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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Preface to the First Edition
(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
Preface to the First Edition
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
Copyright © 1998 IOP Publishing Ltd
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
xx
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
(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
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
13
Mathematical Background
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
Copyright © 1998 IOP Publishing Ltd
=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)$fiMab.
After this redefinition, equation (1.3.5) takes the form T ( N )= exp(K",M,p
+ PfiA,fi)
(1.3.10)
Using the commutation relations (1.3.7), it is not difficult to obtain the Copyright © 1998 IOP Publishing Ltd
16
Idetis
utid
Metliocls
of' Supersymmetry
and Superyravit.v
commutation relations for generators M T aand
Mi,j. These are:
1 [ M , p M,iil=- (c,;,Mpa + E , ~ M P ; . + & ~ ; . M , ~ + E / ~ ~ M , ; . ~
2
[ M r p , Mi,jI = 0
(1.3.11 )
1
-
[A+,A??<<]= - jc~M,j[~+&~dAaq+EPI,A?id+E,jdIITl~~}.
2 It is now a simple exercise to check that the commutation relations of the operators M,,j ( M , , , M I , = M , , and M,,) coincides with those for standard generators of 4 2 , C).The same assertion is true for the algebra of generators A?i,!.So, the Lorentz algebra is isomorphic to a direct sum of two (mutually conjugate) sl(2, C) algebras. By virtue of equations (1.3.8)and (1.3.9), the generators and Akgact on undotted and dotted spinors according to the rules 1
Mzp(+;.) = j(E,&
+ E;'&,) ( 1.3.12)
These relations serve as the main motivation for the introduction of generators (1.3.9).We see that the undotted generators M U pmove undotted spinor indices only, while the dotted generators Aig give non-vanishing results only when acting on dotted spinor indices. Let us consider operators
c,= M @ M , ~ c2 --li;idjg.. Xp.
(1.3.1 3)
Using equation (1.3.1l), we find [C,, M,$ = O and [C,, R&g] =O. On the other hand, C, trivially commutes with M6g and C, commutes with M X p . So, C, and C, are the Casimir operators of the Lorentz group. We are going to calculate values of these operators in the (n/2, m / 2 ) representations. It is sufficient to find C , in the case of some (n/2,0) representation. Given a totally symmetric tensor I),,,?, ,T,>,from equation (1.3.12) we obtain ,
where Bi means that the corresponding index is omitted. Raising the indices I and p in the first relation (1.3.12) gives M'pI), Copyright © 1998 IOP Publishing Ltd
=
1
- - (S;$/' + 6fI)"). 2
Mathemuticul Buckground
17
is a Lorentz invariant tensor, we then have
Taking into account that 1
or, . ~ ; ' c ' M ; ~ ~ j ,+, rxc =-(c ,l r ~ , ,M;'"lj/. 2
,
.1,., , x,,
+ E,,S M;'% ,
. .%. . . Z")
I),
The last term here vanishes since $,l,,,a,z is totally symmetric. O n the same grounds, the first two terms give the following contribution 1
The above calculation leads to the final results: 1 M x ' ~ , p I ( n / z , m / Z ) = -- n(n + 2 ) ~ 2
a@fi .)
(1.3.14)
ip ( n / Z , m / Z )
1 --m(m+2)0*
2
Our final comment concerns the fact that the set of (n/2,m/2) representations, where n, m=O, 1, 2, . . ., actually embraces all irreducible finite-dimensional representations of the Lorentz group. As is well known, finite-dimensional representations of d(2, C) are parametrized by an integer n =0, 1, 2, .. .. The dimension of the corresponding representation is equal to ( n 1). Since the Lorentz algebra is the direct sum of two conjugate d(2, C ) copies (generated by M,p and its irreducible representations are parametrized by pair of integers n, m=O, 1, 2, . . ., and have the dimensions ( n l)(m 1). But the ( 4 2 , m/2)-sequence is characterized by just the same property.
+
+
ail),
+
1.4. Two-component and four-component spinors Following P. Dirac, half-integer spin particles are usually described in terms of four-component spinor fields. However, when working with field theories in a superspace, one has inevitably to break the habit of using the four-component spinor formalism and to develop the habit of using the two-component spinor formalism. The reason is in the fact that all operations Copyright © 1998 IOP Publishing Ltd
with two-component spinors are, as usual, much simpler than with the corresponding four-component ones. Now, we are going to show how two-component and four-component spinors are related, as well as how to translate two-component expressions into four-component language. To begin with, we list the main facts about two-component spinors. In accordance with the spin-statistics theorem, we shall assume all spinors to be anticommuting (odd elements of some Grassmann algebra), i.e. *zzp
= -zp*z
$jl;c/j
=
*& = -%p*,
- ZfiQjl
(1.4.11
for any spinors and xl. The detailed treatment of ‘anticommuting numbers’ will be given in Section 1.9. 1.4.I. Two-component spinors Let $?be an arbitrary undotted spinor, and 2’ be an arbitrary dotted spinor. Infinitesimal Lorentz transformations act on them by the rule:
(1.4.2a) Index raising and lowering gives
(1.4.2b) where we have used the observation that (oab),p and (cab)&$ are symmetric in their spinor indices. Finite Lorentz transformations are given by *i=(exp
K),P$~
jk=(exp R ) P j g .
(1.4.2~)
We define spinor bi-linear scalar and vector combinations
**
= $>;+24 = -pj, q2= $$ $ 0 ~ 2=(aa)zjl$z2k 2Zo$ = ~ , ( 6 ~ ) “ $ ~ .
$x = *“x,
=
-$ a x 1
*2
(1.4.3)
By virtue of equation (1,4.1), the following identities
$x=x$
$j=j$
$ a n i = --jZa$
(1.4.4)
hold. The difference in the definitions of $x and is reasonable since we wish to have a conjugation rule of the form: (+x)*=$j. But spinor conjugation should be understood as Hermitian conjugation (conjugation Copyright © 1998 IOP Publishing Ltd
Muthemutical Buckground and transposition of places), so ($x)* difference, we have
=($’zz)*
=(x,)*($”)*
= j#.
19
Due to this
*,x/,= $/d.+ C,/,$% -
$.? .-$.‘ - p/[+ - E?/>$Z. Yq.p
( I .4.5)
The product of two spinors can be reduced by the rules: 1
$ ~ X / , = - &2~ / ~ I C I Z - ~
1 (oah)z/<$aabX
(1.4.6a) (1.4.6b)
(1.4.6~)
and then use equation (1.2.33). Equation ( 1 . 4 . 6 ~is) a trivial consequence of equation (1.2.27).In group theoretical language, equation (1.4.54 means that (i,0)0(i,0)= (0,O)0( I , 0), while equation (1.4.5~)means that (i,0)0 (0,i)=
(4,$1.
Finally, we list the Fierz rearrangement rules. For arbitrary spinors $;, I&, $: and $2 we have ( $ 1 $2)($3$4)
=
-($I
$3)($2$4)-
-2
1
($1$2)($3$4)
=
(IC/ l$4)($2$3)
($ 1°a$4)($20a$3)*
(1.4.7)
One can easily prove these identities with the help of equations (1.4.5) and (1.4.6~). 1.4.2. Diruc spinors Let $z be an undotted spinor, and z a be a dotted spinor. We incorporate these spinors into a four-component column
YJ =
(;-).
(1.4.8)
This double spinor is called a ‘Dirac spinor’ (here and later, four-component Copyright © 1998 IOP Publishing Ltd
spinor objects are denoted by boldface letters). By virtue of equation (1.4.2a), an infinitesimal Lorentz transformation acts on \y by the rule 1
6" = -2 K a h ( :b Now, recalling the explicit form of to introduce 4 x 4 matrices
(1.4.9)
and Zah (equation (1.2.32)),it is natural (1.4.10)
Then, the transformation law ( I .4.9) takes the form 1 6Y =- KabZabY' 2 (1.4.11) The matrices (1 $4.10)satisfy the algebra la)
?b
1 = -2vobl4
(1.4.12)
so ;'a are the usual Dirac matrices (in the special representation). Equation (1.4.11) is the standard transformation law of Dirac spinors. Given a Dirac spinor (1.4.Q we conjugate IC/, to obtain $& and conjugate j* to obtain x'. Let us combine the resulting two-component spinors in a four-component row
q =(x5(, $d.
(1.4. 3)
Its transformation law, using equations (1.4.2b) and (1.4. 0,1l), is (1.4. 4)
PP=
What is more, satisfies the equation: q=Y+yo.So, Y is the ordinary Dirac conjugate spinor of Y . One more four-component object is obtained from by transposition of IC/ and x. Namely, let us consider the spinor
Evidently, it transforms as a Dirac spinor, 1
6\yc = - KabXabYc.
2
Copyright © 1998 IOP Publishing Ltd
(1.4.15)
21
Muthernuticul Buckyround What is more, it satisfies the equation
(1.4.16)
Y , = CYT
where C and its inverse matrix C - ' have the form
It is easy to see that C is a unitary antisymmetric matrix obeying the identity
c-'y,c= -$. So, C is the 'charge conjugation matrix', and Y, is the 'charge conjugate spinor' of Y .
1.4.3.Weyl spinors We introduce the Lorentz invariant matrix ys = -iy,y1y2y3 properties
with the
and define operators P, and PRas follows 1
P , =-(U,+ 2
7s)
1 PR =- ( 0 4 - 7 5 ) . 2
Here U is the 2 x 2 unit matrix and 0 is the 4 x 4 unit matrix. Due to the identities
Pi= P,
Pi = PR
P L P R =P R P,=
O
these operators are Lorentz invariant projectors. Acting with P, and PR on an arbitrary Dirac spinor Y , one obtains four-component objects (1.4.17) which transform as Dirac spinors and satisfy the Lorentz covariant constraints yS'f'L=YL ySYR=-'f'R. (1.4.18) The relations (1.4.17) show that \Y, and YR are the Dirac forms of two-component left-handed and right-handed Weyl spinors $orand jk correspondingly. Very often, the spinors Y , and YRare called Weyl spinors too. 1.4.4. Mujorunu spinors In our combinatorical exercises of subsection 1.4.2, we have not considered = xX. If %)I is an undotted two-component spinor, one interesting possibility: J/, Copyright © 1998 IOP Publishing Ltd
then the four-component object
(;)
(1.4.19)
IM =
transforms as a Dirac spinor, 1 2
=- K a b x a b y M .
Its main property is that I M coincides with its charge conjugate spinor, ( 1.4.20)
I, = CT;.
So YMis called a 'real' (or 'Majorana') four-component spinor. Any Dirac spinor (1.4.8) can be represented as a sum of two Majorana spinors m M and AM by the rule: = +M
+i h *
+U
= pa
+ ]AU
XU = VU -
(1.4.21)
1.4.5. The reduction rule and the Fier: identity The set of 4 x 4 matrices Y A=
{ 0 43 iya, 2izab, ~
0
~
1 5/ s },
forms a basis in the linear space of 4 x 4 matrices. Defining the corresponding set with upper indices, yA = { 0 , iy", 2iCab,yay,, ys}
we have the identities 1 4
,
y A y A = 0 (no sum).
- Tr(,fy,) = dA,
In accordance with these identities, if
r is a 4 x 4 matrix, then
or, in components,
Since
r is arbitrary, one obtains the following completeness relation
=-E (rA)kl(?A)ij. 4A 1
di16jk
(1.4.22)
This relation is a four-component version of equation (1.2.27). Now let VI,Y 2 ,Y, and I, be arbitrary Dirac spinors. Using equation Copyright © 1998 IOP Publishing Ltd
23
Muthemuticul Background
(1.4.22),one can easily show that
v
1
1i v 2 j
=iT1YA‘U,(yJji 1
(v32)(v3%)
=
-4
(~,YA‘Y4)(v311Av2).
(1.4.23) (1.4.24)
A
Equation ( 1.4.23) is a four-component generalization of the two-component reduction rules ( 1 -4.6).Equation (1.4.24)is a four-component version of the Fierz identities (1.4.7).After some practice, one can find that working with equations (1.4.6,7)is much simpler than with equations (1.4.23,24).
I .4.6. Two-component und four-component bi-lineur combinations In conclusion, we consider the connection between two-component and four-component bi-linear combinations. Given two Dirac spinors
we have T I V 2
=X l * 2
+$ 2 2
~1175~2=x1*2-$1Xz v1YaV2 vl*JaY5v, vlzabV2 v1cabY5v2
= X I c a i 2 - $2ca$1 =
- ~ 1 c a ~- J2/ 2 c a $ 1
=XlcabJ/2 xIcab@2
+$18ab22
- $18abi2*
These relations show how to express an arbitrary bi-linear combination of Dirac spinors in terms of two-component bi-linear combinations and vice versa.
1.5. Representations of the Poincare group I .5. I . The Poincare algebra To begin analysis of the Poincare group, we give its realization as a matrix Lie group. Let us associate with every element (A, b ) of the Poincare group a 5 x 5 matrix (A, b ) defined as follows:
(I Copyright © 1998 IOP Publishing Ltd
.s.I )
Since the multiplication law in the Poincare group is given as ( A 2 ,h2)x ( A l , b l ) = ( A 2 A l ,b,+h,h,), thecorrespondence(A,b)+(A,b)isan isomoprhism of the Poincare group on the subgroup of GL(5, R) consisting of matrices of the form ( l S . t ) , where Amnis restricted by equations (1.1.3,4) and h"' is an arbitrary four-vector. So, the Poincare group can be identified with this matrix group. We denote it by IT. Minkowski space can be identified with the hyperplane x4= 1 in R5 (with coordinates xo, X I , .. .,x4). Then, the linear operator (1 5 1 ) acts on this hyperplane as the corresponding Poincare transformation (A, b). The Lie algebra of the Lie group consists of 5 x 5 matrices of the form
n
Kmn = -Knm where K m , is an arbitrary antisymmetric Lorentz tensor and P i s an arbitrary four-vector. We identify this matrix Lie algebra with the Lie algebra of the Poincare group (the 'Poincare algebra' will be denoted by 9). The multiplication law in the Poincare algebra is
C(K2,
62>,
( K I 6~1 > I = ( [ K 2 , K , ] , K 2 6 1 - ~ 1 6 1 )
(1.5.3)
( ~ 2 6 1 ) ~ = ~ ~ , f i .
It is useful for later applications to introduce a basis (jab,po), jab= -jbo,for the Poincare algebra by the rule
(K,
i 2
6) = - Kabjob-@pa (1.5.4)
0 0
0
0
__
__
Po =
1
0
0
Here "Mabare the Lorentz generators in the vector representation (equation
(1.3.8~)). Using rule (1.5.4), we obtain the commutation relations in the Copyright © 1998 IOP Publishing Ltd
Mathemutical Background
25
Poincare algebra: [Pal
Pbl = O
Dab, Pcl = i?acPb - hbcPa tiab, jcdl = iqac.ibd- hadjbc + itlbdjac-
(1 S . 5 ) iqbc.iad*
n can be represented as
Note that every element (A, b ) from
b m = ( KeK-O, )
Amn= (exp K)",
" 6".
( 1.5.6)
n
Now, let T be a representation of the Poincare group, and T(A,b) be some representation operator. In accordance with equation (1.5.6), we have (1.5.7)
where the Poincare generators Jab=dT(jnb) and Pa=dT(pa) provide a representation of the Poincare algebra. Here Pa are the generators of translations and Jabarethe generators of Lorentz,transformations.As a result of equation (1.5.5), the generators satisfy the following commutation relations:
[pa, [Jab, Jcdl
pcl
Pbl
=
= iqucpb - h b c P a
= iqucJbd - h a d J b c
(1.5.8)
+ h b d J a c - iqbcJad*
These commutation relations define an arbitrary representation of the Poincare algebra. The Poincare group has two Casimiar operators (commuting with Pa and Jab)
cz = waw,
c,= -papa
(1 5 9 )
where W, is the Pauli-Lubanski vector ( 1.5.10)
Using equation (1.5.8), one can prove the following properties of the Pauli- Lu banski vector w"P,=o [Jab,
wcl
[w,, Pb]=o =iqacw
b
-h b c W a
[WO,w b] = i&abcdWCpd. Copyright © 1998 IOP Publishing Ltd
(1.5.1 la)
(1.5.1Ib) (1.5.1 IC)
26
I ~ C L Iund S Methocis of' Supersymmetry und Superyruvity
1.5.2. Field representutions
Let @(x)be an arbitrary Lorentz tensor field on Minkowski space (indices are suppressed). By definition, its components in two different coordinate systems, related by some Poincare transformation (1.1, l), are connected by the rule @'(x')= T(A)@(x)
(1.5.12)
where T(A) is a finite-dimensional representation of the Lorentz group. Proceeding from the transformation law (1.5.12),we construct a representation of the Poincare group acting in a linear space of tensor fields as follows (1.5.13) This representation is infinite dimensional. The corresponding Poincare generators are ( 1.5.14)
where Mob are the generators of the Lorentz representation T(A), T(A)=exp(iK"M,,).
1 S.3. Unitary representations In quantum field theory, Poincare invariance means that any Poincare transformation (A, b) induces a unitary transformation (1.5.15) acting in a Hilbert space of particle states. The union of operators U ( A , b ) provides us with a unitary representation of the Poincare group. Operators P, and J a b are the corresponding generators. The generators P, are identified with the energy-momentum operator, P, =( - W,p), where W and p are the Hamiltonian and the momentum of some quantized field. Therefore, the first Casimir operator (1.5.9) coincides with the squared mass operator. In any irreducible representation of the Poincare group, all particle states are eigenstates of this operator with the same mass, P2P,= --m20, m220.Note that there exist Poincare representations with negative mass squared. However, physically, such representations are not admissible. Under the supposition of non-negativity of mass, the Poincare group has one more Casimiar operator in addition to the operators (1.5.9), namely sign (Po). From a physical point of view, we should consider only positive-energy Copyright © 1998 IOP Publishing Ltd
27
Muthemuticul Background representations, characterized by the condition
( W O I W
(1.5.1 6 )
for any non-zero state 1"). The second Casimir operator (1.5.9) describes the spin of the particles. To clarify this assertion, we introduce one auxiliary notion which will be useful also when constructing irreducible representations of the (super) Poincare group. 1.5.4. Stubility subgroup In a Hilbert space of one-particle states with a given mass m, we consider the substance V, of particle states having a given four-momentum qa,
pa 14) = q a 14)
for any state 14)
E
V,. The four-vector qp,lies on the 'mass-shell' surface
papp,= - m 2
po
( 1.5.17)
in momentum space. We define the set H, of group elements (A, b) such that the corresponding operators U(A,b) transform V, onto itself. Evidently, H, forms a subgroup of the Poincartt group. It will be called the stability subgroup for V,. To find H,, note that, according to (1.5.8) an operator exp[(i/2)KabJ,,] transforms some state / q ) into another state Iq'), where q'" =(exp K)"bqb. Demanding q' = q, we obtain the equation Kp,bqb= 0, which has the following general solution: = &abcdqCnd where nd is an arbitrary vector. Therefore, the stability subgroup consists of elements of the form exp[ i( - bapa +-1 E"b"qcnddab)] 2
( 1.5.18)
for arbitrary vectors b, n. Being restricted to V,, elements of H, can be expressed in terms of the Pauli-Lubanski vector, since operator (1.5.18) coincides (on V,) with exp( - ia) exp( - h a w " )
( 1 ,5.19)
where a=baqa. It is worth now recalling the identify (1.5.1 IC). This indicates that components of the Pauli-Lubanski vector form a Lie algebra (so(3) in the massive case and so(2) in the massless case, see below) restricted to V,. So, the stability subgroup acts on V, as the U(1) x SO(3) group in the massive case and as the U(1) x U(1) group in the massless case. All vectors from V, describe particle states with the same momentum. On physical grounds, any two linearly independent states Il), 12) E V, should Copyright © 1998 IOP Publishing Ltd
28
Ideas und Methods of Supersymmetry and Supergravity
correspond to different spin polarizations to be transforming into each other under the action of H,. In accordance with expression (1.5.19), the stability subgroup is generated (up to phase factors) on V, by the Pauli-Lubanski vector. Hence, the spin properties of the particles are characterized by the Pauli-Lubanski vector. For a given particle, the spectrum of its spin polarizations contains only a finite number of values. This means that in any physically admissible irreducible representation of the Poincare group all subspaces V, are finite dimensional and H , acts irreducibly on V,. Now one could ask about the coset space n/H,of the Poincare group over some stability subgroup H,. It is clear that n/H,may be parameterized by a set of Lorentz transformations Cl[ p ] transferring the four-momentum q" into another four-momentum pa from the same 'mass-shell' surface (1.5.17), p"=(Q[p])",q*. Given some such family { Q [ p ] } , we find that a unitary operator U ( C l [ p ] , O ) transforms V, on to V,. In the massive case, the most useful choice for the test momentum q" is the momentum of a particle at rest. Then. a useful candidate for the role of { C l [ p ] } is the family of Lorentz transformations
W I=
( 1.5.20)
In the massless case, the test momentum is usually chosen in the form p" = ( E , 0, 0, E). The corresponding family (Cl[ p ] } is
--(l-a) 2 E
2 E
Wpl=
---(l+a)
--(l-a) 1 P' 2 E
n ' ( p ) m'(p) 1 P'
---(l-a) 1 P2 2 E
n 2 ( p ) m 2 ( p ) 1 P2
1P 3 --(I \ 2 E
-2)
Copyright © 1998 IOP Publishing Ltd
---(I + a )
n3(p) m3(p)
1P 3 ---(I +z)
( 1.5.2 1 )
Mathematical Background
29
Here f i ( p ) and &(p) are three-vectors such that the set { f i , &, B/lBl) forms an orthonormal basis in the space. Now we are in a position to describe the main steps in the construction of unitary irreducible Poincare representations. Since P"P, is the Casimir operator, irreducible Poincare representations are classified by mass. Let us fix some mass value m and choose a test momentum q" from the 'mass-shell' surface (1.5.17). Then one has to find all (finite-dimensional) unitary irreducible representat;9ns of H,, characterized by the condition that any operator (1.5.18) acts in accordance with rule (1.5.19). Suppose that T, is a given representation (with the described properties) of H, acting on a space V,, and let 14, o; m 2 ) be an orthonormal basis in V,; here the variable runs over a finite number of values (spin variable). Further, we formally associate with any point p a # q a on the 'mass-shell' surface (1.5.17) a vector space V, of the same dimension as V,. Let Jp, a; m 2 ) be a basis in V,. As the next step, we define a Hilbert space of one-particle states as a formal direct sum 2 = 0, V p (for all pa, including q") with the following inner product
1 p', 0';m') = E,6,,,6(
(p , o; m'
B -8').
