superfield of Lorentz type (4'2, B/2) proves to have superhelicity

(2.7.27) Every antichiral massless superfield of Lorentz type (A/2,B/2) proves to have superhelicity 1 1 (2.7.28) L~(A/l,B/l). antichiral = - ( A - B) - -. 2 2 Given a massless chiral superfield superfield

,

1 4 h l , ,,

*,,, its secondary antichiral

-

z 4 +, h , . . . & )

=Dal %?...

z,

+

f,(4

,f I . . .

is also massless, and both superfields have the same superhelicity. Copyright © 1998 IOP Publishing Ltd

186

Ideas and Methods of Supersymmetry und Supergruvity

2.8. From superfields to component fields In our opinion, the reader has had the opportunity to become convinced that working with superfields is not much harder than with fields in space-time. The formalism developed in Sections 2.4-2.7 makes it possible to handle a superfield as a simple indivisible object such that at any stage of practical calculations there is no need to think about its explicit construction in terms of the components fields. Unfortunately, at present it has not been possible to extract all the necessary physical information directly from superfields. For example, so far no-one has a clear unerstanding how to formulate canonical quantization on superfield language. It is a quite general situation that in order to do reasonable physical analysis of a supersymmetric theory one must re-express the theory in terms of components fields. Hence one should carry out the reduction from superfieids to components in an optimum way. In this section we describe the reduction technique invented by J. Wess and B. Zumino. 2.8.1. Chiral scalar superfield To fix the ideas let us start with a chiral scalar superfield @(z)= exp [i8aai2,]@(x, 8). Its component expansion is

+

@(x,8,8) = ~ ( x ) eZt)"(x)

i + ezF(x)+ i80a@,A(x) + -8zB8ad,t)(x) 2 (2.8.1)

In this expansion only the first three components are independent fields, the rest are secondary fields. Expression (2.8.1) determines the chiral superfield in terms of the component fields. Now we want to discuss the task of determining the component fields from the superfield. Introduce a mapping projecting every superfield V ( x ,8,8) into its zeroth-order (in 8" and 8') component field: (2.8.2) We will refer to this mapping as the 'space projection'. Obviously, component fields of any superfield can be obtained by first taking some partial derivatives in 8 and B and then applying the space projection. In the chiral scalar case we have

and so on. But proceeding in such a way, we have no explicit realization of Copyright © 1998 IOP Publishing Ltd


187

how, for example, [a,, 3b](€l is expressed through ( € 1. How can one implement this? Recalling expressions for the covariant derivatives, one finds the identities D,Vl = d,Vl

D,vi

= -&VI

DZVI = - P d , V / DZVl

=

[D,, D,] V = - [a,,

-&FV/

a,] VI

(2.8.3)

where V ( z ) is an arbitrary superfield. After this, the independent component fields of our chiral superfield can be defined as follows

It is now evident that, due to the chirality constraint Db(€= 0, the space projection of any number of Ds and Ds applied to @(z) is expressed in terms of the above fields. The given definition is very useful for obtaining supersymmetric transformation laws of component fields. Namely, in accordance with the identity i(EQ

+ a) = -(ED + ED) + 2i(~a"B- daT)8,

we have

i(EQ

+ a ) V /= -(ED + a ) V /

(2.8.5)

for every superfield V(z).Since the covariant derivatives anticommute with the supersymmetry generators, fields (2.8.4) transform according to the rule 6 A ( x ) = i(EQ

+ a)@ = -€DO

6+,(x) = D,{i(EQ =

=

-E$(x)

+ s)@}/ = i(EQ + B)D,@I

-EDD,@I - a D , @ / = - 2 ~ , F ( x ) - 2iF?dd,bA(x) (2.8.6) 1

6 F ( x ) = - -D2{i(EQ 4

i + a)(€}/ = - -(EQ + $)Dz(€l 4

14

= - E D D ~ (= € / - i

Ideas and methods of supersymmetry and supergravity, or, A walk through superspace

Introduction to Supersymmetry and Supergravity

Superspace. 1001 lessons in Supersymmetry

Time and Tide: A Walk Through Nantucket

Time and Tide: A Walk Through Nantucket

Supergravity and Superstrings A Geometric Perspective, Vol 2: Supergravity

A Walk Through the Bible

Walk Through Darkness

Walk Through Darkness

Unification and Supersymmetry

Supersymmetry: basics and concepts

Supersymmetry and Superfields

Fugitives and Refugees: A Walk through Portland, Oregon

Fugitives and Refugees: A Walk through Portland, Oregon

Supersymmetry and cosmology

Supersymmetry and Cosmology

Supersymmetry and quantum mechanics

Unification and Supersymmetry

Supersymmetry: Basics and Concepts

Unification and Supersymmetry

A Random Walk Through Fractal Dimensions

A Random Walk Through Fractal Dimensions

Land's end: a walk through Provincetown

Supersymmetry in Disorder and Chaos

Supersymmetry

The quantum theory of fields. Supersymmetry Supersymmetry

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