(1.5.22)
Finally, let us define a unitary Poincare representation on 2 by four ~ the unitary operator U(A,b) restricted requirements: (1) for any (A, b )H,, to V, coincides with the operator T,(A,b);(2) each subspace V, carries the momentum pa, P,
IP,
0;m 2 ) = P o
IP,
a; m2>
(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
%=(-my
o,o,
(1.5.26)
0)
of a particle at rest. Being restricted to the subspace V,, components of wa=+&,bcddbcpd should have the form
wo=o
W,=mS1
l=l,2,3 (1.5.27)
and satisfy the algebra (1 5 2 8 )
The algebra ( 1 S.28) coincides with the angular momentum algebra su(2) (or so(3)). As is well known, finite-dimensional irreducible representations of su(2) are characterized by the condition ( S 1)2
+ (S# +(S,)Z= s(s+ 1)U
(1.5.29)
where possible values of s are: s=O, 3, 1, f, . . .. For a given s, the dimension of the representation is (2s+ 1). Equations (1.5.27,29) give W W , = ~ * S ( S +1)U
s=O, 1/2, 1, 3/2,
. . ..
(1.5.30)
This relation determines the spectrum of values which the spin operator can take in unitary irreducible massive Poincare representations. The quantum number s is called a spin, Hence, in the massive case, irreducible representations of the Poincare group are classified by mass m and spin s.
1 S.6. Mussless irreducible representations Now we are going to discuss the massless case, where Pap,= 0.
On the upper light-cone surface we choose the four-momentum q.=(-E,
0,0, E )
EZO
(l.S.31)
and analyse unitary finite-dimensional representations of the stationary subgroup H,. When acting on the subspace V,, the components of the Pauli-Lubanski Copyright © 1998 IOP Publishing Ltd
31
Mathematical Background vector are WO= -Ed12
w
1=
W3=E912
E(92 3 + 9 2 0 ) E ER,
W2 = E(J 13
+J
(1.5.32)
10) E ER2
where we have used equations (1 S.10) and (1.5.1 1 a). As a result of equation (1.5.1IC), the operators J I 2 , R, and R2 satisfy the algebra
(1.5.33) which is simply the Lie algebra of the group E2 of translations and rotations on a two-dimensional plane. The Casimir operator of E , is
(W2+ (WZ and in any unitary irreducible representation this operator looks like (Rl)2
+
(R,)2
p2 2 0.
= p20
(1 5 3 4 )
If p2 >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=
Copyright © 1998 IOP Publishing Ltd
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)
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
= 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:-))
Copyright © 1998 IOP Publishing Ltd
e=det(eam)
(1 6 4 3 )
Mathematical Background
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<%d9cd) = 0 as a consequence of equation (1A.46). Any conformally flat metric is a solution of these equations.
1.6.8. Energy-momentum tensor Let S [ 0 , eam] be a model of some field @=(ai} coupled to a gravity background. All informa'tion about coupling to gravity is encoded in the variational derivative of the action with respect to the vierbein
Tma(x)= - 6S/6eam(x).
(1.6.57)
Note that the variational derivative is defined as follows
s
6S 6eam
6s = d4x e - 6eam-, The symmetric part of T,"
1 6S T(ab) = - (eam Tmb+ ebmT,") = 2emaenb (1.6.58) 2 @mn is known as the 'energy-momentum tensor'. We anticipate the model to be invariant under the transformations (1 -6.48) supplemented by general coordinate and local Lorentz transformations of matter fields 0.What are consequences this invariance leads to? Let the matter fields be on-shell, 6S/60 = 0.
(1.6.59)
Then, the Lorentz invariance means that
s
d4X e -
KabebmT,"= 0
for any antisymmetric field Kab(x).So on-shell, Tab has no antisymmetric part, and hence Tab coincides with the energy-momentum tensor. The invariance under the general coordinate transformations, by virtue of Copyright © 1998 IOP Publishing Ltd
equations (1.6.48,49), means d4X e - V(,Kb,Tab =0
j.
for any vector field K"(x).Therefore, the on-shell energy-momentum tensor satisfies the equation (1.6.60)
V ,Tab= 0.
Suppose, in addition, that the action S[@, eam]is invariant under the Weyl transformations
Seam=
seam
6@' =ad(,,@'=o(d@)'
(1.6.61)
where d,,, are constants. Then, the on-shell energy-momentum tensor is traceless, (1.6.62)
T:=O as a consequence of conditions (1.6.61).
1.6.9. The covuriant derivatives algebra in spinor notation For completeness, we now rewrite the algebra (1.6.21) of the torsion-free covariant derivatives in two-component SL(2, C ) notation. This notation happens to be very useful when working with field theories in superspace which will be analysed in detail later. Recalling the one-to-one correspondence (1.2.30) between Lorentz vectors and SL(2, C)bi-spinors with one dotted and one undotted index, we introduce the bi-spinor derivatives
(1.6.63) Then the algebra (1.6.21) under the constraint Fa:= O takes the form cv25,
V,pl
1
= - %,,/{fi
2
cd M c d
= %i,/@,;,6M;'6+ ,%,pp,s.s
A?A (1.6.64)
where we have used equation (1.2.33). Because of algebraic curvature constraints (1.6.15), B2jr.Pfi,,n and .&afi,?8 can be decomposed in terms of their irreducible components as follows:
+ &XpE;a+fi + & i / 3 ( E + p n + E,sEp;.)FB,i,/jfi,:d = E,/jCjl/j?ij+ E $ / ~ E ~+/ EV, / ~j ( E q E f i h + E i d E / j j . ) F .
%k,pfi,;.ii
-
= ~ifiC+s
-
(1.6.65)
Here C,p;'iiis a completely symmetric tensor, and E;.6kfiis a tensor symmetric in its dotted and in its undotted indices. The spin-tensors from equations Copyright © 1998 IOP Publishing Ltd
Mathematical Background
45
(1.6.65) are connected with the curvature components by the rules:
1
+
c(
czP;S = - { (oah)x/l(ocd)yij (oab)y;(ocd)R/l f(cab)xfi(oCd)py} - )abcd
6
-
1 6
c&fi)s =- ((aab)&fi(?cd)>& + (?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 <"(x) is called a 'conformal Killing' vector if it satisfies the equation
1 Vmtn +Vn<m =-gmn(V,tk). 2
(1.7.3)
Any conformal Killing vector induces the general coordinate transformation Copyright © 1998 IOP Publishing Ltd
46
Idem and Methods of SupersynPmetry und Supergrmity
which locally scales the metric J ~ m n=
-2gCEIgmn
1 4
0[tI =- Vktk.
(1.7.4)
It is convenient to introduce some vierbein enm(x)for the metric and to convert curved-space indices in equation (1.7.3) into flat-space ones in standard to <"=<,emn and V, to the Lorentz covariant fashion: one has to change derivatives V,. Then equation (1.7.3) takes the form
<"
Vntb +Vbta=-
1 qnb(vctc). 2
(1.7.5)
Equation (1.7.4) is now equivalent to the identity
(1.7.6)
where i w b c are the Lorentz generators. This identity means that the composition of the general coordinate transformation
(1.7.7~) the local Lorentz transformation Benm=
be^" - ('coCabebm
(1.7.7b)
and the Weyl transformation
8enm= 0[5]eam
(1.7.7~)
where 5" is a conformal Killing vector, do not change the vierbein, and hence the metric. 1.7.2. Conformul Killing fiectors in Minkowski space Not every manifold admits conformal Killing vectors because of global restrictions on the curvature and topology of the manifold. Minkowski space admits non-trivial solutions of equation (1.7.5). Now we find all conformal Killing vectors in Minkowski space, where equation (1.7.5) takes the form
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
47
It has the evident consequences:
1 0(, = - - c?,(d,~) 2
0 d"8,
( I .7.9)
0(ZCtC)= 0.
After applying the master equation (1.7.8) once more, we find ( 1.7.10)
These three equations have the general solution
acadd,l,= 0.
(1.7.11)
We see that conformal Killing vectors in Minkowski space are at most quadratic in space-time coordinates. This observation along with the master equation, leads to the final expression for conformal Killing vectors:
<"(x)= b" + AX"+ Kabxb+ f " x 2
- ~ x " ( xf ,)
( 1.7.12) ( x ,f)E x " f a K O ,= - K b a where b", A, Kab, f" are arbitrary real constant parameters. Evidently, the parameters b" and K,, correspond to the Poincare transformations. The parameters A and f" induce infinitesimal space-time transformations of the form (1.7.1) known as 'dilatations' (or scaling transformations) and 'special conformal boosts', respectively.
1.7.3. The conformal algebra Let g?, ,a, and 5(2j= 5i'2)8abe two conformal Killing vectors in Minkowski space. Their Lie bracket [((I),
4(2,1=5(3)=4?3jaa
4?3)=4Pljab5?2j-
5!2Jab5?1)
gives the vector field 5(3J which also satisfies the master equation (1.7.8). Therefore, the set of all conformal Killing vectors forms a Lie algebra called the 'conformal algebra'. Copyright © 1998 IOP Publishing Ltd
48
Itlrus unci Mrtliotls o f Suprrsymmrtry und Supergravity
We introduce a basis {Pa,Jab,D, V o )for the conformal algebra by the rule:
for any conformal Killing vector ('(x) of the form (1.7.12), where
Pa= -ida
Jab=i(Xbda-xadb)
D = ixada
V, = i(2xaxbdb- x2aa).
(1.7.13)
The basis vector fields satisfy the algebra
[D,Val = iVa
[D,Pa]= - iPa
[va,Pb] = 2iqabD- 2iJab
[Jab,
pcl
-hbcPa
(1.7.14)
= i q a c V b - iqbcva
[Jab, Jcdl
=hacPb
= iqacJbd - iqadJbc
+ iqbdJac - iqbcJad*
All other commutators vanish. These commutation relations define the conformal algebra. So we can forget about its explicit realization (1.7.13) and postulate the conformal algebra as an abstract real Lie algebra subject to two requirements: (1) it has a basis {pa,jab= -jba,d, va> with multiplication law as in (1.7,14); (2) its general element X is of the form (1.7.15) where b", Kab= - Kba, A, f" are real parameters. The 'conformal group' is formally obtained by exponentiation of the conformal algebra. So, nearly all its elements can be written as 1
Kabjab+ Ad - f"va
( 1.7.16)
This definition is rather formal. Now we give two realizations of the conformal group. 1.7.4. Conformal transformations The conformal group can be formally realized as a group of nonlinear transformations in Minkowski space xa
For infinitesimal group elements
Copyright © 1998 IOP Publishing Ltd
-+
$0
= (j* xa.
49
Muthemuticul Background we define the corresponding transformations as follows:
+
( 1.7.17)
xa+ g xu= xu 5"(x)
where ("(x) is the conformal Killing vector (1.7.12). Exponentiation leads to the transformations:
(1.7.18 4 2. L O ~ U I rrcin.sf(~lmutions I: ei(112)K''<j~x . =(exp ~
) " , ~ b
(1.7.18b)
3. Dilatations eiAd,
- A a xa -e x
( 1.7.18~)
4. Special conformal transformutions (1.7.18d) The only comment required concerns the derivation of equation (1.7.18d). For this purpose, let us consider the inversion transformation defined on a domain of Minkowski space: R: xa+ xta= xa/x2
( 1.7.19)
R2 = 1.
It is an easy exercise to show that the inversion locally scales the metric ds2= qabdx" dxb,
ds2+ ds" = ds2/(x2}2. Hence, the inversion is a discrete conformal transformation. Now, one can check that the transformation Re-if'PtR.xa coincides with the right-hand side of equation (1.7.18 4 . O n the other hand, in the case of infinitesimal parameters f",we have R ( I - ifCp,)R.xu = x u
+ ( f U x 2 - 2x'(f,
x)) = (I
- ifcv,).x".
So, we obtain the identity R e-iJfiPc!R=e-if'W.t
(1.7.20)
which proves the relation (1.7.18d). Note that the following identities R eiAdR =,-iAd R , i ( l i 2 ) K i h j t t h ~ =ei(1/2)K"hj,,h also hold. Copyright © 1998 IOP Publishing Ltd
Remurk. The inversion transformation (1.7.19) turns out to be indeterminate for the points of the light-cone surface x 2 = O in Minkowski space. This leads to the fact that the special conformal transformations (1.7.18d) are not defined globally on Minkowski space. Thus, the conformal group is not a true transformation group of Minkowski space; its global action can be constituted only on a compactified version of Minkowski space. When considering below finite conformal transformations on Minkowski space (or finite superconformal transformations on a superspace, see Chapters 2 and 6), our discussion will be rather formal. The reader should keep in mind that only infinitesimal conformal transformations prove to be well defined on Minkowski space; finite conformal transformations are well defined in general on some domains in Minkowski space. 1.7.5. Mutris rea1i:ation of the conformal group We now give an exact realization for the conformal group as a group of linear transformations acting in a six-dimensional space R6 with coordinates ).'=yo, y', . . ., y 5 and the metric
dS2= - ( y o ) 2+ ( Y ' ) +, ~ . . + (y4)2- ( Y ~ )=~yaya+ ( Y ~ -)( y~5 ) *
(1.7.21)
Let us consider the group 0(4,2) of linear homogeneous transformations -y'' = A',yJ preserving the above metric. All 0(4,2)-transformations move the 'light-cone' surface J,'
yay,+(y4)2-(y5)2=0
(1.7.22)
onto itself. 0(4,2) is a 15-dimensional Lie group consisting of four connected components. We identify the conformal group with the component of unit in 0(4,2) denoted by S0(4,2)? (the matrices from S0(4,2)? are specified by the requirements that theii diagonal 2 x 2 and 4 x 4 blocks labelled by the indices 0, 5 and 1, 2, 3, 4, respectively, have positive determinants). Let us comment upon this definition. It is useful to redefine the variables y4 and y s by the rule
Then equation (1.7.22) takes the form yay, = .B.
We parametrize locally the 'light-cone' as follows (1.7.23) where x' will be identified with the coordinates of Minkowski space. Now Copyright © 1998 IOP Publishing Ltd
iMuthemuricu1 Buckground
51
consider particular SO(4, 2)T transformations given with respect to the variables ( j " , a, 8) by the matrices
0 -_0
A,=
(1.7.24~)
___ 1
-:I
A,=
(1.7.24b)
_--
e*
8ad
0
___
_-_
___
___
0
0
where A is an element of the Lorentz group. The transformation ( 1 . 7 . 2 4 ~ ) acts on the surface (1.7.23) by the rule x'x'"
[at')=[
+
a(A",xb b")
a(A;+b)'
so we recognize here some ordinary Poincare transformation in Minkowski space. Analogously, the operators A , and A , act on the space-time as a dilatation and a special conformal transformation, respectively. Matrices of the form (1.7.24a-c) generate SO(4, 2)T. Note also that the six-dimensional transformation y"-Py"
u+p
p+u
corresponds to the space-time inversion (1.7.19). 1.7.6. Conformal invariance The conformal group turns our to be the space-time symmetry group of some massless field theories. One possible way to understand this assertion is as follows. Copyright © 1998 IOP Publishing Ltd
52
Iriem crud Meiliods
of Supersj*mnzetrj und Supergravity
Consider a field theory in Minkowski space and suppose that it can be extended to a curved space so that its curved-space action is invariant under Weyl transformations of the metric (1 -6.32) supplemented by some 0dependent transformations of the matter fields. In detail, let S [ @ ] = j d4x9(@) be an action describing the dynamics of some fields @ = {ai} in Minkowski be a curved space-time metric. If S[@] is a massless field space. Let g,(x) theory, it is possible, as a rule, to define a generally covariant action S[@, gmnl=
where g = det(gmn),such that:
!
d4x
6Y(@,9)
(1.7.25)
1. In the flat-space limit, S[@, g,,]
reduces to the original action S[@], (1.7.26) SC@, g m n = ~ m n l =SC@I
where qmnis the Minkowski metric. 2. The action S [ @ , g,,] is invariant under the following transformations dq,, = - 20qm,
mi=
(1.7.27) which have the form (1.6.61)in the vierbein approach. The constants d(il are known as 'conformal weights' of massless fields ai, = o(d@)'
We shall argue later that the local invariance (1.7.27) is possible in the massless case only. Due to the Weyl invariance and general covariance, the action S [ @ , g,,] does not change under the transformations 1 1 8 0 = lava@- KabMab@ 0d@= t"a,@ - (Kab t"Onob)M,b@ + ad@ 2 2 8eam= -(V,tb)ebm K,,"ebm nea, = team- eatm ( K O b (nu,,b)ebm Beam
+
+
+ +
+
+
+
+
+
where ("(x), Kab(x)=- Kb"and a(x) are arbitrary parameters (it has been supposed above that all @' are Lorentz tensor fields). Now let ("(x) be a conformal Killing vcctor with respect to the metric .g,, Then the composition of the general coordinate transformation (1.7.7a), the local Lorentz transformation (1.7.7b) and the Weyl transformation ( 1 . 7 . 7 ~do ) not change the vierbein and, hence, the metric
+
+
6eam= -(Vacb)ebm K[(lObebm a[t]e," = 0.
So, the Weyl invariance (1.7.27) of S [ @ , ]g,, leads to the fact that every transformation of the form 1 -d@= t"v,@+-K[t]"bM,b@++[(]d@ 2 (1.7.28) 6eam=0 3 6g,=O where (" is a conformal Killing vector, preserves the action S [ @ , g,,]. Copyright © 1998 IOP Publishing Ltd
Mathemuricul Background
53
Taking now the flat-space limit in equation (1.7.28)(gmn-,qmn,V, + d, and so on) and using the boundary condition (1.7.261, one finds the action S[@] in Minkowski space to be invariant under conformal transformations 1 +; KIEIhcMbc@ +~ [ < ] d @
-8 0 =
where is an arbitrary conformal Killing vector (1.7.12). Consider scale transformations with ta= Axa in equation (1.7.29):
8@= - AXada,0- Ad@ or in the case of finite transformations, @'(x)= e - Ad@(e'x).
(1.7.30)
It is clear now that the conformal weight d(i)coincides with the dimension of
Oi. The action S[@] is invariant under (1.7.30). This invariance makes it
possible to change arbitrary scales in the theory under consideration, and therefore anticipates the absence of fixed dimensional constants (otherwise, it implies an infinite number of dimensional constants). On these grounds, the conformal symmetry is zdmissible for massless theories only. To be more exact, a theory may admit conformal symmetry if it is massless or describes particles of all possible positive masses O < m < m.Indeed, at the quantum level, the conformal symmetry means that in a Hilbert space of physical states we have a representation of the conformal algebra (1.7.14) by Hermitian operators { P, Jab, D, WO),where Pa and J,, are the Poincare generators, D is the dilatation generator and V, are the generators of special conformal transformations. Due to the identity [D, P a ] = -iPa
the squared mass operator ( - PaPo) is characterized by the following transformation law e-iAD(-papa)eiAD=e-2A
( - papa)
with respect to the dilatations. So, if 1'3') is some state of given positive mass m,then the state [ Y ' )=eiAe)Y)carries the mass m'=e-'*m. Both states 1'3'') and ]U') must belong, by virtue of the conformal invariance, to the same' Hilbert space. Hence, conformally invariant theories describe only massless particles or particles with all possible positive masses. 1.7.7. Examples of conformally invariant theories Our general consideration will be accompanied by three examples. The first ) the example is the theory of a massless self-interacting scalar field ~ ( x with Copyright © 1998 IOP Publishing Ltd
action (1.7.31) where j. is a dimensionless coupling constant. The unique continuation of S [ q ] in a curved space-time, consistent with the above given requirements, is
where .# is the scalar curvature. Using the transformation law (1.6.33)of the scalar curvature, one can prove that the action S[q,gmn] is invariant under the Weyl transformations bymn= -209,"
6 q = oq.
(1.7.33)
Now equation (1.7.29) says that the action S [ q ] is invariant under the following conformal transformations - 6q = ga?,q
+ -41 (SaIy)q
(1.7.34)
where 4" is a conformal Killing vector. Making use of the master equation (1.7.8),one can explicitly check the invariance (1.7.34). Our second example is the theory of a free massless Majorana spinor field Y(x)with the action
( I .7.35) Its continuation to a curved space-time is given by (
.7.36)
where eam is the vierbein. This action is invariant under the Weyl transformations 6eam= oeam
3 6Y = - o y . 2
.7.37)
Then equation (1.7.29) shows that S[Y] is invariant under the conformal transformations 1 3 -6y= (a d a y +- K b ' [ ( ] C b c y +-(2ata)V. 2 8 Copyright © 1998 IOP Publishing Ltd
(1.7.38)
Mathematical Background
55
Here we have used the Lorentz transformation law (1.4.11) offour-component spinors. The last example is the Yang-Mills theory describing the dynamics of and T' vector fields A, taking values in a compact Lie algebra, A,=iA,'T' are the generators of the algebra, tr(T'TJ)=dfJ. The action is S[A] =- tr 4g2
d4x F,,Fab.
(1.7.39)
'j.
The only continuation of this action to a curved space-time, consistent with the gauge invariance of S[A], is
Here A, carries a curved-space index. The action S[A,g,,] invariant under the Weyl rescalings Sgmn
=
SA, = 0
=3
{
Se,"' = geum
SA,=aA,
is evidently
(1.7.41)
where A, = enmAm, and eamis some vierbein for the metric. Recalling equation (1.7.29),we conclude that the action S[A] is invariant under the conformal transformations 1 1 -6A,=('d,A,+-(~,('-~(,)A,+-(d,~')A,. 2 4
(1.7.42)
1.7.8. Example of a non-conformal massless theory Now we present an example of conformally non-invariant massless field theory. This is the model of a second-rank antisymmetric tensor field BCd(x) with the action
(1.7.43) 1
L,(B) = - &,bed abgcd. 2 It is a simple exercise to show that the theory S [ B ] cannot be continued to curved space-time in a way consistent with Weyl invariance. Moreover, in the next section, we give a direct proof of the fact that the Poincare group is the maximal space-time symmetry group of S[B]. Copyright © 1998 IOP Publishing Ltd
Idcus und Methods of' Supersymmetry und Superyruvity
56
1.8. The mass-shell field representations In Section 1.5, the irreducible unitary representations (massive and massless) of the Poincare group have been described. Now, we give their realizations in terms of tensor fields restricted by some supplementary conditions. We shall also consider irreducible massless field representations of the conformal group.
1.8.1. Mussice j e l d representations of the Poincard group Recall that the irreducible massive Poincare representations are classified by mass and spin. For definiteness, we fix some mass value m>O. The irreducible massive spin-zero representation is seen to admit the only realization in terms of a scalar field @(x) under the mass-shell equation
P a = -iZ0.
Pap,@=- m Z @
(1.8.1)
However, in the case of some non-vanishing spin s, the irreducible massive spin-s representation admits several realizations in terms of fields. Let us consider the linear space q;A.B) of (A/2, B/2)-type spin-tensor fields @,I,?, , , z , h 1 4 ~ , , , h B ( ~totally ) symmetric in their A undotted indices and independently in their B dotted indices, A + B = 2s, and satisfying the following supplementary condition 2%Sc,
,
. .x L q
I&*,.
,
(1-8.2)
.b g . I ( 4 =0,
and the Klein-Gordon equation (0- mZ)@,,,?, . ,
,
(1.8.3)
*,(4=0.
Here a, =(crLl)Sctida.As we shall show, the supplementary condition (1.8.2) is needed to select the spin-s representation. Note that this condition is absent in the cases A = 2s, B = 0 and A = 0, B = 2s. One more restriction should be added to obtain a positive energy representation. Namely that in the 'momentum space' decomposition for O y , y >, ,, i l + : , ,*,j.~), only positive frequency plane waves should be kept ,,
,
,,
i8( PI
(1 3.4)
This decomposition should be extended to include negative frequency modes when the discrete space-time symmetries (time reversal and parity) are taken into account. In momentum space, the restriction (1.8.2) takes the form P%ZI
Copyright © 1998 IOP Publishing Ltd
...
, h h , ., .hn
, ( P )= 0.
(1.8.5)
Mathernutical Background
57
Note that the supplementary conditions (1.8.2) and (1.8.4) and the mass-shell equation ( 1.8.3)are invariant under Poincare transformations. has ( A + E + 1) independent components. To prove Any field from qA,B) this, it is useful to employ the momentum space decomposition and to make the transition into the rest frame, in which pZf=pa(aa)uf= mdXc(;then equation (1.8.5) means that the rest-frame field components are totally symmetric in all their indices. There is another, more elegant, way to prove the above statement. Let us consider the following operator
invertible through the equation (1.8.3), A ~ A I=~J{; +
~
1
~
=~.8 :
1
Now, if E#O, we can define a one-to-one map of the rule: @ Z I . . . 1,
f , ,,
.&4
-+@XI
...1,,1,,+,jll...jl,I
(1.8.7)
8
on Yi;A+I,B-l,by
(x)
.
= A x 4 + l i a @ z ~ .,
.. c
( ~ ( ~ )
+ I,B-
1).
(1*8*8)
Applying this operation E times, we obtain the one-to-one map of A!&B) on iYf&o,. But all fields @ a l . , , J x ) from qz,,o, are totally symmetric in their indices, so they have (2s+ 1) independent components. Rather beautifully, this describe equivalent qzs- . . .! qo,zs) shows that all spaces q2s,o), representations of the Poincare group. Now we are going to demonstrate that the spin operator Wawa,where W, is the Pauli-Lubanski vector, is a multiple of the identity operator on qA,B). Owing to the explicit expressions (1.5.14)for the Poincare generators, the Pauli-Lubanski vecicir is (1.8.9)
It is worth changing here the Lorentz generators with vector indices to generators with spinor indices according to the rules (1.3.9) and (1.2.33): L
Making use of properties (1.6.67) for the o-matrices, we obtain W, = i(a,,),pMZ~’i3b+i(8,,),gA?’/lidb.
Finally, we convert the vector index and undotted, resulting in
‘U’
W S k =- i @ . M Copyright © 1998 IOP Publishing Ltd
into a pair of spinor indices, dotted /jl
+ ic?,fiRgf.
( 1.8.1 0)
58
Idem und Methods of Supersymmetry and Supergravity
Then, the spin operator takes the form
wa wa = _ _1 wziwwi= - m2 ( M Z / ~rBM+ R ~ B A ~ +~ M) , ~ Ri p. c-xi dPP. -
2
2
(1.8.11)
Here we, have used the mass-shell equation (1.8.3). The values of M Z p M Z B and &?‘”Ripin the ( 4 2 , m / 2 ) representation series have been calculated in Section 1.3 (see equation (1.3.14)).So, we only need to determine the last term in (1.8.1 1). Recalling how M X Pand A,,i act on spinor indices (equation (1.3.12)),one obtains B
?”‘S”flM,,A,p@‘;.,
,,,;’
,$,
,,,IH= k= 1
~“i.,c~~~~,B@;I,,,,i~l ,.,;,r,,.:a
Here the first term vanishes due to the supplementary condition (1.8.2). For the same reason, the second term can be rewritten as
where we have used equation (1.8.3). The relation obtained together with equation (1.3.14) lead to the final result s = A/2
W a w a,/I,,=,, m2s(s , + l)U
+ B/2.
( 1.8.12)
To summarize, we have shown that the massive spin-s Poincare representation can be described in terms of ( A / 2 ,B/2)-type fields, A B = 2s, restricted by equations (1.8.2,3). Note that the irreducible integer spin representations are usually described by choosing A = B and considering in 3(is,s) the subspace of real spin-tensor fields
+
-
@?,Z?.
. . Z > i ) i ? .. . k ( X ) = @’?,??.. . Z , h , i , .
. . %(XI.
( 1.8.13)
All spinor indices here can be converted into vector ones @ a i a 2 . . .a,(’)=
1 2
( - 1)” < (‘al)ilxl(‘a?)h?Z2,,
. ( ~ a J i ~ z ~ @ x l x.x,i,*>, 2,, , ,i\(.x)
(1.8.14)
obtaining a real tensor field which is totally symmetric and traceless: @ a l a > . . . a , = @ ( a l a z . , .a,)
.
@‘aa,a?. . a > 2
=O.
(1.8.1 5 )
The supplementary condition (1.8.2) is now
da@wl a ? .. . a , ,(x)=O. Copyright © 1998 IOP Publishing Ltd
( 1.8.16)
59
Muthemutical Buckyround
The irreducible half-integer spin representations are usually described by choosing A - 1 = B or B - 1 = A . In the first case, we write any field from + 1 . E ) as 6%
Y?ZI..
. ? , J i,Ii.,.J ( x )
and then convert each pair (zi,+J, i = 1, . . ., B, into a vector index,
obtaining a spin-tensor which is totally symmetric, traceless and a-traceless:
The supplementary condition (1 3.2) can now be written as d"Ya4,. ,.U8
~
,?(4 = 0.
( 1.8.1 9 )
The second case, B - 1 = A , is treated analogously. Any field from should be represented in the form Y ? .,. .l A h I. .&) ,
and then each pair (xi,hi),i = 1, . . ., A , transformed into a vector index, 1 y a l . , . a 4 i =l)AA(k?l)hlY' (" . ( ~ a 4 ~ ~ ' ~ 4 ~ a l . , . z , i i l , (1.8.20) .,i,
2
The resultant spin-tensor satisfies the following algebraic
and differential (1.8.22) constraints. In conclusion, note that, if A # B, the Poincare representation on qA,B) is irreducible until considering the operation of complex conjugation.
1 J.2. Real massive field representutions We would like to continue analysis of the massive case and to consider real field representations. Each space Af&B)is assumed here to describe massive fields possessing both positive and negative frequency modes in their Fourier decomposition. Recall that the operation of complex conjugation provides us with a one-to-one mapping * (1.2.19) of the linear space of Copyright © 1998 IOP Publishing Ltd
60
Ideas and Methods of Supersymmetry and Supergravity
(A/2, B/2) tensors on the space of (B/2, A/2) tensors and, as a consequence, of the mass-shell space qA,B) on On the other hand, making use of operator A%&(1.8.6), we can define, by analogy with (1.8.8), the one-to-one as follows mapping A of qA,B) on QY]
,,
,
h B ( 4 + @XI
. . ' Au:BAi'lhl . . . A.%,@yl,
.. , Z B k I .. , h , ( 4 =
..,AS'I,.
.&).
Now we select a Poincare invariant subspace 3y;,,, in qA,B), which is characterized by the coincidence of the maps * and A on it. Any field from X'opA.s,satisfies the equation
k 4 ( ~ ) = A y , 3 1. A. .U B h B A y l k l. ,A. 1 ' A k A ~ ) 7 1 . , , ; . 4 1 / l , . , j . B(1.8.23) (~).
@)II,,,xBj!
We shall call such fields 'real massive fields'. To clarify what equation (1.8.23) means, let us consider some particular cases. In the case A = B = s, we have ,2,+,,
AYl$'. . . A2,9'AY1 . . . A"k5QYl,, ,?$, ,?,(x)
Jx)=
,,
, ~ u l . . . x , . ; l,,:~ ~ ( . ~ ) = Q ~ l . . . Y , h l . . . h ~ ( x )
where we have used the supplementary condition (1.8.2) and the mass-shell equation (1.8.3). So, if A = B, equation (1.8.23) gives nothing more than the reality condition (1.8.13). In the case A - 1 = B, equation (1.8.23) together with equations (1.8.2,3) lead to
Yx,, .+k, ,,
,,
. . . AgB$BA'hAylhi. . . A;'B&BYg71,,
, k B ( x= )
,
= A ; , , ? I. . A +r B.~ B A * ~ A... ~ IA+ ~, B ~ ~ Y ~ ~ ~ , , , ~ ~ ~ ~ , . , = A ' k ~ B I I .. %. B & I . ,
,
kB(X).
In terms of the spin-tensor field (1.8.17), this result is ,
-myal
. .:(XI=
~ " ~ a l ,aBy(x) , .
*
( 1.8.24)
. . . , B ~ ( ~ ) = ~ a ~ ~ a ~ . . . a ~ ( x ) .
The equations (1.8.24) are elegant in three respects. First, the Klein-Gordon equation (1.8.3) is a consequence of the first-order differential equations (1.8.24). Secondly, the fields 'Pal, . . a B u and Ta,.,. under the Klein-Gordon equation, describe two different massive particles having spin ( B + i).When equations (1.8.24) are imposed, however, we have only the independent field 'Pal.,.aBu, which corresponds to a single massive particle with spin ( B +$). Thirdly, let us make the replacement y a l . . .agu
--$
-
e(314)77iy aI...agm Val.,. a B & ~ e - ' 3 ' 4 ) 7 7 i ~ a l . . . ~ B &
and incorporate the resultant fields into a four-component column (1.8.25)
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
61
which is a Majorana spinor and rank-B Lorentz tensor. Then equations (1.8.24) coincide with the Dirac equation (iybab
+ m)‘yu,
( 1 -8.26)
., ,U B ( ~ = )
where the y-matrices were defined in (1.4.10). Finally, if IA-B1#0, 1, the reality condition (1.8.23) can be represented in the form of a differential equation independent of the Klein-Gordon equation but having the same or higher-order IA -BI. 1.8.3. Massless field representations of the PoincarP group To describe the massless case, we consider a (A/2, B/2)-type spin-tensor field . , , a c B ( xtotally ), symmetric in its A undotted indices and in its B dotted indices, satisfying the following supplementary conditions
(1.8.27) As will be shown, the supplementary conditions are sufficient to select a single helicity state. The equations (1.8.27) can be rewritten in the following equivalent form a7>Gzl., .z A h l . ,.k,(x) = ~
.
n r ~ G, .~, b a A ,I,. z 4 h l, , , h B ( X )= d y t q G z , . , aAl/jrl...
where k = 1, . . ., A and I = 1,
&.. . i , ( X ) (1.8.28)
.. ., B. If A # 0 or B # 0, the on-shell equation
0GI.. .U,&]. ..h , ( X ) = 0
(1.8.29)
follows from the supplementary conditions. If A=B=O, there are no supplementary conditions, and we have only the on-shell equation
(1.8.30)
uqx)=o.
Note that the supplementary conditions (1.8.27) are invariant with respect to the Poincare transformations. The massless analogue of decomposition (1.8.4) is
(1.8.31)
P”=(Po,b)
Po=lbl.
In momentum space, the conditions (1.8.27) take the form PYYGya,.. . a A -
. .&)
,&I.
=0
(1.8.32) PYLGa l . . . a A yhl...hB , ( p ) = O . It is now an easy task to prove that the fields under consideration have only Copyright © 1998 IOP Publishing Ltd
62
Ideas and Methods of Supersymmetry and Supergravity
one independent component. Indeed, choosing a reference system, in which pfl = ( E , 0, 0, E ) , we have
Then equations (1.8.32) mean that the only non-vanishing component is
--
G1, , , I i . . . i (PI. LC
A
B
Therefore, we have found that any (A/2,B/2)-type field under the supplementary conditions (1 3.27) (or under the on-shell equation (1.8.30) when A = B =0) describes a massless particle having helicity j. = ( A - B)/2. 1.8.4. Examples of massless fields It is our purpose now to demonstrate how massless fields, described in the previous subsection, arise in field theories. To start with, we consider two different field realizations of a spin-zero massless particle. The first realization is built in terms of a scalar field cp(x). The action
'5
S [ q ] = -- d4x d"cp Z0cp 2
(1.8.34)
leads to the on-shell equation (1.8.30). So, the field cp(.x) carries helicity j. = 0. Copyright © 1998 IOP Publishing Ltd
Mathematical Background
63
’I’hesecond realization is described in terms of a second-rank antisymmetric tensor field Bed(x).Its classical dynamics are dictated by the action (1.7.43) being invariant under the following gauge transformations Bed(X)
--t B:d(X)
=
+ a[cj”d]
(1 3.35)
where Evd(x)is an arbitrary vector field. It is the field strength L,(B) that is invariant under these transformations. The equation of motion (1.8.36a) and its consequence B,,Ld,(B)= 0
(1.8.36b)
can be rewritten in two-component spinor notation:
a”L,&q = 0
37‘iLT)(B) . = 0.
(1.8.37)
To derive equations (1.8.37) from (1.8.36), one has to d o the same steps as in deriving equation (1.8.10) from (1.8.9). The equations (1.8.37) mean that L,,(B) is a massless field. In accordance with (1.8.33), La@) carries helicity i = O . So we may conclude that the field models (1.8.34) and (1.7.43) are equivalent, since they describe the same Poincare representation. The equivalence can be seen also as follows. The equation (1.8.366) means that L,(B)=d,cp, where q ( x ) is a scalar field satisfying, by virtue of (1.8.37), the on-shell equation (1.8.30). Therefore, the models (1.7.43) and (1.8.34) lead to the same dynamics. Now consider the model (1.7.35) describing the dynamics of a massless Majorana spinor field Y . It is useful to rewrite the Dirac equation y”d,Y = 0
in two-component spinor notation:
p y a= 0
= 0.
(1.8.38)
It is seen that Yu,(x) and Tdx) are massless fields. Recalling equation (1.8.33), we conclude that our model describes two massless particles having helicities E.= ki. One more example is given by electrodynamics. The action (1.8.39) F o b = d,A,
- abA,
is invariant under the gauge transformations 6 A , = ab;. Copyright © 1998 IOP Publishing Ltd
64
Idem und Methods
of' Supersymmetry und Supergrut.ity
where 3.(x) is an arbitrary scalar field. So the only physical observable is the field strength Fab. We rewrite the first -ybcd
2
8F h
cd=O
and the second a b F a b =0
Maxwell's equations in two-component spinor notation:
ZY?F,, = 0
aY3F.v;, = 0
(1.8.40)
where
+
F,+,jb= ((7a)m+((7b)p,jFab = 2&zpF@ 2 ~ + g F , p . The equations (1.8.40) imply that F , & x ) and F&,j(x)are massless fields carrying helicities E, = & 1, respectively. Our next example is the Rarita-Schwinger model
'S
s [ y a ] =- d4x & a b c d q a Y b Y 5 a c v d 2
(1.8.41)
where Y a ( x ) is a Majorana spinor and Lorentz vector field,
In terms of the two-component spinors, the action is
's
S['Y,] =- d4x EabCd\jFlai?bYcd 2
(13.42)
This model is a gauge theory since S[Ya]is invariant under the following transformations Ya,(x)
+
Y'bAx)= ' y a m ( x ) + a a E m ( x )
where E,(x) is an arbitrary spinor field. Evidently, these transformations do not change the field strength Y&. To analyse the equations of motion &abcd(i?b)bPYcdp = 0
(1.8.43)
it is useful to rewrite Yablin spinor notation introducing the spin-tensor Yfi& = (d)fi(ad)&$Y,,, Y 96%= Y Copyright © 1998 IOP Publishing Ltd
+
= 2&@Y3& 2E$Y',&
Yi.g,=
( 1 3.44)
65
Muthenraticul Buckground
Then equation (1.8.43) is nothing more than the following two equations (1.8.45)
Therefore, YcsIz vanishes on-shell, and Yzyvis totally symmetric under the equations of motion. What is more, Yz,jp satisfies some differential constraints. Indeed, starting from the obvious relation Eabcd d b y c e
=0
and using ( 1 .8.44,45), one obtains (1.8.46)
Therefore, Yz[+;(x) and 'T',B.;(x) are massless fields and, in accordance with (1.8.33), their helicities are f5, respectively. In conclusion, let us consider linearized gravity. Its action is obtained from the Einstein gravity action (1.6.47) by representing the metric in the form K
T
gm,(x) = V m n - H A x )
where H,, is a small fluctuation, and keeping in SG only terms quadratic in Hmn.This gives
's
S [ H ] = - - d4x{8 H a baCHab- 8 H : d c H i + 2dbHab(a,H~ - dcHRc)}. (1.8.47) 2 The action does not change under the linearized gauge transformations -)
Hab(X)
+d ( a l b ) ( x )
=
where (,(x) is an arbitrary vector field. Linearized field strengths, invariant under these transformations, are obtained from (1.6.30) by keeping only H o b terms and setting V, = 8,. Then one obtains &abed = a a a c H b d
-abacHad
+ abadHac-aaadHbc (1.8.48)
Imposing the equations of motion dab= 0
( 1 3.49)
we work with the linearized Weyl tensor f R b c d . In two-component spinor notation, this is described by totally symmetric spin-tensors CEBYa and C,fi+d: Copyright © 1998 IOP Publishing Ltd
66
Iiieus und Methods of Supersymmetry and Superyravity
which satisfy. by virtue of equations (1.8.49) and (1.6.68), the constraints
s ?+k,s are massless fields describing, due to equation Therefore, ~ l 1 9 ~and ( I .8.33), the helicity states & 2, respectively. The given examples illustrate the general situation: having some gauge theory S[Q,], the dynamical fields Q, do not belong, as is usual, to the family of massless fields described in Section 1.8.3. What is more, the original fields @ cannot be treated as physical observables because of gauge arbitrariness in their choice. Rather, it is the gauge invariant field strengths G(@) built from the dynamical fields which play the role of physical observables. Namely these objects satisfy the usual criteria imposed on massless fields. To clarify the spin content of a theory, one must analyse the corresponding field strengths.
1 J . 5 . Massless field representations of the conformal group Let S[@] be a massless field theory invariant with respect to conformal transformations (1.7.29).This transformation law defines a field representation of the conformal group. The corresponding generators {Po,Jab,D, V,) are introduced by the rule
1
KnbJ,b+AD-~V0
and their explicit expressions are
They satisfy the commutation relations (1.7.14) with arbitrary conformal weights d. One can construct, with the help of the dynamical fields 0,different secondary fields: do@, i?$b@ and so on. In general, their transformation laws have a structure other than that of equation (1.7.29). For example, considering 2,Q, one finds
- d(d,@) = ( b a b ( d , @ )
+ -21 K [(]b'~kdb,(a,@) + a[ (Id'( do@)
+ ( a b o [ t l ) { M b o @ + I?abd@)
(1.8.51)
where d' = d + 1. The difference from equation (1.7.29) is the presence of some Copyright © 1998 IOP Publishing Ltd
Mathematical Background
67
non-homogeneous terms. But gauge invariant field strengths G(@), being secondary fields, should transform, at least on-shell, homogeneously - (iG(@)= (“S,G(@)
+ -21 K[(]’”M~,G(@)+ a[{]d,G(@)
(1 -8.52)
where d, are the scale dimensions of G(@).This assertion may be considered as a principle. But there are some obvious physical arguments. Indeed, let us anticipate that the gauge transformations (internal, as a rule) in the theory S[@] commute with transformations from the conformal group (space-time group). Then any conformal transformation should transform G(@) into a gauge invariant object. So, the right-hand side of equation (1.8.52) should be represented in the form of some operator acting on G(@). Under this assertion, the right-hand side of equation (1.8.52) is restored uniquely. Now one has to clarify what the requirements are for the off-shell transformation law (1.8.52) to be consistent with the on-shell equations (1.8.27). The consistency conditions are (1.8.53) for any conformal transformation (1.8.52). Note that the Poincare transformations satisfy these requirements. So we are to study the dilatations and the special conformal transformations. As applied to these two cases, the conditions (1.8.53) can be rewritten in the form: (1.8.54~)
(1.8.546) where the generators D and Vp,j are the same as in (1.8.50) but with conformal weight d,: 1
.
D = - - x1ua,2 + id, 2 V,,fi = - ix,tx’pd(jg
+ 2id,.u,p
- 2i(x’6 M,i,
+ xpdRp).
(1.8.55)
Here we have converted all vector indices into spinor ones. It is not difficult to check that equations (1.8.54~)are fulfilled identically (under equations ( I .8.27)). A more interesting situation arises when considering the restrictions (1.8.54b). To start the analysis, let us suppose Copyright © 1998 IOP Publishing Ltd
68
Iiieus and Methods of Supersymmetry and Supergravity
that AZO. Using equations (1.8.55), one obtains V/i/jGzI.,. x d ? l . ,, 4 H
Taking into account equations ( I .8.27), one then finds 2z1'v/j,jGxlx:,,
, x ,4,,
,,
+
4g=4i(A/2
- d G ) 8 $ G / ~ x ~ . . . z A 4 1 ,.4g ,
B
+ 2i
';hG[jx:.
, .?
~ p b ,(.$,, ,
,. & H
k= 1 B
-2i
'b,flGB%>.. , ? A i { ,
k= 1
.
,
,&si;
where we have used the identity 81jbxmb= - 2635;. Since A #O, this expression vanishes if and only if B=O. (1.8.56) d G =1+ A / 2 Then, the second equation (1.8.54b) is absent since B=O. Otherwise, in the case B #O, one obtains the following restrictions dG=1+8/2
A=O.
(1.8.57)
Finally, in the case where A =B=O one can find d G = 1. We conclude that admissible tensor types for field strength, arising in a conformal field theory, are (A/2,0) or (0,B / 2 ) only. The corresponding scale dimensions (1.8.56) and (1.8.57) are known as 'canonical dimensions'. The main results (1.8.56,57) show, in particular, that the antisymmetric tensor field model S [ B ] (1.7.43) is not a conformal theory, while the model S[@] (1.8.34), classically equivalent to the first one, is a conformal theory. The statement about equivalence of these theories does not contradict the fact that S [ @ ] possesses conformal symmetry but S [ B ] does not. The equivalence means coincidence of the two dynamics. But we have seen that L,(B) = a,@ on-shell. Hence, since @ is a conformally covariant field with the transformation law (1.7.34), L,(B) transforms in a non-covariant way (equation (1.8.51)). 1.9. Elements of algebra with supernumbers
Supersymmetric field theories (as well as all fermionic field theories) are formulated most naturally in the language of supermathematics based on Copyright © 1998 IOP Publishing Ltd
Matliematical Background
69
the concept of Grassmann algebra. Now we are going to describe the main ideas of supermathematics and obtain some results which will be explored in the following chapters. To make our consideration self-contained, it is worth starting by recalling some trivial definitions from algebra theory. A linear space d (complex or real) is said to be an ‘algebra’ (complex or real) if d is provided with a binary operation of multiplication ( - )
(a, b ) - + a b E d
V a,bEcd
which satisfy the axioms
+ (ctb+ pc)a = ctba + pca
a(ab pc) = aab + pac
V a, b, c
~
d
where a and fl are arbitrary numbers (complex or real; for the time being, we restrict outself to the complex case only). If the multiplication law is characterized by the property
ab=ba
Va,bEd
the algebra is called commutative. If the multiplication law is characterized by the property
a(bc)= (ab)c
V a, b, c E d
the algebra is called associative. If d contains a unit e with the property
ea=ae=a
Vacd
the algebra is called a unital algebra (or algebra with unit). An example of associative commutative algebra with unit is the algebra C“(M) of smooth functions on a manifold M . The multiplication law is defined as the product of functions: iff, cp E Cm(M),then ( f a cp)(p)=f(p)cp(p), p E M . An example of an associative non-commutative unital algebra is the algebra Mat,(C) of n x n complex matrices. The multiplication law in Mat,(@) is the matrix multiplication. All Lie algebras are non-associative algebras (without unit). Let d be an associative algebra with unit e and B c d be some set of elements. The algebra is said to be generated by B if every element u ~ d can be represented as a finite-order polynomial of elements from B: P
a=cte+
1 1 k = l ii,i2.
...,ia
Ci,i?,,,iibilbi2.. . bik
where CI and C i l i 2 . , .are i k complex numbers, and all b’ lie in B. Then, B is called a ‘system of generating elements’ for A Now we are in a position to define Grassmann algebras. Copyright © 1998 IOP Publishing Ltd
70
Ideus und Methods qf Supersymmetry and Superyravity
1.9.1. Gsussniunn ulgebrus AN unci A x The ‘Grassmann algebra’ A,v is an associative unital algebra generated by a set of N linearly independent elements c‘, i = 1, 2, . . ., N , which anticommute with each other: i i i j + ij(i
=0
i , j = l , ..., N .
(1.9.1)
In particular, (i’)2 = 0. Every element a E A, can be represented in the form a =a
1 + C“ CiIi?, k!
, , ik(i1[i2
k=l
. . . (‘A
(1.9.2)
where summation over all repeated indices is to be understood, a and C i l i 2 ,i r, are complex numbers, and the Cs are totally antisymmetric in their indices (it has been supposed that the unit element coincides with the number 1 EC). It is clear that elements ,
1
i’ ii1ci2
i ,
p i = .. (N form a basis for A,. So, the algebras A, are finite-dimensional, dim AN = 2 N . We shall mainly be interested in an ‘infinite-dimensional Grassmann algebra’ A,. By definition, this is an associative unital algebra generated by an infinite set {Ci}, i=1, 2, . . ,, of linearly independent, anticommuting elements i’ i‘(j+Cji‘=O
i , j = I , 2, ...
(1.9.3)
Elements of A, are called ‘supernumbers’ and Am is called the ‘space of supernumbers’. Every supernumber Z E A , can be represented in the form
z = ZJj + zs (1.9.4) ZB, Cili?., , ir E @,
Here the coefficients C i l i 2 ,ii, are totally antisymmetric in their indices, and only a finite number of them do not vanish. Following B. De Witt, zB will be called the ‘body’ and zs the ‘soul’ of z. Because of equation (1.9.3), (z,)” = 0 for some integer n. If i ’ z = 0, for all Ci, then z = 0. Every supernumber z can be decomposed into the sum of its ‘even’ z, and ,
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
71
‘odd‘ o: parts defined by
z = 2, + 2, (1-9.5)
If 2“ =0, z is called a ‘c-number’; if z,=O, z is called an ‘a-number’. By virtue of equation (1.9.3), e-numbers commute with all supernumbers. The set of all c-numbers is denoted as Cc and forms a commutative sub-algebra in A,. By virtue of equation (1.9.3),a-numbers anticommute among themselves and commute with c-numbers. Given two a-numbers, their product is a bodiless c-number. For every a-number z, we have z2=0. The set of all a-numbers is denoted as e,. We shall call e-numbers and a-numbers pure supernumbers. Our consideration shows that pure supernumbers are characterized by the following properties:
@,‘c,= c, @,
*
e,=@;
c, a=, c c, @, = ca.
These properties mean that A * is a Z,-graded associative algebra (see Section 2.1). Together with the algebra Am, one can consider its extension (closure) A, with respect to the norm 11 11 defined as
(1.9.6) The elements of A, are arbitrary linear combinations of the form (1.9.4) having finite norms. Since lIz+wll ,< IIZII + I
l~lI
and llli~li~..~lG i ~ IIzIl zll
*
IIZ 0 1 1G
=j
llzll . llwll
for any elements z, w E A,, the set A = is an algebra. It is clear that A, c Am, and every element Z E A , may be given as the limit of a sequence {z,}, a = 1,2,.. ., of supernumbers z,EA, such that lima-rm~ I z - z , ~ ~ =O. The elements of A = are also called supernumbers, and A = is said to be the ‘full space of supernumbers’. For every Z E A , , one finds limn.+=(zS)n=O.In this book we deal with A m and formally extend results to the case of A,. It is useful to connect with every pure supernumber z its ‘Grassmann Copyright © 1998 IOP Publishing Ltd
12
Iiletis
~ i n dMctliods
of’
Supersymmetry and Superyruvitjs
parity‘ E ( : ) by the rule E(=) =
0 1
ifzE@,
(1.9.7)
if?€@,.
Then, the properties of pure supernumbers described above are encoded in thc relations 1. E ( ~ . w ) = E ( z ) + E ( ~(mod2) ) - l ) i ( - ) i ( ( ) ) w . z. 2. 2 .
(1.9.8)
To define real supernumbers, we introduce the operation (*) of ‘complex’ conjugation (involution) in A, as follows:
(Ci)*= < i
i= 1, 2, .
( x z ) = x*z* (z +o)*= z*
+ w*
Iy
E Q:
(1.9.9)
( z o ) * = w*z*
where :and o are arbitrary supernumbers. Then for every supernumber
(1.9.4), we have
A supernumber z is said to be ‘real’ if z* = z, ‘imaginary’ if z* = - z, and ‘complex’ otherwise, The set of all real supernumbers in C,will be denoted by R,. The set of all real supernumbers in @, will be denoted by R,. Having two real c-numbers, their product is a real c-number. The product of a real c-number and a real a-number is a real a-number. The product of two a-numbers is a bodiless imaginary c-number.
I .9.2. Supervector spaces Supervector spaces are linear spaces in the usual sense, but supplied with the additional operations of left and right multiplication by supernumbers. Following B. De Witt, we define a ‘supervector (superlinear) space’ as a set 9 of elements, called ‘supervectors’, together with a binary operation of addition ( +), operations of left and right multiplication by supernumbers, and a mapping of complex conjugation (*), which satisfy the axioms: 1.
B+t=k+2
vz, k € Y
2. (z++)+2=z+((k+2) v 2 , T 2 E 3. There exists an element d E 9 such that
z+Cj=z Copyright © 1998 IOP Publishing Ltd
9
V2€Y
Muthemaricul Buckyrnund
73
4. For every 2 E -% there exists an element ?E Y such that
X+Y=6 and every 2,PE-%we have
5. For every n, PEA, (n+
2 ( a +/I) = 8 a +2/1
~ )=2n 2 +p2
=aji+(r? (Z+k)a=2a+%
(r(i+f)
2(ap)=( 8 a ) p
(ap)X = a ( p 2 )
Bl=X.
lX=B Then one finds
&j=ga=a
08=20=(7
X +( -
l)X = s + B(- 1 ) = 6
6. Left and right multiplications are related as follows: V 8 E 2’ V a, p E A, (a) ( n T ) / 3r(8p) ~ (b) a X = X a VaEC, VXEY (c) For every Z E there ~ exist unique supervectors such that
‘2,‘2~2’
Z=Oji+’B +
a’x= -‘?a
a0B=OXa
v EEC,.
The ‘8 is called the ‘even’ part of 8, and ‘3 is called the ‘odd’ part of 2.If ? = O 8 or s = ’ B , it is called a ‘pure’ supervector, of ‘C-type’ in the first case and of ‘a-type’ in the second case. Associated with every pure supervector 2 is its Grassmann parity ~ ( 2 defined ) by
if 2 even ifgodd.
“)={:
For any pure a and
(1.9.11)
8, we have = (- p w k ) 8 a ,
7. For arbitrary a E Am and
(1.9.12)
s,‘LE we have s**=z (2+ Y)*=s* + ‘i*
(EX)* =?*a*
( s a ) * =a*%*.
It is not difficult to see that if 2 is an even or odd supervector then g* is also an even or odd supervector, respectively. A supervector 2 is called ‘real’ if %* = 2,‘imaginary’if k*= -8 and ‘complex’otherwise. The product Copyright © 1998 IOP Publishing Ltd
74
Ideas and Methods of Supersymmetry and Supergravity
of a real c-number and a real supervector is a real supervector. The product of a real a-number and a real C-type supervector is a real a-type supervector. Finally, the product of a real a-number and a real a-type supervector is an imaginary C-type supervector. The set of all C-type complex (real) supervectors in 2' will be denoted by '9(OYR), and the set of all a-type complex (real) supervectors by '2'('YR). Each of '9and '9is a linear space in the usual sense, but supplied with the operation of multiplication by c-numbers and the mapping of complex conjugation. Each of OYRand 'YRis a linear space in the usual sense, but supplied with the operation of multiplication by real c-numbers. Associated with a supervector space 2 'is the 2'-parity mapping 8:9+ 2' acting on an arbitrary supervector 2 = ' 2 + '2 by the law
~(Oji+12)=0~-12, In the case of a pure supervector
( 1.9.13a)
8, the formula reads
9(2)=(l)e(X)X.
(1.9.13b)
Basic properties of the 9-parity mapping are:
p =0
(1.9.144
9(2+ t)= P(2)+ 9(k)
( 1.9.14b)
P ( a 2 ) = ( - lr%8(2) Here
2 and k are arbitrary
g ( z a ) = ( - l)E(')8(2)a.
(1.9.14~)
supervectors, a is a pure supernumber.
1.9.3. Finite-dimensional supervector spaces Let (8,) be a set of elements of a supervector space 9 These supervectors are said to be linearly independent if and only if the requirement that a finite linear combination XMe, (or P,XM), X,EA,, vanishes is equivalent to XM=O, for all XM. The space 2' is said to be finite dimensional if it possesses a finite system of linearly independent supervectors (e,), where M = 1, 2, . . ,, d, such that every 2 E 9 can be expressed in the form
x = E,XE,
= Xfl_,d,v
XEl~Az.
( 1.9.15)
Then, the system {e,} is called a 'basis' for 9 Clearly, each of the ( X p ) )are called the decompositions in equation (1.9.15) is unique. The Xt",, 'left (right) components' of 2 with respect to the basis (e,). Any two bases of a finite-dimensional supervector space Y have an equal number of elements, which is called the 'total dimension' of 9 To ground the assertion, it is useful to introduce the notion of supermatrices. A 'supermatrix' is a matrix with elements being supernumbers. Given a supermatrix F , one can decompose its matrix elements onto their bodies and souls, resulting with the body F , and the soul F , of F , F = F , + F,. F , is an Copyright © 1998 IOP Publishing Ltd
Mathemutical Buckground
75
ordinary complex matrix, The 'rank' of F is defined to be the rank of its body F,. If F and G are m x n and n x p supermatrices, then the body of their product FG coincides with the product of their bodies, (F.G),=FB*GB. This implies that all statements about matrix rank transfer to the case of supermatrices. In particular, if F and G are m x n and n x m supermatrices such that FG=0,
GF=0,
0, (0,) being the unit m x m ( n x n) matrix, then m = n and G = F the former relation can be rewritten as F,G,
I.
Obviously
+ FsGB+ F,G, + FsGs =.,0
Since 0, has no soul, we deduce that FBGB =. ,0 Similarly, the latter relation leads to F,GB = ,U, which confirms our assertion. An n x n supermatrix F is said to be 'non-singular' if its body F , is non-singular. Every non-singular supermatrix has a unique inverse. We can write F =F,(U, + FB I F s ) where supermatrix F B ' F , has no body, hence (FB1Fs)P=O for some integer p (recall, for every supernumber z there exists an integer q such that (z#=O). Now one obtains F - ' =F B 1 +
P
( - l)k(FB ' F J k F B
'
k= 1
the power series being terminated at some finite order. It is clear that a square supermatrix has an inverse i f and only i f its body is non-singular. The inverse supermatrix is unique. Let { Z M } , M = 1,2,. . .,d , be a basis in Y It is readily seen that supervectors
2 a = ~ a Ma ~ = lM , 2 , ..., d' are linearly independent if and only if the supermatrix GaM has rank d', in particular, d ' 6 d . As a result, any two bases in 9 have the same number of elements. Remark. A% is a one-dimensional supervector space. Every supernumber with non-vanishing body can be taken in the role of a basis for A,. Remark. In contrast with ordinary vector spaces, not every subspace of a finite-dimensional supervector space Y has definite dimension. As an example, one can consider a subspace z z where z E Ca. Given a d-dimensional supervector space we may always choose a basis {EY}consisting of pure supervectors only. Indeed, let {e,} be some basis. Copyright © 1998 IOP Publishing Ltd
76
Idem und Methods of Supersymmetry and Supergravity
We decompose every P, into the sum of its even and odd parts
P, On the other hand, each of ,2'
= OP,
and '2,
+ I;,. can be decomposed with respect to
iPM)
0e,=f,"P,
e,=cpMNP,
f M N ,qMMNeA,.
1-
One can easily see that the rank of 2d x d supermatrix ( f M N , cpMN) is equal to d. Therefore, there are p supervectors from the set (oP,) and q supervectors from the set { ' P m } , p + q = d, that form a basis which will be denoted by {E,) and called a 'pure' basis. It is convenient to label C-type elements of the -basis by small Latin letters and a-type elements by small Greek letters, + &=(E,,,, E,), where m = 1, 2, . . ., p and p= 1, 2, . . ., 4. Together with the total dimension, the numbers p and q turn out to be invariant characteristics of the supervector space and are called the 'even' and 'odd dimensions' of 64 respectively. Then 9 is said to have dimension ( p , q). Remark. A x is a supervector space of dimension (1,O). Any c-number with non-zero body can be taken in the role of a pure basis. Every C-type supervector 2 E 9 can now be written as
X
= ymE,
+ e@,
ymE
c,
e, E e,.
(1.9.16)
Therefore, the set '9of C-type supervectors in 9 is in one-to-one correspondence with points of the space CPlq C,P x e:: W = ( ( y l , y 2 , ..., p, el, e2, ..., eq), y m ~ ce = wo, ) .
(1.9.17)
This implies, in particular, that the numbers p and q are invariants of the supervector space. Every finite-dimensional supervector space 9 proves to have a pure basis {EM), which at the same time is real, (E,)* =E,. With respect to such a basis, real C-type supervectors are given in the form
X = .xmEm+ iehE,
x m E R,
e, E R,.
(1.9.18)
Hence, the set Od;CR of real C-type supervectors in 9 is in one-to-one correspondence with points of the space RpIp= R,Px R:: R P ~ ~ = { z ~ = ( xx2. ,~..,, ,
xp,
e', e',
..., P), x ~ E R , ,W E R ~ ) . (1.9.19)
Looking at equation (1.9.18), it would appear more convenient to exchange where = Em and b, = ig?. the pure real basis (E,} with the basis Every pure basis (2,) with the properties (8m)* and = -2, IS known as a 'standard basis'. defined by (1.9.16), is known as a 'complex Supernumber space superspace of dimension' ( p , q), supernumber space RpIq, defined by equation (1.9.18), is known as a 'real superspace of dimension' ( p , 4).
{z,},
Copyright © 1998 IOP Publishing Ltd
z,,, =z,,, (z,)*
Mathemutical Background
77
Pure supervectors from Y are characterized by the following properties
2 = EMX& = Xi"_,E, &(Xg))=&(Xp,)=&(2)+&M
(1.9.20)
X(")=(-l)cM(l+c(%)XM (+) where we have introduced the notation (1.9.21) As may be seen, the right components of every a-type supervector coincide with the left ones, but this is not the case for C-type supervectors. Let {EM} and {Eh) be two pure bases. Each supervector PM may be decomposed with respect to {EM]: PM=
(1.9.22)
G,~E,
where GMN=(
A,", D,Y E @, B,', C," E Ca.
:L: :;:)
Supermatrix GMNshould be non-singular, that is its body
is a non-singular matrix, hence AB and DB are invertible matrices. As a result, the supermatrices A and D are invertible, and the unique inverse of G can be expressed in the form G-'= &- - : ; : ( ; , - ; - (
1
- A - B(D - C A - B ) - l ) . (1.9.23) ( D - C A - B )- '
It is worth pointing out that, because of relation (1.9.22), the supermatrices BD-'C and C A - ' B are bodiless, therefore ( A - B D - ' C ) and ( D - C A - l B ) are invertible.
1.9.4. Linear operators and supermatrices Let Y be some supervector space. A mapping linear operator' on Y if
4: Y+Y
is called a 'left (1.9.24a)
Copyright © 1998 IOP Publishing Ltd
78
Ideas and Methods of Supersymmetry and Supergravity
A mapping gr: 9+ Y is called a ‘right linear operator’ on 9 if 1.
v 2,Y € Y
(Z+Y)%=(?)%+(Y)%
2. (d)%=ci(?)% V~EA,
(1.9.24b)
V ? E ~
The set of all left (right) linear operators on 9 will be denoted by € n d ( + ) Y (End(-%’). As will be shown, the spaces E n d + ) 9 and E n d ( - ) 9 have non-empty overlap but do not coincide in general. Now, we are going to discuss in detail the properties of End+)LZ Specific features of E n d ( - ) 9 will be commented upon later. The simplest elements in € n d ( + ) Yare linear operators of left multiplication by supernumbers: given some supernumber z, the corresponding operator i ~ E n d “ ’ 9 is defined as follows
e(?)=zZ
(1.9.25)
V ? € 2
E n d ( + ) Yis naturally provided with the operations of 1. addition
+
(gl F z ) ( i = ) Fl(27,+ 9 6 ; ( i )
(1.9.26~)
2 . multiplication
(F1 .9z)(x) = F1(9;(8))
(1.9.26b)
3. left multiplication by supernumbers 2 9
= 2 . 9 0(ZF)(S) =z F ( 8 )
(1$ 9 . 2 6 ~ )
4. right multiplication by supernumbers
Bz= 9
* k ( 9 2 ) ( X )= 9 ( z ? )
(1.9.364
Introduce a pure basis (EM}in 9and decompose every supervector 8 E 9 with respect to this basis in two different forms (1.9.27a) (1.9.27b)
{zM}.
introducing the left and the right components of 3 with respect to The former decomposition in expression (1-9.27) will be used when dealing with left linear operators on the latter when dealing with right linear operators. To any left linear operator 9 on 9 we associate a supermatrix Ff by the prescription
s(E,)= EM^
(+
F
~ !) ~
E
~ (1.9.28)
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
79
Then, in accordance with (1.9.24a),the operator 9acts on every supervector (1.9.27~)according to the rule
S(Z)= X'= E M X '( M +I
Xi?) = F M N X y + , .
(1.9.29)
Note that the operators (1.9.25) of multiplication by pure supernumbers are characterized by supermatrices (1.9.30a) (1.9.30b)
a(p,q,
The supermatrix will be called the 'graded unit supermatrix'. The above results can be combined in a compact expression, (i)MN=
z( - 1
)C(Z)CM
pN .
(1.9.30~)
Our consideration shows that every left linear operator on Y is uniquely determined by its supermatrix with respect to a given pure basis in 2' Conversely, having a supermatrix F M N ,the prescription (1.9.29) provides us with a left linear operator on $cl Thus E n d ( + ) Y is in one-to-one correspondence with the set of all supermatrices of the type (1.9.28),denoted , If F,and F2are left linear operators on Y and F , and by M a t ( + ) ( pqlAm). F 2 are the corresponding supermatrices, then the operators (Fl+F2) and 9, *F2 lead to the supermatrices F1 + F 2 =
( 1.9.31a)
and
respectively. If a left linear operator F on Y is characterized by the supermatrix F , then the operators z 9 and Fz, with z being some pure supernumber, lead to the supermatrices iF and Fi, respectively, where the supermatrix i is defined by equations (1.9.30). Explicitly, we have
iF=
(- l ) E ( r ) ~ C( - 1)E")zD
(1.9.3IC)
and (1.9.31d ) These relations determine the operations of left and right multiplication by supernumbers in M a t ( + ) ( pq, 1 Am). Copyright © 1998 IOP Publishing Ltd
80
Ideas and Methods of Supersymmetry and Supergravity
Now, we are going to describe even and odd elements in End'''9 and M a r ( + ) ( pq, I A x ) . Given a left linear operator 8 on Y: we represent it as $ = 0 8 + 1 9
9 being the Y-parity mapping (1.9.13), and show that '9 ('8) is a left linear operator commuting (anticommuting) with all a-numbers. Choosing an arbitrary 2 E Y and pure x E Amand using equation (1.9.144, one obtains a s P ( X a )= ( - l y ( W 8 ( P ( Z ) x )=( - l y W ( F ( Y ( 2 ) ) a )= ( P F P ( Z ) ) a
which implies that
'8E End'+'XNext, '9and '8possess the properties: (1.9.33)
Let us prove, as an example, the former relation. If '2 E '9and '2E 'Y: then
='
s++ ( 1 X) '
0-
2
1 F(lZ)--98('Z)= '(S(O2)) E '2 2
Using relations (1.9.33),it is easy to prove that a 0 8
= OFa
where x is an arbitrary pure supernumber. In the case of we have (1
(1.9.34)
x'g-=(-l)w$Tr
g-a)(2)= '8(&) = ( - 1) E ( M 18(jQa =( - 1
' E for example, 'F(fi)
3 being a pure supervector.
In accordance with the relations (1.9.33,34), a linear operator 9E E n d + ) Y is said to be of C-type, if it does not change supervector type, and a-type, if it changes supervector type: C-type: a-type:
E(F(Z)) =~ ( 2 ) ~(8(?)) = 1+~ ( 2 ) (mod 2)
(1.9.35)
for every pure 2 E X Linear operators of definite types will be called 'pure' and endowed with the Grassmann parity
{;
&(F) =
for C-type 8 for a-type 5
(1.9.36)
Every left linear operator on 9 is uniquely represented as the sum of its C-type and a-type parts by the law (1.9.32). Copyright © 1998 IOP Publishing Ltd
Muthemutical Buckground
81
Equations (1.9.28) and (1.9.35) show that a pure operator F on 9 is characterized by a supermatrix F with the following entries: C-type: a-type:
A",,
Dpl,E @,
BmV, CL',E C, A",, DhII, E @,
B",,, Cl', E @,.
( 1.9.37a)
(1.9.376)
Making use of the notation (1.9.21) and the Grassmann parity function (1.9.36), these properties may be collected in a compact expression: E ( F ~ , ) = E ( ~ ) + E(mod ~ + E2).~ (1.9.37~) It is instructive to check explicitly that equation (1.9.37~)guarantees the desired properties (1.9.35) of 9 Let 2 = E M X z )be a pure supervector, X g ) as in equation (1.9.20). Then, from equation (1.9.37c), we obtain (mod 2) E ( X ; ' ~ ) ) = E ( F ~ ~ ~{XE~ (+F ) ))=+ +E,, E+ ~ E~+E(?)) ={E(F + E) , ~ E(2))(mod 2) = { E(F) +dX:,>] (mod 2)
+
which is in agreement with relations (1.9.35). A supermatrix FEMat(+)(p,q I A,) is said to be of C-type (a-type) if it satisfies the conditions (1.9.37~)((1.9.376)).Supermatrices of definite types are called pure. A left linear operator 9 on 9 is pure if and only if the corresponding supermatrix F is pure and of the same type as 9 The Grassmann parity E ( F )of pure supermatrix F is defined to be equal to the of the corresponding linear operator 5 From Grassmann parity ~(9) equation (1.9.34) we obtain the following important identities: 1F = ( - 1)E(Z)dF)Fi
(1.9.39a)
ZF= ( - l)E(z)c(F),F~ 0F(z@ = ( - 1)&@)EtF)~9(?) (1.9.39b) for any pure z E Am, F EMat(+)(p,q I A,) and 9E End(+)S?So, c-numbers commute with all supermatrices (linear operators), while a-numbers commute with C-type supermatrices (linear operators) and anticommute with a-types. The equations (1.9.26a,c,d)and (1.9.39b) show that End(+)5?is a supervector space (with an undefined operation of complex conjugation); the set OEnd(+)9('End"'9) of even (odd) supervectors in E n d ( + ) 2coincides with the set of C-type (a-type) linear operators defined by relations (1.9.35). Analogously, equations (1.9.3la,c,d)and(1.9.39~)show that Mat(+)(p,q I Aw) is a supervector space (with an undefined operation of complex conjugation); the set 'Mat'+)(p, q 1 A,) ('Mut'+)(p, q I Am)) of even (odd) supervectors in Mat'+)(p,q 1 A,) coincides with the set of C-type (a-type)supermatrices defined by relations (1.9.37). Each of E n d + ) 9 and Mat(+)(p,q 1 Am) has dimension (p2+ q2,2pq). Finally, E n d + ) 9and Mat(+)(p,q 1 A,) form a unital associative algebra with respect to the operations of multiplications (1.9.26b) and (1.9.31b), respectively. The product of two C-type supermatrices is a C-type Copyright © 1998 IOP Publishing Ltd
82
Ideas and Merhocis of' Supersymmetry und Supergrauity
supermatrix, the product of c-type and a-type supermatrices is an a-type supermatrix, the product of two a-type supermatrices is a c-type supermatrix. Analogous results take place for linear operatons on Y: In fact, there is a natural way to introduce operations of complex conjugation in supervector spaces End(+)dp and Mat(')(p, q I A%).For the former space this is done as follows. To every pure operator FE E n d ' + ) Y we associate the mapping 9*: 9-+ 9acting on pure supervectors 2 E dp as
(9 2( *))* .
F*(X)=(-)")]
(1.9.40~~)
In the case of arbitrary 8E End(+)dp and 2 E Z one is to represent 9 and *$ as superpositions of their c-type and a-type parts and then to apply the above definition, assuming that
(PI+9,2)*=9y+.!q
(1.9.40b)
8 * ( 2 ,+2,)=9*(2,)+9*(2,). Let us show that the operation '*' is a complex conjugation in End(+%? If is a pure supernumber, then
E
8*(Xa)=(-11)E(")CE(X)+E(CoI - ( - 1)8(F)E(X)( E
-
( 0/ -( a * - * 1)*
x
* * ( X- *))* - F* (8)&
therefore 5"* E E n d ( + ) zNext, one finds
($zcX)*(2)= ( - ])C&(~)+E(r)ldX.)(~;(X*))* which implies
(Fa)*=a*$* for every F ~ E n d " ) d p and every &EA,. It is also easily verified that (F*)* =% for every left linear operator on 2 As a result, the operation (1.9.40) satisfies all the properties of complex conjugation. We take this operation in the role of complex conjugation in E n d + ) 9The linear operator 9E E n d ( + ) Yis said to be real if F* = E imaginary if F*= - and complex otherwise. As is seen from equation (1.9.40a),these are real c-type operators which map every real supervector on to a real one. Operation (1.9.40)induces a lot of complex conjugations in Mat(')(p, q 1 Am), depending on the pure basis chosen in 9 to identify End(')Y and Mat(+)(p,q 1 Am]. Let us choose, for convenience, a standard basis {gM)in 5$ & =( - lTh'gM. If F is the supermatrix of a pure operator R with respect then the supermatrix of F*will be denoted as Fs* and called the to supercomplex conjugate of F . From the definition, we have
(zM},
F ( g M ) = Z N F N M F*(dM)= ZN(F")NM. Copyright © 1998 IOP Publishing Ltd
Mathematicul Background
83
On the other hand, the use of equation (1.9.40) gives 0(c 4 (-* 8 , )*)-- ( - l y w ( l + T ( F ) )(d (&A)* * - l)r(~Xr~+i~)+T~(+TwThl)iar(l+r(F))(p Fh "N M) - ( 6N(FN,)*.
c"2*(~M)=(-1)"Fl'"
--( -
Therefore, we obtain ( F s * ) M N = ( - 1)t(FHt $1 + i
\ ) +I
\ + I
wi h
( F MN ) *.
(1.9.41~)
Suppressing indices, this definition can be rewritten in the form: C-type
(1.9.41b) a-type FS*=(
A* -C*
'*)
(1.9.41~)
D*
A supermatrix F is said to be real if F S * = F , imaginary if F s * = - F , and complex otherwise. Pure real supermatrices satisfy the equation (f'MN)*=(-
1)C(F)(CM+Clr)+C\+Ehle\
In the C-type case, this means (FMN)*=( - 1 ) C % + i
MEV
FN '
(1.9.424
FMN
A", iB",. A",, Dp,, E R, F M N = ( C p n Dp,,) B",, CpnE R,.
(1.9.42b)
In the a-type case, we have ( F M ~ ) *= (
- 1yw +
(": ::)
F M N= iCpn
CVCI
FMN
A",, Dp,, E R, B",,, Cp, E R,.
(1.9.42~)
Operation 's*', being rather unusual, proves more acceptable than the naive operation '*' defined by (F*)MN= (FMN)*,since the latter does not respect axiom 7 of supervector spaces while the former does: one finds (iF)S* = FS*i*
(Fi)S* =Z^*FS*
(1 -9.43)
for arbitrary z E A m and F E M a t ' + ) ( p q, 1 Am). An important property of supercomplex conjugation is that ( F 1 F 2 )S* =( - I)c:(FI)I:(FdFS*f'S*. 1 2
(1.9.44)
This relation shows, in particular, that the product of two real C-type supermatrices is a real C-type supermatrix, the product of real C-type and Copyright © 1998 IOP Publishing Ltd
84
I d w s und Methoils of’ Supersymmetry and Supergravity
u-type supermatrices is a real type a-supermatrix , and the product of two Li-type supermatrices is an imaginary c-type supermatrix. Analogous results hold for left linear operators on 2 The set of all real c-type (a-type) elements in E n d ‘ + ’ Y and M a t ‘ + ’ ( p , q / A , ) will be denoted by ‘Endk+’9 (‘Endkf’9) and ‘Mark+ ) ( p ,q 1 A x ) (‘Matk+)(p , q 1 A=)), respectively. Now, we briefly comment upon E n d - ) T and its connection to End‘”14 Let 9 be a right linear operator on 2 In contrast to equation (1.9.28), its action on the basis is represented as (EAw)Y = G,NE,
(1.9.45) Then 9 acts on every ~
Ewritten Z in the form(1.9.27b), by the rule
(.QY=$’=X’p)EM
X;M,=Xr-)GNM.
(1.9.46)
Then G,;’ is said to be the supermatrix of 9 in the basis {EM}.It is clear that every right linear operator is uniquely determined by its supermatrix. So, End‘-’9 is in one-to-one correspondence with the set of all supermatrices of the type (1.9.45), the latter being denoted by M a t ( - ) ( p ,q 1 Ax). Supermatrices from Mat‘+’(p,q 1 A = ) and Mat‘-’(p, q I A = ) differ in the position of indices enumerating columns and rows. The supervector space structure on End‘-’Y or Mat‘-)(p, q I A = ) is introduced in perfect analogy with the cases of E n d ( + ) 9 or Mat(+’(p,q I AJ. In particular, c-type supermatrices from M a t ( - ) ( p ,q I A x ) look like those in equation (1.9.22). It follows from equation (1.9.39b) and its right analogue that every left c-type linear operator on Y turns out to be a right c-type one and vice versa. But this is not the case for a-type operators. Nevertheless, there exists a simple way to associate with any left a-type linear operator a right a-type operator. The point is that every operator of the form
Fp = - YpB
9E l E n d ( + ) 9
9 being the 9-parity mapping (1.9.13), proves to be right linear, Fp(&=~%.ls(Z) V CXEA, V 2 ~ 2 Setting x and
(1.9.47) (1.9.48)
to be pure, one obtains
@.ls(&)=( - 1 ~ ‘ ~ ) + ~ ( ~ ) 9 ( ~ ~ ) = ( - 1 ) & ( ~ ) ~ ~ ( ~ ) = M 9 ~ ( 2 ) .
It is evident that 9 8 ~ ’ E n d ‘ - ’ L Z ’ As a result, the correspondence 2 E n d ( + ’ Y + E n d ( - ) T defined by %;=0y++’B+O~+f’9p
(1.9.49)
O 8 and ‘9 being the c-type and a-type components of % determines a one-to-one mapping of E n d + ) Y on to End(-%’. Now, for every left linear
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
85
operator 9= '3+ '9 we can define its right action on Y by the rule
(ii)SEOS(X)+19~(2) When 9 and
v ii€X
2 are pure, this definition reads as (2)s ( - 1)i.(FM2)y(2).
(1.9.50a) (1.9.50b)
Since every element of OEnd(+]9is at the same time an element of ' E n d ( - ) Y and vice versa, these spaces coincide and can be denoted simply as 'End2 However, we shall not identify the C-type supermatrix spaces ' M u t ' + ) ( pq, 1 A%) and OMar'-'(p, q I A = ) used to represent operators from O E n d ( " 9 and o E n d ( - l X The reason for keeping this difference will be described in the next subsection. Up to now we have studied the operator spaces E n d ' ) 9 associated with some supervector space Y Given two supervector spaces 9 and 2 one can associate with them not only the spaces E n d ( * ) Y and End("$ but also the spaces End"'(6c: 8)of all left (right) linear operators from 9 to 8 and the spaces E n d ( * ) ( $ 9 ) of all left (right) linear operators from 8 to 2 Let us briefly discuss, as an example, E n d ( " ( Z 8 ) . By definition, a mapping 9'9 :+ 8 is called a left linear operator acting from 2 to 8 if it satisfies the properties (1.9.24a). A supervector space structure on E n d ( + ) ( Z8)is introduced with the help of the operations (1.9.26a,c,d). In particular, pure linear operators from €nd(+)(Z 8)are characterized as follows: C-type
a-type (1.9.50)
The operation of complex conjugation in E n d ( + ) ( Z8)can be introduced through rule (1.9.40).Finally, the transformation (1.9.49) maps End(+)(%8) on to End(-)(6c:8). The overlap of End(+)(%8)with E n d ( - ) ( Z8 )is given by the set of all C-type linear operators from 9to $denoted by 'End(% 8).
I .9.5. Dual supervector spaces, supertransposition Consider some supervector space Y Supervector spaces *2= E n d ( + ' ( ZA,) and 3* End(-)(% Am) are said to be the 'left dual' and the 'right dual' of Z respectively. Elements of *9(9*) are called 'left (right) super 1-forms' on 9 (or 'supercovectors'). In the remainder of this section, we label super I-forms by capital Latin letters in boldface and use the notation
J , ( ~ ) = ( J I ~ ) E A ,(VJ,E*Y
V ~ S E )
(X)J,=(~IJ)EA, ( V J , E ~ * VXEY). Copyright © 1998 IOP Publishing Ltd
86
I ( l w s und Metiiods of' Supersymmetry and Supergravity
Basic properties of left super 1-forms are:
(J i X+P) = ( J 1 X ) + ( J 1 P) (J+LIX)=(JI
(J ( X r ) = ( J
X)+(LI2) ( J x I X ) = (J I&)
(XJI
12)~
X)==(J 12)
(1.9.51)
for all J, L E *Y and 2, ? E 9 and r E Aa. To obtain their right analogues, one replaces supervectors by right super 1-forms and left super 1-forms by supervectors. Pure super 1-forms are characterized as follows: c-type
a-type (1.9.52)
Similarly to (1.9.40), complex conjugation in *dp and
(j*/ji)-(-i)e(JMZ) (J I
( 2 1 J*) - (
- I)c(J)e(%)
Y*is defined by
z*)*
(x*1 J)*.
(1-9.534 (1.9.53b)
Given a pure left super 1-form J, one can obtain a right one through the prescription
(2 ( J ) =(- l)E(J)E(Z)(J 12).
(1.9.54)
The set of all c-type super 1-forms 0 2 * EO ( * Y )
= 0(6p*)
is said to be the 'dual' of '2The set of all real c-type super 1-forms oy~~o(*Y)R=o(Y*)R is said to be the dual of O P R . Every J E '9* is a c-number-valued function on OdZ: every J E ' ~ ;is a real c-number-valued function on O6PR. When Y has dimension ( p , q ) , the same is true for both *dp and 9*. If we introduce a pure basis in dc: {EM),then every super 1-form is completely specified by its values on E,:
(Jig) = (JIEM)XK)
(ilJ) = x c l ( i ~ I J ) .
In particular, there exist a unique set of left super 1-forms {(EM/}and a unique set of right super 1-forms {(EM)}such that
(EM1 EN) =
(EM 1 EN)= aMN.
(1.9.55)
The {(EM()prove to form a pure basis in *Y called the 'left dual' of (EM}. Similarly, the set {(EM)}forms a pure bases in dp* known as the 'right dual' of {E,}. It follows from equation (1.9.53) that the left (right) dual of a real pure basis in Y is a standard basis in *Y(9*) and vice versa. Let 9 be a pure left linear operator on 2 It can be considered as a right Copyright © 1998 IOP Publishing Ltd
Mathematical Background linear operator
87
e on *2defined as ((J)& 1 2)= (J 1 F(2))
or as a left linear operator FsT on *9defined as
(FsT(J) 12) E ( - l)"(s)"(J)(J I F(2)).
(1.9.56)
Evidently, if a is a pure supernumber, one obtains (9sT(ja)l
F(m
~ ) E ( . ~ ) C E ( J ) + & ( ~aI (I J
&=(-
- ( - 1)d.f)dJ)(J -
1 F(&))= (FST(J)a12).
The operator FST is said to be the 'supertranspose' of 9?Given an arbitrary left linear operator on Z F= O F + ' E its supertranspose is defined as p T =
( 0 F ) S T +( 1 F ) S T .
Now, the operation of supertransposition determines a one-to-one mapping of E n d + ) 9 on to End'+)(*$p). Its inverse mapping is defined to be the supertransposition on End'+)(*$p).Basic properties of supertransposition are:
(Fl+ F J S T = F G ; T + 9 ; T ST=( - ~ ) ~ . ' W E ( ~ Z1) F ~ T2. F ~ T (a . y ) S T = a.p-ST (,ST)*
(9.@)ST = ~
=(F*)ST
P T . ~
(9sT)sT
(V aEA,)
(1.9.57)
=g
The correspondence 9 -,FsTdescribed induces an operation of supertransposition which maps every supermatrix F from Mat'*'(p, q I A m ) into the supermatrix FsTin Mat(-)(p,q 1 A,) and vice versa. Introduce the with respect to a pure basis {E,) in 9 and its supermatrices of F and FsT dual {EM}in *Y: respectively,
S(B,)
= ENFN,
FsT(EM)= E N ( F S T ) N M . Note, supermatrices from Mat'+)(p,q I A m )are used as left linear operators on 64 while supermatrices from M a t ' - ' ( p , q l A , ) are used as left linear The use of equations ( 1 . 9 . 5 5 , 5 6 ) leads to operators on *9 (FsT),N
= (- 1 )R(FWEM + E ~ ) + c M + E H B Y FN,.
(1.9.58~)
It is instructive to rewrite this definition in more detail:
F"=( -BT AT Copyright © 1998 IOP Publishing Ltd
DT cT)
(1.9.58b)
88
Itletis crnd Melhods of Supersymmetry und SuperyraGify
FsT=(i:
- DT). CT
(1.9.58~)
The supermatrix FsTis said to be the ‘supertranspose’ of F. Supertransposition can be extended in an obvious way to the full Mat‘+’(p,q I A%) resulting in a one-to-one mapping of Mat(+’(p,q !A%)on to Mati-’(p, q ! A%). Its inverse mapping will be taken in the role of supertransposition on Muti-’(p, q I A%). One readily deduces from (1.9.58~)that the supertranspose of a pure supermatrix G , ” E Mat‘-)(p , q 1 A x ) reads ( G # T ) M ~ = ( _ ~ ) E ( G ) ( E M + E N ) + E N + E w EGN ~ M ,
(1.9.59)
Comparing equations (1-9.58) and (1.9.59), one finds that supertransposition is sensitive to the positions of indices! That is why we do not identify OMar(+fp,q I A x ) and OMati-’(p, q I A%).Basic properties of supertransposition are: ( F , +F2yT=FsT+FiT (FsT)sT=F (F1. F 2 ) s T = ( - 1) I ( F I ) ~ ( F 1? ) F ~ T . (2. F)sT = 2 . p
( F . $)ST = p.1 2 s T --
(V
ZEA,)
(1.9.60)
z. A
This can be supplemented by the relation (FsT)s*
=( ~ s * ) s T
(1.9.61)
provided complex conjugation in M u t ( - ) ( p ,q 1 A,-) is defined in the manner ( GS*)M~v = ( - ~ ) F ( G ) ( F v+ r i ) + FII + r M F I ( G M N ) * (1.9.62) different from equation (1.9.41).The point is that the dual of a standard basis in Y is a real pure basis in *2 Let XM be components of a C-type supervector, e ( X M ) = eM. One readily finds that FMNXN=XN(FsT)NMV F E o M a t ‘ + ’ ( p ,q \ A m ) (1.9.63) , X N G ~ *= ( G S T ) * ~ X NVG E O M ~ a t ‘ - ’ ( pqIAo3) These relations will often be used below. 1.9.6. Bi-linear forms Bi-linear forms on supervector spaces may be introduced in different ways. For example, associated with every right linear operator &: Y + *Y is the following bi-linear form on 2
(2, ?),H= ((2)s ! t) Copyright © 1998 IOP Publishing Ltd
(1.9.64)
Mathemutical Background
89
which possesses the property (1 -9.65) (22,Pp)q =r.(ii,Y ) , q for arbitrary supernumbers Q and 8. When H is pure, we easily construct
its supertranspose HsTas the left linear operator YF: 2'+2'* defined by (21 & m T ( Y ) ) ~ ( - ~ ) I O Y ) C I ( ~ ) + I ( ~( I(+t I) ~H~1 2) ~ ~ ~ I (1.9.66)
2 and being pure supervectors, thus obtaining another bi-linear form on 2 ' (1.9.67) ( X , Y ) f ' T E (21xsT (?)I -
4
possessing property (1.9.65). The latter bi-linear form is called the supertranspose of the former. Bi-linear forms (1.9.64) and (1.9.67) are said to be C-type or a-type or mixed depending on the type of 2 A C-type bi-linear form (1.9.64)is called 'supersymmetric' if it coincides with its supertranspose, that is
(2, p)f = ( -
(1.9.68)
(X 8 ) K
1pId7
and 'antisupersymmetric' if
(8,y)K= -(- 1 p I o l(2 8)x.
(1.9.69)
It is worth pointing out that every C-type bi-linear form takes c-number values on even supervectors,
(8,Y ) , E
v z,Y €O L Z
c,
(1.9.70)
Next, if 2:9+ *9is a real C-type linear operator,
( H ( 2 ) )= * H(2*)
v 8€ 9
(see also equation (1.9.40)), then the corresponding bi-linear form takes real c-number values on real even supervectors,
(2,P),E
R,
v 8, Y € 0 9 &
(1.9.71)
A real C-type linear operator H:9+ *9is said to be 'self-adjoint' if the corresponding bi-linear form is supersymmetric. Below we discuss only C-type bi-linear forms and consider their values only on even supervectors. Let us fix some standard basis in 2' and its left dual {EM},which is a real pure basis in *yand introduce left components for supervectors in '9and right components for super l-forms in O Y * :
{aM}
z=2 , X M J = JMEM
& ( X M )= &M
&(JM)=E
~
.
As a result, we have
(1.9.72) Copyright © 1998 IOP Publishing Ltd
90
Ideas and Methods of Supersymmetry and Supergravity
for all J E '9* and 2 E ' 2 Real supervectors and super 1-forms satisfy the constraints (XM)*= X M
(JM)*
= ( - 1)""J.
(1.9.73)
Given a C-type linear operator X : 9+*sC: we represent its action as
(E,)8
= ( - I)CMH,,E~
(1.9.74~)
hence
( 2 )= x,H,,E, ~ H,,
(1.9.74b)
being some supermatrix under the conditions &(HM,)=&M+&N.
( 1.9.75)
Then, the induced bi-linear form reads
(2,Y ) x= XMHM,YN.
(1.9.76)
Is supertranspose (1.9.67)is given by (1.9.77) If our bi-linear form is supersymmetric, then its supermatrix satisfies the equation H,,=(-
HNM.
1)EU+Eh'+&Y&h
(1.9.78)
Such supermatrices are called 'supersymmetric'. Finally, the reality condition (1.9.71) means that (HMN)*=(- l)EU+EEN+EUElHMN.
(1.9.79)
Suppose the operator X is invertible. Introducing its inverse X - ': we define the bi-linear form on *9:
*9 4
(J,L)~-IE(J~(L)X-').
(1.9.80)
It is instructive to obtain its explicit form. Since the supermatrix HMN is non-singular, there exists a unique inverse ( H - l ) M N , HMR(H-' ) R N = aMN ( H - ' )MRHRN = aMN. (1.9.81) Resolving equations (1.9.74) gives ( E ~ ) x -=(~
I)EN(H-~)~~E~
(J)%-' = E M J N ( H - ' ) , ~
therefore the expression (1.9.80) takes the form (J, L),*-I' J M L N ( & - ' ) " ~ Copyright © 1998 IOP Publishing Ltd
(1.9.82)
9I
Muthemuticul Buckground
which should be compared with equation (1.9.76). In conclusion, we note that the inverse of supersymmetric supermatrix (1.9.78) is characterized by the property:
(H-1 ) M N = ( - ~ ) E M O N ( H - 1)NM.
(1.9.83)
1.10. Elements of analysis with supernumbers 1.10.1. Superfunctions We are going to introduce a generalization of the usual smooth and analytic functions in R" or @" to the case of supernumber-valued functions. Instead of R", we consider a real superspace Rplq= IF! x IF!: (1.9.19) parametrized by p c-number coordinates xm, m = 1, 2, . . ., p and q a-number coordinates OP, p = l , 2, . . . ) q. We start by generalizing the concept of analytic functions of ordinary real variables. A supernumber-valued function on RPl4
f: RP1*_+AX is called 'superanalytic' if it can be expanded in a Taylor series in zM: f(z)=f(x', * .
9 )
xp,
P,. . ., 6 4 )
To see what this definition leads to, we consider some particular cases. First, when p=O, q= 1, we deal with functions of one real odd variable 8. Since a-numbers anticommute between themselves, 8' = 0, and equation (1.10.1) means
f(8)=cp+@
cp, $ E A = .
(1.10.2)
On the same grounds, in the case of q Z 0 , 1 odd variables 8 P only, every product of more than q 8s vanishes, and the most general expression for a superanalytic functions on is
(1.10.3)
with f P , P L I . . , P A being totally antisymmetric in its indices. It is clear that the set of all superanalytic functions on [w01q= R: forms a supervector space of dimension ( 2 4 - l , 2 4 - l ) . Life is not so simple when p Z 0 . In particular, in the simplest case p = 1, Copyright © 1998 IOP Publishing Ltd
92
Ideas and Methods of Supersymmetry and Supergravity
q = O all terms in the Taylor expression I
(1.10.4) survive and we are faced with the problem of finding restrictions on fk providing existence of the expansion. But for a wide class of superanalytic functions, it is really a problem of ordinary analysis. Indeed, let us split the variable x into its body and soul, x = x B + x s , and let f: R-, R(@) be an analytic function of an ordinary real variable, XI
y(xB)
=
1
Jk(XFJ)k
3 k E R(c)*
k=O
Then, the supernumber-valued function on R, (1.10.5) where y(k) denotes the kth derivative of 7,is superanalytic on R,. Since ( x J = 0 for some integer n, only a finite number of terms in the expansion (1.10.5) are non-vanishing. Then f(x) is said to be the super extension of f ( x B ) . A remarkable result is that the general form of a superanalytic function on R, is ( 1.10.6)
where C' are the generating elements of A, and all the f i , , , , i k ( xare ) functions of the type (1.10.5). Another important observation is that equation (1.10.5) may serve as a prescription for constructing 'supersmooth' functions on R,. Having a smooth function y o n R, we introduce its super extension using rule (1.10.5)resulting in a supersmooth function on R,. The nth derivative of f ( x ) with respect to x is defined to be
By definition, the general form of a supersmooth function on R, is given by equation (1.10,6), where all the fi,,, , i A ( x are ) super extensions of a smooth function on R. Every supersmooth function f ( x ) is smooth on the body and analytic on the soul of the c-number variable x . Our previous consideration is easily extended to the case of supernumbervalued functions of c-number variables. In particular, the most general form of a superanalytic function f: R: -+A%is
Copyright © 1998 IOP Publishing Ltd
Mathematical Background
93
with the fs being totally symmetric in their indices. Finally, if p # O and q # O , every supersmooth or superanalytic function f: Rp14-+A m can be expressed in the form
.
f(Xl,.. , x p , e l , .
. .,e4)=fo(xl, . . ., x p )
+
4
1
k=1
k.
. . ., x p ) e w p z . . . e g h
f [ p l K L 2 . ,, p k l ( x l ,
( 1.1 0.9)
with the fs being supersmooth or superanalytic functions of 03;. Note that all supersmooth functions f: R p t q + A s are supersmooth with respect to their c-number variables and superanalytic with respect to their a-number variables. Let f: lWP14- A, be some superfunction. For every point z M €Rpt4, the image f(z) may be decomposed into its even and odd parts
f ( 4=f C ( 4 +fM
fC(4
E
c,
fA-4E @ a .
Thus, it is reasonable to consider purely even
f: RP‘4+CC
(1.10,loa)
f:[ W p ’ 4 + c a
(1.10.lob)
or purely odd mappings. Superfunctions of the type (1.10.loa) will be called ‘bosonic’ or ‘even’, and superfunctions of the type (1.10.10b) will be called ‘fermionic’ or ‘odd’. Bosonic and fermionic superfunctions will also be called ’pure’. It is useful to associate with any pure superfunction f its Grassmann parity ~ ( f ) by the rule 0 1
iffbosonic iff fermionic.
(1.10.11)
Having some pure supersmooth function (1.10.9>, the coefficients in the right-hand side of ( I . 10.9)carry the following Grassmann parities E(fo)=ECf)
E(fg,g,...gJ=E(f)+k
(mod 2).
(1.10.12)
From here we will consider pure superfunctions only. Let f ( x ’ , . . ., x p , e’, . . ., OB) be a supersmooth or superanalytic function. We define ‘partial derivatives’ o f f with respect to its variables as follows
Copyright © 1998 IOP Publishing Ltd
94
Idem und Methods of Supersymmetry and Supergravity
where partial derivatives of functions of the type (1.10.8) are defined in the ordinary way:
We have introduced left z / a O p and right @aep partial derivatives with respect to the odd variables since any infinitesimal displacement -,Op + d P results in the changing o f f df(z)=f(x, e+d@-f(x, 0) which can be represented in two forms: (1.I 0.144
The left and right derivatives are related:
(1.10.14b) So, one can work with the left or the right derivatives only. In what follows, we shall use mainly the left derivatives which will be denoted (1.10.15)
Looking at equations (1.10.13), one finds that differentiation with respect to odd variables (by virtue of (1.10.12)) changes the type of superfunction. Namely, the operator a/aOP transforms every bosonic superfunction into a fermionic one and vice versa,
):(-
E
= 1 + ~ ( f ) (mod 2)
( 1.I 0.16a)
In contrast, differentiation with respect to even variables leaves the superfunction type unchanged, E
(83 -
=&(f).
(1.10.16b)
Introducing the following useful notation EM
= E(ZM) =
Copyright © 1998 IOP Publishing Ltd
0 1
M=m M=p
(1.10.17)
Mathematical Background
95
(see also (1,9.21), and (1.10.18) one can unify equations (1.10.16a)and (1.10.16b)into a single equation: (mod 2).
~(a,f)=tM++ECf)
(1.10.16~)
Further properties of the partial derivatives are:
1
a,xn = 6,"
1.
8MZN
(1.10.19a)
= 6 M N 0 d,8' = 6,"
8,8" = d p X "
2.
8Ma,Y
= ( - 1 )E'"'
1
=0
zman= anzm
8 N a M 0 (?mav
a,a,
3. ' h f ( f V )= (8Mf)q
+ (-
=avdm
(1.10.19b)
= - Q,
(1.10.19~)
l)"""'f'f(a&)
where f and cp are superfunctions. We see that the odd derivatives anticommute between themselves and commute with the even derivatives. What is more, pulling the operator a/aOp through a fermionic superfunction gives a minus sign. The equation (1.10.19~)can also be rewritten as
8M .f-(-
1)EMdJ)
f.8, =(8,f).
(1.10.20)
Finally, we clarify how the operation of complex conjugation (1.9.10) acts on partial derivatives. Given a superfunction of the form (1.10.9),one obtains
=f:+
4
(k= 1
=so*+
4
(--
k=l
where we have used equation (1.10.12).Making use of equation (1.10.13),we then obtain
Copyright © 1998 IOP Publishing Ltd
96
Idem und Methods of Supersymmetry and Supergravity
Now we change the order of operations,
Xf%',...>i
11P
...A' ,
I,
Comparing Zpf* and (?,$)*, we find (?,f)* = -( - 1)E'f) 2,f*.
(1.10.21)
This beautiful result can be unified with the trivial relation (zmf)*
=a m f *
as follows
(8M f )* -- ( - 1 ) E M ( l + E ( S I ) ,
Mf*,
(1.10.22)
What does equation (1.10.22) mean? Let us consider 'real' bosonic f:
RP'4-
R,
( 1.10.23)
f:
R P ' 4 -+
R,
(1.10.24)
and fermionic superfunctions. Then equation( 1.10.22)tells us that an odd partial derivative of a real bosonic (fermionic) superfunction is. an imaginary fermionic (real bosonic) superfunction. I . 10.2, Integrution over RPIQ Our goal is to develop an integration theory on superspaces WI4. Of course, the theory of integration over RIB"will guide us. We shall consider in detail the simplest cases p = 1, q = 0 and p = 0, q = 1, and then generalize the results to the general case. Let f ( x ) : Rc-+Az be a supersmooth function. The main question is how to define the integral ["'f(x) dx J
xl, x2 E R,
(1.10.25)
XI
where the integration is performed over a contour x ( t ) : [0, l]-R,
x(O)=x,, x(l)=x,.
The problem we are faced with is that, in contrast to R where integrals like Copyright © 1998 IOP Publishing Ltd
Muthemutical Buckground
97
( I . 10.25) are usually performed over a line there are now many more possibilities in the choice of integration contour. Indeed, splitting x ( t ) into its body and soul, x(t)= XB(t)
+ x&)
and fixing the body, for example, as in equation (1.10.26), we still have wide freedom in our choice of the soul. Therefore, it would be desirable to take as our integration prescription one which will guarantee that the integral ( 1.10.25) will not depend on the particular choice of integration contour (which implies that the soul is measureless). The second requirement of integration theory over R, is a correspondence with the integration theory over R. Namely, if ,f(x) is the super extension of a smooth function J(x,) on R (equation (1.10.5) ) and the end points in equation (1.10.25) are real numbers, then the integral should coincide with the ordinary integral
Xi
The above requirements are satisfied by an integration theory which is characterized by the property ( 1.10.27)
where q ( x ) is an arbitrary supersmooth function. Let us prove this statement for the example of a superfunction f(x) which is the super extension of a smooth function f(x,) on R. We introduce another smooth function F(x,) on R,
J O
and construct its extension F(x) on 02, by the rule (1.10.5):
=F(x,)+
" 1
-f'k
k=i
k!
As a result of(1.10.7), we have
F'(x)= f(x). Copyright © 1998 IOP Publishing Ltd
98
Ideas und Methods of Supersymmetry and Supergravity
Applying equation ( I . 10.27), we have
61
f ( x ) d.u = F(.u2)- F(x,).
This completes the proof. Note that our consideration happened also to be successful for the following reason: every supersmooth function f(x) is analytic with respect to the soul of x. Since the integrals (1.10.25)depend only on the end points of the integration contour, one may immediately extend distributions on R (like the &function) to distributions on R, to develop Fourier transform theory on R,, and so on. In particular, the &function on R, can be represented in a standard way, namely
where the integrals here can be performed over every contour p(t): R! -+ R, such that the body of p ( t ) goes to 00 when t goes to F CO, respectively, lim p e ( t ) = & CO. r-+m
For every supersmooth function f(x): R,-+Am, we have (1.10.29)
In perfect analogy with integration over RP, the integration over R P l o = R: will be understood as multiple integration consisting of sequences of ordinary integrations over R, [dpxf(xl, x2, . . ., x p ) = j z dx' -m
dxf'
jm ...jm dx2
-m
dxPf(xl, x2, . . ., x")
-m
= dx' d x 2 . .. dxP
(1.10.30)
where the supersmooth function f ( x ' , x 2 , . . ., xp) should satisfy some obvious restrictions to ensure existence of the integral. We do not explicitly define integrals over finite domains in R: since every such integral can be extended to an integral of the type (1.10.30).The &function on R!: is defined by the rule BP(Xm)
= S(X')S(X2). . . qxp)
and has the same properties as the &function in Rp. We proceed to integrate over R,. Given some superfunction f(0) on R, which is expressed in the form (1.10.2), one must define an integral
Copyright © 1998 IOP Publishing Ltd
Muthemutied Buckground
99
How do we do this? When considering integration over R, it was possible, since R, contains R, to make use of our experience of integration over R. Now any experience is absent. All we are able to do is to postulate some reasonable axioms determining the properties of integrals. Following F. Berezin, we postulate ( 1.10.3 1a)
s[,f(O)+h(H)] dH=
:s s
f(0)dz f(H)dH=O Ode= -1.
ZEA,
(1.10.3lb) ( 1.10.31c)
(1.10.3 Id)
Let us comment on the axioms. The equations (1.10.31a,b) express standard properties of ordinary integrals. Equation (1.10.31~)is a generalization of the well-known integral property (1.10.27). Since f(0) d8 is understood as an integral over the whole space R, and R, is boundless, the right-hand side of property (1.10.3IC) should vanish. Equivalently, the axiom demands that every integral be invariant with respect to translations on R,
This axiom can also be treated as the rule for integration by parts:
Recalling the most general form (1.10.2) of superfunctions on R, equation (1.10.31~)can also be expressed as the statement that R, has vanishing 'volume'.
s
di?= 0.
(1.10.34)
After imposing the axioms (1.10.31a-c), the remaining task is to normalize the integral JOdO. It is the last axiom (1.10.31d) which plays the role of the normalization condition. Since the right-hand side in axiom (1.10.31d) is a c-number and the integration variable 8 is an a-number, the measure d e Copyright © 1998 IOP Publishing Ltd
100
Iileus and Met/io(is of’ Supersymmetry und Superyruvity
should be treated as an a-number object anticommuting with all a-numbers, d & = -2d6 +
(V
ZE@,)
sd88= -
( 1.10.35)
de.
Then equation (1.10.31d) can be expressed in the form ( 1.10.36)
Therefore, the integration over R, is equivalent to differentiation. We introduce the &function on R,: 6(6) = 8.
( 1.10.37)
I t possesses the standard &function properties
s(e)f(@=f(o)
{dB
(1.10.38) as well as some unusual properties:
6(8)S(B)rO,
It is not difficult to obtain an analogue of the representation (1.10.28) for the &function on R,. It is
s
6(e)= d i e A e
(1.10.40)
where the integral is performed over R,. By analogy with integration over Rf, integration over RX will be understood as multiple integration:
s
ss s
dqef(e1, ..., eq)= de,
deq+..
delj-(el, ..., e q ) (1.10.41)
where de, is the measure on R, with coordinate e”, f
( 1.10.42)
The measures de, are assumed to anticommute with all a-numbers and Copyright © 1998 IOP Publishing Ltd
Muthemutical Buckground
101
between themselves
do,, do,,+ de, do, = 0.
(1.10.43)
For any superfunction ,f(O', . . ., e*),which is expressed in the form (1.10.3), one obtains
s
d40,f(f11,. . ., 0 4 ) = ( - 1)4"+"'s') f 1 2 , . . q .
(1.10.44)
Therefore, only the last term in the power series (1.10.3)gives a non-vanishing contribution to the integral. The &function on rWg is bq(e1, 8 2 ,
. . ., 0 4 ) =
. .q e 4 ) .
s(e1)s(e2).
( I . 10.45)
Its properties can be easily read off from expressions (1.10.38, 39). Finally, integration over RPl4 is defined as multiple integration over Rf and R8,4:
s
ss
dpf4zf(zM)= dPx d48f(x1, . . ., x p , 01, . . ., P).
(1.10.46)
1.10.3. Linear replacements of variables on RPIq
Consider a linear one-to-one mapping of RPl4 on itself
z M + ztM= FM,zN FM -
A",
B",
(1.10.47)
where F is a C-type non-singular supermatrix (recall that a supermatrix is non-singular if and only if it has non-singular body); F has to be a real supermatrix of the form (1.9.42b) for the zfM to be real. Indeed, rewriting the transformation explicitly as
xlm= Amnx" + B",l3"
+
8'" = C",xn DW,Bv
and demanding (x'")*= X" and (efp)*= e'p,one finds that the matrices A, D and C have real entries while the matrix B is purely imaginary. Evidently, the set of all invertible transformations (1.10.47) forms a group. Let us treat the transformation (1.10.47) as a replacement of variables in integrals like (1.10.46). It is clear then that the measures d P + %and dP+qz'in the old and new variables, respectively, differ by the transformation Jacobian J(F) d P f 4 z ' = J ( F )d P + 4 ~
(1.10.48)
which is some function of the supermatrix F. Our main goal is to determine Copyright © 1998 IOP Publishing Ltd
I02
Ideu.s und Methods of' S u p e r s p m e t r y and Supergravity
J ( F ) . We divide the solution of the problem into parts, first calculating the Jacobians of some particular transformations and returning to the general case at the end.
I . We start with the replacement
j
X" = Amnxn* F = (
A
0
0" = 0'
0
0, .
(1.10.49)
This is a linear change of variables in R,P. The Jacobian can be calculated in the same way as for Rp. The result is
J(F)= det A .
(1.10.50)
2. Now consider the replacement (1.10.51) To find the Jacobian, note that the variables 8'" and the corresponding measures de" should satisfy the same basic identity (1.10.42) as in the former variables. So, we have
I
S
d P v = de; 8 ' = Dvi de:, OA +
(1.10.52)
del = dO,(D- ')OH.
By virtue of equation (1.10.43), one then has del de;
. . . deb = (det D ) - ' de, de, . . .de,.
Therefore, the transformation (1.10.51) is characterized by the Jacobian J(F)=(det D)-l.
(1.10.53)
3. Consider the replacement ( 1.10.54)
In this case we have ( 1.10.55)
J(F)= 1
Indeed, let us understand the integral (1.10.46) as follows Id"".
f(zM)= [dpO Id'.
Copyright © 1998 IOP Publishing Ltd
f ( x ' , . . ., xp,
e',
,
. ,, 04).
103
Mathematical Background
For fixed Os, the transformation (1.10.54) is a shift in R:, which keeps the measure dPx invariant.
4. Finally, consider the transformation ( 1.10.56)
Now we understand the integration over R P I 4 as in (1.10.46). For fixed xs, the replacement (1.10.56) is a shift on Rz. But in accordance with equation (1,10.32), the measure d48is translationally invariant. Therefore, the Jacobian is unity, (1.10.57)
J ( F ) = 1.
Now we are in a position to find the Jacobian of the transformation (1.10.47). To do this, we represent the transformation as a product of some replacements, each of which belongs to one of the four types considered above. The matrix F can be expressed as )(Ip [I,
D-'C
O ) [I,
(1.10.58a)
or as
f-'").
(1.10.58b)
Therefore, the replacement (1.10.47) is characterized by the Jacobian J ( F )= det(A - BD - 'C) det - '(D) = det A det - '(D - C A - ' B ) (1.10.59) where we have used eqiiations (1.10.49-58). The-Jacobian J ( F ) proves to be real, J ( F )E R,,because F is a real supermatrix of the form (1.9.42b).
I . 10.4. C-type supermatrices revisited After obtaining some experience in analysis with supernumbers, it is worth returning again to an algebraic subject. We find it convenient to collect here the main operations with C-type ( p + q ) x ( p q ) supermatrices, elements of OMat(+)(p,q I Am) or ' M u t ( - ) ( p ,q I Am). Every supermatrix F ~ ' M u t ' + ) ( pq ,I A=) is represented in the form (1.9.28), where the matrix element F', are pure supernumbers with the following Grassmann parities
+
c(F',)
= cM
+E,
(mod 2 )
(1.10.60)
(see also equation (1.9.37~)).Every supermatrix G E 'Mut(-'(p, q I Am) looks Copyright © 1998 IOP Publishing Ltd
104
Ideus und MPtitotls of Supersymmetry und Supergrmity
as in equation (1.9.22). Supermatrices from Mat(+’(p,q 1 A = ) and Mati-’ ( p , q 1 A x ) differ in the positions of indices. Both spaces ‘ M a t ( + ’ ( p q, 1 A x ) and OMar(-’(p,q I A = ) are closed with respect to: 1. 2. 3. 4.
the operation of addition (1.9.31~); the operation of multiplication (1.9.316); the operation of multiplication by c-numbers (1.9.31~); the operation of supercomplex conjugation defined on OMat(+’(p,q I A,) by equation (1.9.41b) and on ‘Mat(-’(p, q 1 A x ) by
..*=i -C* A*
B*) D*
(1.10.61)
the specialization of equation (1.9.62) to the c-type case; 5. the operation of taking the inverse (1.9.23), which is defined for non-singular c-type supermatrices only. The spaces are connected to each other by the operation of supertransposition:
Recall, supertransposition is characterized by the property (1.9.63). The operation of supercomplex conjugation selects i n O M a t F ) ( p , / A m ) the subset oMutk+)(p,q 1 A x ) of real supermatrices which satisfy the equation F5* = F and have the form (1.9.42b). OMatk+)(p,q 1 A,) is closed with respect to the operations of addition and multiplication of its elements and the operation of multiplication by real c-numbers. The supertransposition maps OMutF’(p, q 1 A = ) onto the real subset OMutk-’(p, q I A,) in OMat‘-)(p, q I A%). The set of non-singular supermatrices in oMat‘C)(p,q I A,), evidently, forms a group denoted by GL‘+’(p,q 1 A=). If F E OMatk+)(p,q I A,) is a non-singular supermatrix, then, due to (1.9.23), its inverse supermatrix F - is real. Since the product of real supermatrices is a real supermatrix, the set of all non-singular supermatrices in OMark+)(p,q I A,) forms a group denoted by GLk+’(p,q I A=). Supertransposition maps GL‘+’(p,q I A,) (GLk+’(p,q 1 Am)) onto the group G L ( - ) ( p ,q 1 A=) (GLL-)(p,qJA,)) of non-singular (real non-singular) supermatrices in OMut(-’(p, q I Am). Now we are going to introduce a supergeneralization of the determinant. Recall that the determinant of an ordinary n x n matrix A can be defined as the Jacobian of the replacement of variables x’ = A x on R”. It is this definition which will be generalized to the super-case. In the previous subsection, we found the Jacobian of the linear replacement of variables (1.10.47) in RPl4 Copyright © 1998 IOP Publishing Ltd
Muthemutical Background
105
associated with a non-singular supermatrix F . The Jacobian was given by equation ( I . 10.59).So it would be desirable to postulate the ‘superdeterminant’ or the ‘Berezinian’ of a non-singular supermatrix F E GL(+)(p,q I A x ) as
’
sdet F - Ber F=det(A -BD- C) det - ‘D=det A det- ‘(D-C A - ‘ B ) . (1.10.63) and similarly for supermatrices from GL(-)(p,q I A=). The notion of superdeterminant was introduced by F. Berezin. In accordance with (1.10.63), the Berezinian can be calculated using two different prescriptions. The statement that the first and the second lead to the same final result is a consequence of our consideration in subsection 1.10.3. (In fact, the supermatrix F in (1.10.47) was real. To generalize equation (1.10.59) to the case of complex supermatrices, one has simply to consider linear replacements of variables in the complex superspace CPl4.)On the same grounds, since the Jacobian for the composition of two replacements of variables is equal to the product of the Jacobians, one immediately obtains Ber(F,F,)=Ber F, Ber F ,
*
Ber(F - I ) = (Ber F)-
( 1.10.64)
We also have Ber(F”) = Ber F .
(1.10.65)
Indeed, due to (1.9.58b) and (1.10.63), one can write Ber(F“) = det[AT + CT(DT)-’BT] det-’ DT = det[(A - BD det DT= Ber F where we have used the fact that the entries of the submatrices B and C are a-numbers. Note that relation (1.10.65) is in complete agreement with the fact that the transformation(l.lO.47) can be written in two equivalent forms =p
Z’M
f $ N
= ZN(FST),M
in accordance with (1.9.63). Finally, we introduce a supergeneralization of the trace of a matrix. Recall that det(0 + A ) = 1 + tr A , for every ordinary matrix A with infinitesimal elements. We take this identity as the starting point in defining a supertrace. Let F be an infinitesimal supermatrix (this means that the matrix elements !F are infinitesimal supernumbers with respect to the norm (1.9.6)). Then the supermatrix ( O ( p , q ) F) is non-singular and, due to (1.10.63),we have
+
Ber(O(,,,,+F)= 1 + tr A -tr D .
So we postulate the ‘supertrace’ofa supermatrix F E Mat(+’(p,4 I A x ) to be str F =( - I)””FMM Copyright © 1998 IOP Publishing Ltd
( I , 10.66)
106
Ideus and Methods of Supersymmetry and Supergruuity
and similarly for supermatrices from OMat'-'(p, q 1 A=), The main supertrace properties are str(FST)= str F
(1.10.67)
str(FlF2)= str(F2Fl). The proof is quite trivial str(FsT)=(- ,)EM(F"T),M str(F 1 F2 I-(=(
=( - 1 y v ( - I ) E M + E M E M FM M-(-lywM
1 y v ~ MF N -(.-1yv(1N 2 M -
~ ) ( E M + E ~ ) ( E M + E \ ) FFNM 2M 1N
- l)NF;MFyN.
The superdeterminant and supertrace are related as follows: Ber(eF)= estrF
(1.10.68)
where F is an arbitrary supermatrix. To prove (1.10.68), one has simply to follow the ordinary proof. One introduces the c-number-valued function on R f(t) = Ber(etF)
f(0)= 1.
For an infinitesimal displacement dt, we have
+
+
+
f ( t dt) = Ber(etFedfF)= Ber(etF)Ber(O(P,4) dtF) = f(t)( 1 dt-str F } .
Then, df(t)/dt = (str F ) f ( t )
* f ( t )= c
Taking into account the initial condition, one arrives at (1.10.68).
1.1 1. The supergroup of general coordinate transformations on Rp14 We would like to continue the study of real superspaces RPl4 parametrized by p even coordinates X" and q odd coordinates 8'. As the next step it would be very interesting to generalize the concept of the general coordinate transformation group on manifolds which, as is well known, plays an important role in all gravity models. Let us consider a one-to-one mapping of RPl4on itself. In the coordinates on RpIq, this is Z M -+ Z ' M = f M ( z )
i
0
x" -+ X" = f"(X, e) 8' -+ 8' = fyx,9).
(1.11.1)
We restrict ourselves to the consideration of supersmooth transformations with f" and f" being supersmooth functions. The necessary condition for the transformation being invertible is that the supermatrix of partial Copyright © 1998 IOP Publishing Ltd
Mathematical Background
107
derivatives G M N ( z=) aMfN(z)
is non-singular at each point zME [WPIq. Indeed, infinitesimally, every displacement dz'M should be the image of a unique displacement dzNrelated to each other by the rule dz'M= dzN(aNfM). The set of all invertible supersmooth transformations (1.11.1) forms a group. It will be called the 'general coordinate transformation supergroup' (the term 'supergroup' will be described properly in the next chapter). In the present section, our treatment is based mainly on special technique developed by S.J. Gates and W. Siegel in their study of superfield supergravity'
I
I . 1I.I . The exponential form for general coordinate transformations An infinitesimal coordinate transformation on Rp14can be written as ZIM
= ZM
- zKM(z).
(1.11.2)
Here t is an infinitesimal real parameter, K M ( z )=(K'"(x, e), K p ( x ,e)), where K"(z) are real bosonic superfunctions and K'jz) are real fermionic superfunctions. We introduce the first-order differential operator K =~ ~ ( z ) = a ~, y x e)a, ,
&(KM)=tM
+ ~ ' ( xe)a, ,
(KM)*= K M
( 1.11.3)
which will be called a 'real supervector field'. Then equation (1.1 1.2) takes the form ZIM
= (1 - TK)ZM.
Exponentiating this infinitesimal transformation, one obtains a finite coordinate transformation, (1.11.4)
fM=,-K,M
or, more explicitly,
Evidently, the transformation (1.11.4) is one-to-one. A sufficient condition for it to map RPI4 onto RPI9 is that the components of K M vanish when the bodies of the c-number coordinates xm go to infinity,
-
IXBI+a
KM I
S. J. Gates and W.Siegel, Nucl. Phys.
Copyright © 1998 IOP Publishing Ltd
B 163 519,
0.
1980.
108
Ideus und Methods of' Supersymmetrj and Superyrucity
Therefore, existence of the exponential form (1.1 1.4) for a transformation (1.1 1.1) guarantees that the transformation is invertible. Let us give two simple examples of coordinate transformations which admit the exponential form. First, consider a translation on IwPl4:
zM+ zfM= z M
+ bM
a,bM =O.
This is expressed in the form (1.1 1.4) with K=-b'aM. Secondly, consider a linear transformation of the type (1A0.47). This admits the exponential form if F = e R for some supermatrix R. Under this condition, one can prove that (1.11.5) Not all general coordinate transformations (1.1 1.1)possess the exponential form (1.1 1.4), only contractible ones. A coordinate transformation z ' =~fM(z) is said to be 'contractible' if it may be deformed continuously to the identity transformation, i.e. there exists a sequence of transformations Z'M
=fM(z, t )
t E [O, 13
such that f M ( z , t ) are continuous functions of a real parameter t, and f M ( z ,1) = f M ( z ) ,fM(z,0)= zM. The set of all contractible transformations on RPI4 forms a subgroup of the general coordinate transformation supergroup. From there we shall consider only the contractible coordinate transformations. The representation (1.1 1.4) is very useful in practice. Indeed, let Wz) be a supersmooth function on Rp14. In the variables z ' ~ it, takes the form (1.11.6)
W ( 2 ' )= @(z).
A natural question to ask is: how does the function W(z) differ from the former one @(z)?To answer this we use the observation that, by virtue of equation (1.10.20),
[ K W)l=(K"'(z)) where
"(2)
is an arbitrary superfunction,
$0
e-K"(z)eK = ( e - V ( z ) ) e-K
M
z e
*
K
(1.11.7)
Then, the left-hand side of (1.1 1.6) can be written as @'(e-% eK)=eTK@'(z). As a result, we have
W(z)= eK@(z).
(1.1 1.8)
To see further properties of the representation (1.11.4), we now discuss in detail supervector fields. Copyright © 1998 IOP Publishing Ltd
Mathematical Background 1.11.2. The operators K and
109
k
Let K = KY(z)a, and K, = KF(z)d, be real supervector fields. In accordance with equations (1.10.19, 20, 22), their commutator [K,, K ~ I = K , = K ~ ( Z ) ~ M KM-KNa K"-K"d 2 N KM 3.1 N 2 1
(1.1 1.9)
is a real supervector field. Therefore, the set of all real supervector fields forms a Lie algebra. According to the Baker-Hausdorff formula, this implies that the transformation ZtM = e K ~ eKzZM
may be represented in the form (1.1 1.4). Another consequence is that, if K and I/ are real supervector fields, the operator x,
1
(1.1 1.10)
is a real supervector field. For every supervector field K , we define the first-order differential operator k by the rule:
R E ~ " = (8- I )~E % I $ +(~ K ~I ) " " ( ~ , K ~ ) where bv
(1.1 1.1 1)
gM is defined to be related to the right partial derivative a / a z M in zM t
t
(1.11.12a) c
The K acts on a superfunction @(z) as follows
Lemmu I . (1 *e' )(e'@) = (@.e')
where @(z) is an arbitrary superfunction.
Proof. Define the superfunction @(z,T ) ~ ( z7,)= ~ ( zer') depending on a real parameter
T . @(T)
~ ( z0), = @(z) satisfies the equation
d@(T)/dT= @(T)R. Define another superfunction
"(2, T )
~ ( z? ),= ( I .er')(e%(z)) Copyright © 1998 IOP Publishing Ltd
by the rule ~ ( zo)=@(z). ,
(1.1 1.13)
1 10
Iiicus unci Methods q f Supersjlmmetry and Superyracity
We have
= [ K ( I *e")+(-
I ) " ~ ( ~ ~ K M ) *e")](e"~,)+(~ (I -e")K(e'KQ)
= K Y ( T ) + ( - l)"Z'(~MKM)y(r)=y(r)R.
) the same equation and the same boundary Therefore, @(r) and Y ( Tsatisfy condition, hence they coincide. Leriinza 2.
(1-ek)-I=eK(I-e-k),
Proof. Choose
Q, = ( 1 *e- ')
(1.11.14)
in equation (1.1 1.1 3).
Lemma 3.
[E,P]=
-[ K ,VI= c _
t_
- LKV
(1.11.15)
where K and V are arbitrary supervector fields. Proof.
[ R , V ] = KNZNNMZM - V"ZNKMZM =( -
~ ) E T P ~- ( K~ ~ ~~ , ~V ~Z) &~
- (- l)EME** V N K M a N a M + ("KM)& = V N K M { 8 ~ $ , -( 1 ) E M E N 8~ 8{ ~( K}V')-(
v K M ) }=8-~( L K V ) ~ ~ M .
Here we have used the fact that the derivatives 8, satisfy the same identity (1.10.19b) which was fulfiled by the derivatives 8 , . The equation (1.1 1.15) shows that the set of all operators R, where K is a real supervector field, forms a Lie algebra, and the correspondence K + R is an anti-isomorphism of the Lie algebras. Lemma 4.
kKl/e-K=e-E Vek.
(1.11.16)
Proof. The statement follows from equations (1.11.10) and (1.1 1.15).
Now we are able to prove an important result. I .I 1.3. Theorem
zfM=eWKzM Copyright © 1998 IOP Publishing Ltd
Ber(i3,z")
=( 1.e- '),
(1.11.17)
11 1
Mathematical Background
Proof. Introduce the partial derivatives ah corresponding to the variables :,z'
(1.1 1.18)
In accordance with equations (1.10.1 9a,b), the operators the equations
a,:
should satisfy
dhz" = 6,N 1pE.a'N 8' M --0.
aha;-(-
The solution is unique and has the form
,a;
= e - K a, eK.
(1.11.19)
On the other hand, in terms of the supervector fields G, (1.1 1.18), we have l)"""a;dh=G,RaRG,QdQ-(-
o=aha;-(-
1)""""GNRaRGMQdQ
=(c,GNQ)aQ +( - 1)"8@\+ & Q ) G , ~ G . ~dRdQ-( Q - l)"MEN(GNG,Q)dQ -(-
1 ) ~ v y -~ ) M c M + + F ) ) G ~ R G , Q dRaQ
= ((G,G~Q) -(-
1)"w8v(GG,G,Q)}aQ
+(- l ) E ~ ( E+E~)GNRG,Q{aQ~R-(h l)""~dRdQ} = ((c,cNQ)-(
- l)"""(GNG,Q)}dQ
= {(cMcNQ)-( - I ) " ~ ~ ~ ( G ~ G , Q ) } ('lQRGR. c-
Therefore,
((G,GNQ) -( - 1)EMgN( GNG,')}
(G - I)QR= 0.
In this relation, we take the supertrace over the indices N and R (contract both sides with (- 1)""sRN). This gives str{(G,G)G-') =(
- l)Eh('
(dNGMN).
Evidently, the left-hand side coincides with G,(ln(Ber G)}.We recognize the right-hand side as (l*G,). So, we obtain G,(ln(Ber G)} = (1 *eM).
(1.11.20)
c
But (1. G M )may be transformed further making use of equations (1.1 1.19) and (1.1 1.16): (1
.e,)=(~.e-Ka,eK)=(l *---
-e'
a,
e- ')=(d,(l
*e')) e-'.
Now apply equation (1.1 1.13) (1
.eM)=(1 .e- ')e-
Copyright © 1998 IOP Publishing Ltd
a,(
1 .e- ')
= (1 * e -')G,(e-K(
1 *e')).
1 12
Idem and Methods of Supersymmetry und Superyrmity
All that remains is to apply equation (1.11.14). This gives (1
*eM) = - GM In( 1 *e-
-
K).
Comparing with (1.1 I .20) gives BerG=C(l.e-')-' where C is a universal constant which can be calculated by considering some special transformation. In the case of the identity transformation, Ber G = 1 and K = 0, so C = 1. Finally, since Ber G = Ber - '(d,~"), one obtains (1.11.17).
Remark. Making use of equations (1.1 1.15) and (1.1 1.17), one can prove the following statement: Z~M=e-Ke - L Z M => Ber(8,z") = (1 e-K e-L)=( 1 .e- e- '). (1.1 1.21) c _ _
1.11.4. The transformation law for the volume element on RpIq Let us treat the general coordinate transformation as a replacement of variables in an integral over RPlq
dP+qzf@(z') of some superfunction under reasonable boundary conditions. The replacement (1.1 1.4) changes the integration measure leading to the appearance of the Jacobian J ( z )
Jd r
r
P+q
'@(z') = dP+"z J(z)@(z'(z)).
(1.11.22)
J
The technique developed above makes it possible to determine the Jacobian of the transformation.
Theorem. Z~M =e
- KZM
dP+qz Ber(a,z")@(z'(z)).
(1.1 1.23)
Proof. Recall that @(e-%)=e-%(z). So, the right-hand side of equation (1.11.22) is
s
d p f q zJ ( z ) e-%@)=
s
dP+qze-K[(eKJ(z))@(z)].
We insert into the integral the decomposition of unity
-
-
-
1 = ( I .e-K)(1 ae-KI-1 = ( I .e-K)e-KeK(l Se-Kl-1. Copyright © 1998 IOP Publishing Ltd
L
(1.1 1.24)
Muthemuticul Background
1 13
Then, ( 1 , 1 1.24) takes the form
= jdptqz[Q(z) eK.(J(z)( 1 -e-
‘)-‘)I
e-
(1.11.25)
-
where we haveusedequation(1.11.13). B y v i r t u e 0 f ( l . l l . l 2 ) , e - ~ = l + ..., where dots mean terms giving rise to total derivatives in the integral ( 1 . 1 1.25). Therefore, equation ( I . 1 1.25) can be equivalently rewriten as ]dP+qz ~ ( z ) e “ i J ( z ) ( l . e - i i ) - l l . This integral should coincide with the left-hand side of expression (1.1 1.22). Since Q(z) is a completely arbitrary superfunction, one demands that e“(J(z)(1 .e- ‘1-
1
=1
~ ( z=)( 1 *e-1‘.
The equation ( 1 . 1 1.17) then leads to equation ( 1 . 1 1.23). Remark. The Jacobian in ( 1 . 1 1.23) presents the Berezinian of the supermatrix
which is the supertranspose of L
Here we have used the identity
which generalizes equation (1.10.14b). Because of the relation Ber F = Ber (F”) and equation ( 1 . 1 1.17), we can write the transformation Jacobian in three equivalent forms: ( 1 . 1 1.26)
1.11.5 Basic properties of integration theory over W 1 2 q We would like to finish this section by listing the properties of integration theory over [wPl24, which are assumed to be valid (properly generalized) for functional integration in quantum field theory. A chief reason why we choose Copyright © 1998 IOP Publishing Ltd
the number of a-type coordinates to be even here is the existence of non-singular supersymmetric bi-linear forms. Explicitly, consider a real supersymmetric bi-linear form on R P l 4 (see subsection 1.9.6) s2
(1.1 1.27)
= p'lMn"N
where 'laf,v = ( - 1 y:u + t : \ + <:\I<:\
(1.11.28) VNM
('lWN)*
= VNM*
The requirement for non-singularity ' l M N Z N = 0 e= ZN = 0
(1.11.29)
implies that the bodies of the even-even and odd-odd blocks of q are invertible matrices. Since the odd-odd block is antisymmetric, owing to equation (1.11.28), its body can be invertible only if q is even. Note, every bi-linear form (1.11.27) under equations (1.1 1.28,29) can be taken in the role of a supermetric on the superspace. Integration over RP12Y is characterized by the following properties: 1. Linearity
J^
+
d P + 2 4 ~ { ~ f l ( ~ )Pf2(z)}
=2
s
s
dP+2qzfl(z)+ /l dPfZqzf2(z) (1.11.30)
where f l ( z ) and f 2 ( z ) are arbitrary functions on RPI2q, a and p are arbitrary supernumbers. The measure dP+'¶z is an even object commuting with all supernumbers. 2 . Inregration bji parts (three equivalent forms) (1.1 1.31~1)
1
s
d p + 2 4 z f (+~ 4)
=
s
d ~ + ~ q z f ( z ) d M t N= o
(1.11.31b)
J
dP+2qz(dyf1(z))f2(z) = - ( - 1)E,"'(f1) dP+2qzf1(~)(8Mf2(~)). (1.11.31~)
3. Replacement of variables Making a replacement of variables in integrals over by the appearance of the transformation Jacobian: dp f 2 4 z ' f ( z ! )
-s
Z' = z' Z
( )
W12q
is accompanied
dPf2qzJ(z)f(z'(z))
Copyright © 1998 IOP Publishing Ltd
(1.11.32)
Mathematicul Background
4. E.\-istiJnccof deltu ,function There exists the unique delti function 6 p + 2 q ( z ) on pro pert y
[WP1'4
possessing the
{dp+2qzdp+2q(z) f (z) =.f(O) f being an arbitrary function on
RP''4.
form
bP+2q(z)
=N
s
1 15
(1.1 1.33)
The dpf2'Q) can be expressed in the dP+2qWel(fJwZ"
(1.1 1.34)
where N is a normalization constant and the integral is over Rp12y parametrized by real bosonic and imaginary fermionic coordinates, (oj.w)* =
( - I)' " W M .
Rernark. A formal proof of equation (1.1 1.34) is as follows. Consider a superanalytic function f(z) on [wPI'~, x
and integrate it over
R"'q
with the weight (1.1 1.34):
J
hence
In the above, validity of integration by parts has been assumed. 5. Gaussian integruls Choose a fixed supermatrix q N M under the requirements (1.1 1.28) and (1.1 1.29).For every supermatrix H M Nunder the same requirements, we have N'
s
[:
dP+2qzexp -zMH,,zN
Copyright © 1998 IOP Publishing Ltd
1
= Ber-'((~-')MPHp,)
(1.11.35)
where N ' is the universal normalization constant defined by ( 1.11-36)
Remark. Relation ( I . I I .35)generalizes the well-known integral formula (1.1 .37) i being a non-zero real number, and its a-number analogue
,381 where O1 and O2 are real a-number coordinates on R:. To prove (1.1 1.35), let us represent H,, in the manner = (FsTT)MKVKLFLN
H,~N
(1.11.39)
for some supermatrix F E G&+'(p, 2ql Am).Changing the integration variables in (1.1 1.35) according to zM -+ zIM = FM,zN
and using equation (1.11.36), one obtains (1.11.40) On the other hand, equation (1.11.39) can be rewritten in the form
R M ,= (
~ - ' ) ~ ~ = ~ (p q, - vl ) M P ( ~ S T ) p K q K L ~ L N
or, in indexless notation, =q
- l ~ s T T ? ~
hence Ber f7 = Ber2(F). This relation and equation (1.1 1.40) leads to equation (1.1 1.35).
Copyright © 1998 IOP Publishing Ltd
2 Supersymmetry and Superspace Within this restless, hurried, modern world We took our hearts' full pleasure - You and I, And now the white sails of our ship are furled, And spent the lading of our argosy. Oscar Wilde: M J , Voice
2.0. Introduction: from
IwpI4
to supersymmetry
We have spent a lot of time studying supernumber spaces Wi4.It was beautiful mathematics; and now let us reveal its relation to physics. A set of dynamical fields { q'}, arising in some classical field theory, should be treated as mappings from Minkowski space into a superspace q'(x): R4 + Rip14
(2.0.1)
where p is the number of bosonic fields and q is the number of fermionic fields. The situation is quite clear in the fermionic case. In quantum theory half-integer spin particles are described by Hermitian operators $(x) on an appropriate Fock space which obey some anticommutation relations {@'(x), +'(Y)) = om
with h being the Planck constant. Taking the point of view that a classical theory corresponds to the limit h + 0, classical fermionic fields are anticommuting. Anticipating fermionic fields to be valued in the odd part of a finite-dimensional Grassmann algebra AN, one finds that all combinations of the form ( y l A lv2)(q3A2y4)
' ' ' (q2k
- 1A k v 2 k )
117 Copyright © 1998 IOP Publishing Ltd
vanish for sufficiently large k; here A , , A,, ..., A, are arbitrary operators constructed from the ;l-matrices and space-time partial derivatives. Therefore, we have to look on fermionic fields as a-number-valued functions on Minkowski space. For example, the undotted spinor field @,(x)is a mapping The Majorana spinor field Y, is a mapping The Dirac spinor field Y(s) is a mapping Y(x): R4 -+
a=:
2 RB,s.
Classical bosonic fields are usually considered as real number-valued functions on Minkowski space. This description is correct only in the case of a purely bosonic theory. However, having a system of interacting bosonic and fermionic fields, we must inevitably think of bosonic fields as c-numbervalued functions on space-time. For example, in electrodynamics
the vector field equation of motion
PF,,
=
-eTy,,,\Y
tells us that the body of Am(x) has free Maxwell dynamics and the soul of A , ( s ) is propagated by the charged bodiless current j&) = eVy,,,Y. As a result, ordinary electromagnetic radiation is absent in this classical picture; radiation is a purely quantum effect (of course, one may generate non-trivial dynamics by switching on some external source). This may look rather unusual from an ordinary point of view. Nevertheless, the treatment of classical fields as mappings of the form (2.0.1) is most convenient when understanding the classical theory as a starting point for quantum field theory. According to B. De Witt, it would be more precise to speak about a 'superclassical' starting point. Now let us return once more to equation (2.0.1). Note that every set {cp'(x)) of smooth fields on Minkowski space can be continued uniquely to a set {$(x)) of supersmooth fields on the c-number space R.: The resultant mapping [(y:R4 + [Wplq = RBPc x R41 al (2.0.2) looks more symmetrical than equation (2.0.1). Of course, Minkowski space should be identified with the soulless surface in R,: R4 =
{(Xm)€
R:, x; = 0, m = 0, 1,2,3;.
After mapping (2.0.2), it remains to perform one more step Copyright © 1998 IOP Publishing Ltd
-
to consider
1 19
Supersymmetry and Superspace supersmooth mappings of the form
ai:~ 4 i -k, [WPIY.
(2.0.3)
Can such generalized fields, which it is natural to call 'superfields', be interesting from the physical point of view? To answer this question, let us consider one particular case k = 4. We parametrize R4I4 by four real c-number variables x", two complex a-number variables 0" and their complex conjugate (ea)* I!?'. Consider a superfield (2.0.4)
T/: R414-, R,.
This can be written explicitly as
v ( ~e, 8) , = A(X) +
+
+ POzi.l(x)+ e28,p(x)
+ e2F(x)+ BzF(x) + + d2PG(x)
(4.0.5)
where A , F , F, V , and G are bosonic fields and t+hz,$', Aa and I' are fermionic fields. Interpreting odd coordinates 8" and 71" on R4I4 as two-component Weyl spinors makes it possible to define an action of the Poincare group on R4I4 as follows: xm-, X" = (exp K)",,xfl
[ (i
8%-, 0; = exp
+ b"
Knb~,b)]a'Op
(2.0.6)
Finally, demanding the Poincare transformation law
vpi, e', F) = v ( x ,48,
one finds that A , F ,
(2.0.7)
F and G are scalar fields, V , is a vector field, and
are Majorana spinor fields. Therefore, every superfield of the form (2.0.4) is equivalent to a whole family of ordinary bosonic and fermionic fields. The above example can be further generalized. Namely, every superspace R414N, N = 1,2,. . ., admits a natural action on it of the Poincark group. So in these cases as well, superfields (2.0.3) are equivalent to a whole set of ordinary fields. We are faced with a very interesting situation. We have found a mechanism to unify bosonic and fermionic fields in a multiplet. But our consideration seems rather formal. It would be seen as less formal if an underlying principle leading to such unification existed. Remarkably, the key Copyright © 1998 IOP Publishing Ltd
to the problem lies in the above example. The equation (2.0.5) tells us that V ( s ,0, g) contains an equal number of bosonic and fermionic fields. This clearly applies in the general case of superfields on R414N,N = 1 , 2 , .. . . This observation becomes a simple physical consequence under supposition of the existence of some symmetry between bosons and fermions. Indeed, imagine some hypothetical field theory possesses an invariance 'rotating' bosonic fields into fermionic ones and vice versa. Then the number of bosonic and fermionic partners should, in general, be equal. At first sight, such a symmetry may look very strange and unusual, since it changes particle statistics (the reason it is called a 'supersymmetry'). On the other hand, if God created two types of particles, He could also create a mechanism to transform one type into another, couldn't He? Namely supersymmetry is able to serve as the desirable underlying principle. One can easily invent supersymmetry transformations having made some reasonable assumptions. To this end recall that the Poincare group acts on [w4" by means of the linear transformations (2.0.6).They include the Lorentz rotations of superspace coordinates and the xm-translations (since the set of translations forms a commutative group, it is not possible to supplement space-time shifts 6x" = b" by Iinear transformations of 8, 8 of the type: 60, = k-hJ?, 68, = I;.b,&', where K is a complex constant; the only possible choice is k- = 0). However, while the even and odd coordinates of R4I4 are on an equal footing, @translations are not present in equation (2.0.6). To overcome this discrepancy, it would be natural to consider in superspace not only the Poincare transformations but also shifts in 8, 8 supplemented by some transformations of x". Assuming linearity, the most general form for such a displacement, consistent with the needed Lorentz transformation laws for .xm and e", 8&,reads
66" = E*
SOb = Ti
where E" is an arbitrary undotted spinor, and k is a complex parameter. The case k # 0 is most interesting, since it provides us with non-trivial mixing between even and odd superspace coordinates. Without loss of generality, it is possible to fix k as k = i obtaining the supersymmetry transformation
(2.0.8) Finally, if one adapts the transformation law (2.0.7) to supersymmetry, this gives
e,@= qX, e, 8) - qX,o , ~ ) = i(come- 80" E)a, ~ ( x6,P) , - (~"2,+ ?&dah) V ( X ,8,g)
~ ( x ,
Copyright © 1998 IOP Publishing Ltd
Supersymmetry und Superspuce
121
or, in terms of the fields from which V(x, 8,8) is constructed, 6 A ( x ) = -E$(x) - @(Xj S@,(x) = - 2E,F(x) - (a?),{ V,(X) 6Q.w) = -G.(X)
SV,(x) = EoJ(xj
+ d,A(x)j
1 + -a,$(x)a"c 2
+ %(xja,E + i~oh8,d,$(x)- id,&x)8,abG
i 6i,,(x) = - 2~,G(x) + -(a"8bE),d,v,(x) 2
- i(a'E),d,F(x)
i SG(x) = -a,{j,(x)o"E - Eo"I(x)). 2 As a result, we have constructed a symmetry mixing bosonic and fermionic fields. Note that if some component field from V(x,8,8) was zero at the beginning, it becomes non-vanishing after making a supersymmetry transformation. The set of Poincare transformations (2.0.6) and supersymmetry transformations (2.0.8) form a group. But it is a rather unusual group, since the supersymmetry transformations are built of anticommuting parameters E' and .? ! Before studying supersymmetry it would be instructive to familiarize the reader with such algebraic objects.
2.1. Superalgebras, Grassmann shells and super Lie groups Superalgebras present a natural generalization of the concept of Lie algebras. Every Lie algebra 9 is a non-associative algebra in which the multiplication law [. . .,. . .]: B x 9 -+ 9 is characterized by two basic properties: 1. CU.61 = - [ h a ]
(2.1.1)
2. [a, Cb, cl1 + Cb, cc, a11 + CG Ca, 613 = 0
(2.1.2)
for all elements, a, 6, c E B. The bi-linear operation [.. . , . . .] is usually called a Lie bracket. There is a universal way to construct Lie algebras. Namely, given an associative algebra 42, one can introduce a Lie algebra structure on '11 by taking a commutator of elements: [x,y]
= x-y-y - x
Vx,yE'#.
(2.1.3)
Here the Jacobi identities (2.1.2) are evidently satisfied. In particular, if JU is some matrix algebra, then the prescription (2.1.3) provides CIS with a matrix Copyright © 1998 IOP Publishing Ltd
122
Idens und Methods of Supersvmmetry und Supergrmity
Lie algebra. In fact, every finite-dimensional Lie algebra proves to be isomorphic to some matrix Lie algebra. So, we may look at Lie algebra as a set of linear operators closed under Commutation. Another algebraic operation, often used in applications, is anticommutation of operators. For example, to formulate particle dynamics in quantum field theory, one has to choose commutation relations for integer spin particles and anticommutation relations for half-integer spin particles. Naturally, the question arises: is it possible to generalize the Lie algebra multiplication law to include commutators and anticommutators as well? 2. I . I . Superalgebras To define superalgebras, we will need one more auxiliary notion. A vector space (real or complex) L is said to be '72,-graded' if it is decomposed into a direct sum of two subspaces OL and ' L , L= ' L O ' L . It is useful to call OL-elements and 'L-elements 'even' and 'odd vectors', respectively. Even and odd vectors in L are also called 'pure'. By analogy with the Grassmann ) on the set of pure algebra A x , we introduce a 'parity function' ~ ( u defined vectors in L as follows: K(U) =
ifuEoL ifuE'L.
0 1
(2.1.4)
Every vector U E L may be decomposed uniquely into the sum of its even and odd parts, U = OU
+ 'U.
Using the parity function makes it possible to define a one-to-one linear mapping 8:L+ L by the rule: P(U)
= ( - 1)"(U)U
9 2
= 0,
(2.1.5)
for every pure vector U E L . We will refer to P as the 'L-parity mapping'. A superalgebra is a Z2-graded linear space Y = 'Y 0 'Y and an algebra with respect to a multiplication [. . ., . . .}: Y x Y + Y which is characterized by three basic properties: (2.1.6) (2.1.7) 3. Super-Jacobi identities ( - 1)K(U)K(C)[ U , [b, c}} ( - l)~(h'K'"'[b, [c, U})
+
+ (-
l)K'c'"h'[~,[ U , b } } = 0.
(2.1.8) Here U, b and c are arbitrary pure elements from Y.The bilinear operation [. . . , . . .} is called a 'Lie superbracket'. Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
123
Before considering properties of superalgebras, it is instructive to describe a universal rule by which they may be constructed. Let % be a Z,-graded associative algebra, i.e. (1) JU is an associative algebra; (2) J2 is a Z,-graded vector space, 4 = OJL! 0 'JU;(3) the +?-parity mapping 9 is an automorphism of the algebra, ,Y(x y ) = Y(X)9 ( y ) . Then 4Y becomes a superalgebra with respect to the multiplication law a
[,Y,
y'l
E
x.y
- (-
l ) h ( Y ) h ( 5 .
x,
(2.1.9)
where .Y and y are arbitrary pure elements. The super-Jacobi identities (2.1.8) turn out to be satisfied identically. In particular, having some Z,-graded matrix algebra, the prescription (2.1.9) will provide us with a matrix superalgebra. Examples are given in the next subsection. So, one can imagine a superalgebra as a set of linear operators closed under (anti)commutation. From here on our consideration will be confined to the case of superalgebras obtained in accordance with the prescription (2.1.9).So, we have
bE'9
[U,
b } = - [b, U } = [ U , b]
U,
[U,
b ) = - [b, a ) = [a, b]
a € 0 9 ,b E ' 9
[a, b } =
(2.1* 1 0)
a, b E '9.
[b, a } = {a, b )
For arbitrary elements a, b E 9, we have [a, b ) = p a , Ob]
+ [Oa, ' b ] + ['a, Ob] + { ' a , %).
Now let us analyse the axioms. Equation (2.1.7) tells us that (2.1.11 a) (2.1.11 b) {19,
' 3 ) c OY.
(2.1.11c)
Therefore, ' 9 is a Lie algebra. Equation (2.1.11b) shows that with every element O U E ' 9 we can associate a linear operator ~ ( O U ) : ' 9 4 ' 9 on ' 9 defined by
B(0a)'c
[Oa, ' c l
V'CE
'9.
(2.1.12)
Choosing in relation (2.1.8) a = 'a, b = Ob and c = ' c leads to
F(Oa)F(ob)'c- 9 ( % ) 9 ( o a ) ' c = 9 ( [ 0 a , Ob])'c.
(2.1.13)
As a result, the set of all operators F('a) forms a representation of the Lie algebra ' 9 in the linear space '9. The other superJacobi identities (2.1.8) give some restrictions on the possible type of this representation. To reveal them, we anticipate the superalgebra 9 to be finite dimensional, and let {ei) be a basis in '9, i = 1 , 2 , . . , , dim '9, and {e,) be a basis in '9, r = 1,2,. . . , dim "3. Then, the set e, = { e , e,> is a 'pure basis' in 9. Every pure basis in 9 is said to be a 'set of generators' of the superalgebra. Copyright © 1998 IOP Publishing Ltd
124
Ideas and Methods of Supersymmetry and Superyravitj,
Introduce 'structure constants' of the superalgebra defined by f l J L E C(or R).
le,, =.fI J L e L
(2.1.14)
They are (anti)symmetric in their lower indices, fJ,L
= -(-
(2.1.15)
1)h'IK.lfIJL
in agreement with equation (2,1,6), where we have denoted K! rc(e,). By virtue of equations (2.1.1 l), the only non-vanishing components are f IJ..' = -,f
,
.fix''
J,I, l
=
f
-.fxi"
r,/
= .f pri.
(2.1.16)
The super-Jacobi identifies lead to the bilinear equations in the structure constants: ( - l ) h ' J X ' If J K f L K M
+ (-
l)h.'KifJ1dKf,KM + (-
1)'IKJ
f
LIKfJKM
=0
(2.1.17)
From these equations, those essential for us are the following
;.,;
.f ..'f I //
-ji17
f,//fi;(5
+ .f ,$.ifi;$ + f ; , ; f i l ;
I/
f
-f
ip;' f I;
= 0.
(2.1.18)
= 0.
(2.1. 1 9)
What do these identities mean? First, note the generators 9 ( e i ) of the '%-representation (2.1.12) are given by matrices [9(ei)Ixs = f i l T . Secondly, the generators ad(e,) of the '%adjoint representation are given by matrices [ad(ei)]'j = f i i ' . So, equation (2.1.18) means nothing more than that f , ; is an invariant tensor of the Lie algebra ' 9 transforming in the adjoint representation with respect to the index i and in the representation being contragredient to 9 with respect to each of the indices a and j.What is more, not every invariant tensor with such a structure may be chosen, only that which satisfies equation (2.1.19). The above consideration makes it clear that not every Lie algebra admits an extension to a superalgebra; admissible Lie algebra should have an invariant tensor f = f & subject to constraint (2.1.19). 2. I .2. Esumples of superalgebras Let us consider the set of all ( p + q ) x ( p q ) complex matrices. It is a Z,-graded associative algebra with respect to matrix multiplication and the grading defined as follows. For every matrix
+
(2.1.20) where A, B, C and D are p x p , p x q, q x p and q x q complex matrices, respectively, the decomposition into its even and odd parts reads R = OR A Copyright © 1998 IOP Publishing Ltd
+ 'R 0
O B
(2.1.21)
Supersymmetry and Superspace
125
This algebra is denoted usually by Ma@, q / C ) .In accordance with (2.1.9), Mat@, q /@) becomes a superalgebra with respect to the Lie superbracket [ R , S } = ['R,OS]
+ [OR, 'SI + ['R,OSJ + { ' R , ' S } .
The resultant superalgebra is denoted by gl(p, q /C).Restriction to the case of real ( p + q) x ( p + q) matrices gives the real superalgebra gZ(p, q )R). By analogy with supermatrices, one can define in Mat(p, q1C) the operation of supertrace by the rule: str R = tr A - tr D. It is a simple exercise to check that str ( [ R , SI) = 0 for arbitrary matrices R , S ~ M a t ( p , q J @ )As . a result, the set of all supertraceless matrices, str R = 0
(2.1.22)
forms a superalgebra, denoted by sl(p, q 1 @), which is a subalgebra in gl(p, 4 1 C). The set ofall real matrices in sl(p, q I C)presents the real superalgebra sl(p, q )R). The last example is the orthosymplectic superalgebra os&, 2q)C).It is a subalgebra of sl(p, 2qIC) selected by the conditions
+ AT = 0
A
JD
+ DTJ = 0
B = iCTJ
(2.1.23)
where J is the 2q x 2q symplectic matrix
"). 0
J = ( -0,
(2.1.24)
Equation (2.1.23)means that A E so(p, @)and D E sp(2q, @), so that 0
W ( P , 29 I a=)
= so(p,
a=) 0 sp(24, a=).
2.1.3. The Grassmann shell of a superalgebra Every complex &-graded vector space L(C)= ' L ( @ 0 ) 'L(C)of dimension ( p + q) can be made into a supervector space L(A,) of dimension ( p , 4) defined as follows. Let e, = { e , e'}, where i = 1,2,.. . , p and o! = 1 , 2 , .. . ,q, be a pure basis in L(@), U
= 0U
+ ' U = xiei + x'e,
= x'e,
X'EC
for every U E L ( Q Consider the set L ( A x )of all formal linear combinations U = ziei
with z', I = 1,2,.. . ,p the operations
+ z'e,
= zIeI
z' E A x
(2.1.25)
+ q, being arbitrary supernumbers. In L ( A , ) we define
Copyright © 1998 IOP Publishing Ltd
Ideas und Methods of' Supersj~mnzetryunci Supergruvitj,
126
1. addition U,
+
U2
= (2:
+ z:)e,
(2.1.26a)
2. lcfr niultiplicarion by supernumbers X'U
= (rz'je,
XEA,
(2.1.26b)
3. ricqhr uiultiplication by supernurnbers U*
x = ( - 1 )h"(")(zIX)er
(2.1.26~)
where 2 is a pure supernumber. As a result. every into its even and odd parts by the rule:
OU
+ 'U = ('e, + <"er
'U
= q'e,
U
UE
L(A,) is decomposed
= OU
('E
+ q"er
e,,
$E@,,
(2.1.27)
4" E E, ~"EC,.
It is readily seen that L(A,) becomes a supervector space of dimension ( p , q), in which the operation of complex conjugation is undefined. Every even (odd) vector in L(@)is, at the same time, an even (odd) supervector in ,!,(A,). Following F. Berezin, the supervector space L(A,) is said to be the 'Grassmann shell' of the Z2-graded vector space L(C) (originally, Berezin used the term 'Grassmann shell' for the set OL(A,) of the even supervectors in L(A,)). From now on, we make no distinction between the parity functions E on L(A,j and li on I,(@), using the same notation E for both. Now let 9(C)be a complex superalgebra with pure basis { e r } . We take the Grassmann shell $(A,) of the Z,-graded vector space $(a=). Then, the operations of multiplication (2.1.10) in 9(C) and multiplication by supernumbers (2.1.26b, c) in %'(A,) induce a multiplication in 9 ( A , ) defined by [U,
v ) = [OU,
Ov]
+
[OU,
'VI
+ ['U, Ov] + {
'U, 1.
(2.1.28)
for any supervectors U, v ~ $ ( i Z , ) . Choosing in (2.1.28) the supervectors U = ('er and v = iJeJ to be pure, E((') = &(U) + and c(iJ)= E(V) + E ~ one , obtains [U, v ) = uv - ( - 1)T(u)f(V)vu = ( - 1)"UH"V) + ~ J ) " J 6 5' Eel, (2. .29) = (-
~ ) i ( u ) ( f (+ v )f i i J I
5 s' f l J L e L
where equation (2.1.14) has been used. The main properties of the multiplication (2.1.28) are: 1.
[U,
v} = - ( - Iy(")r(v)[v,U)
2. E([u,v}) = &(U) + E ( V ) (mod 2) Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace 3. ( -
1)i(UlI(Wl
[U, [v,
w}}
+ (-
for any supervectors
4.U.[
+ PV,W ) = &[U,
l y ( " ) ' q V ,[w,U}}
+ (-
127
lpr(")[w, [U, v}} = 0 (2.1.30)
U,
W}
v, w E %(A,)
+ P[v, W}
Va, P E A%
VU, V, w E Y(AJ
To prove property 2, one has to use equations (2.1.7) and (2.1.29). The properties 1-3 show that %(A,) forms a superalgebra. Property 4 means that the Lie superbracket [...,...} of %(A,) is linear with respect to multiplication by supernumbers. The supervector space ??(A,) with the multiplication (2.1.28)is said to be the 'Grassmann shell of the superalgebra' $(C). We shall also call objects like $(A,) 'Berezin superalgebras' to distinguish them from ordinary superalgebras over C or R, which have been considered in subsection 2.1.1. By definition, a Berezin superalgebra B is a supervector space Y together with an operation of multiplication [. . , ,, , .}: Y x Y + Y of elements from B such that the axioms (2.1.30) are satisfied. When Y has finite dimension, all algebraic information is contained in the 'structure constants' defined by Le,,
=flJLeL
&(LJL) = E,
+ t J+
flJLEAcs EL
(mod 2)
(2.1.31)
with {e,} being a pure basis of 9. A Berezin superalgebra is said to be 'conventional' if it admits a pure basis in which the structure constants are ordinary numbers, i.e. as in equation (2.1.14); otherwise it is called 'unconventional'. Every conventional Berezin superalgebra is the Grassmann shell of some complex superalgebra. From now on we will discuss only conventional Berezin superalgebras. Consider the set 'S(A,) of even supervectors in Y(A,). Property 2 from (2.1.30) tells us that OY(A,) forms a subalgebra in the Berezin superalgebra %(Am).In accordance with equations (2.1.28,29),the multiplication on 'S(A,) is given by [U, VI = iJtlfrJLeL. (2.1.32) for any supervectors U = ('e, and v = tJeJ from '%(A,), it is characterized by the properties
1.
[U, v] =
and due to (2.1.30),
- [v, U]
2. [U, cv, w11 + [v, CY U11 + CY [U, VI1 = 0 3.U.[ + Pv, W] = a[u, W] + P[v,W] Va, p E C,
VU,V, w E '%(A,)
1
(2.1.33) Therefore, we obtain a Lie algebra structure on '%(A,) which is induced by the superalgebra structure on S(C). Of course, the reason for this transmutation lies in the heterotic nature of supernumbers: c-numbers are Copyright © 1998 IOP Publishing Ltd
commuting objects while a-numbers are anticommuting. The algebra OS(A,) is said to be the 'complex super Lie algebra associated to the superalgebra' B(@). More generally, a complex super Lie algebra is defined to be the even part ' 9 of a Berezin superalgebra 9;the multiplication on ' 9 is characterized by the properties (2.1.33). There are three basic differences between ordinary Lie algebras and super Lie algebras. First, Lie algebras are endowed with the operation of multiplication by ordinary numbers only, while for super Lie algebras it is defined for arbitrary c-numbers. Secondly, generators of Lie algebras satisfy commutation relations, while generators of super Lie algebras are subject to (anti)commutation relations like (2.1.31). Finally, generators of a Lie algebra are elements of the algebra. But not all generators of a super Lie algebra belong to the algebra; the even generators belong, whereas the odd generators do not. Let Q be a Berezin superalgebra and ' 9 be the corresponding super Lie algebra. Suppose that the supervector space 9 can be provided by an operation of complex conjugation such that the subset 03Rof real supervectors in ' 9 forms a subalgebra of the super Lie algebra ' 3 . Then OQR is said to be a 'real super Lie algebra'. The operation of multiplication in 'gRrespects the properties 1, 2 (2.1.33),but the third property should be substituted by [UU
+ Pv, W] = U[U, W] + P[v, W]
VZ,
P E R,
VU, V, w E O 3 R . (2.1.34)
It is not difficult to find restrictions on the structure constants of Q (2.1.31) (with respect to a standard basis (e,}, e: = ( - lyre,)to ensure that the subset ' 9 , is subalgebra of ' 9 , The result is ( f I J L ) * = ( - 1 ) h + t i ) ( l + CL) + F l u f I J L . (2.1.35) When the structure constants are ordinary numbers, equation (2.1.35) takes a simpler form ( f i J L ) * = ( - 1 yf, JL.
(2.1.36)
2.1.4. Examples of Berezin superalgebras and super Lie algebras First, let us construct Grassmann shells of the complex superalgebras presented in subsection 2.1.2. We begin with the superalgebra gl(p, yl@). Taking the convention that matrix indices are positioned in the manner
R
=
(RMN)
one easily finds that the Grassmann shell of the Z,-graded matrix space Mat(p, ql C)coincides with the supermatrix space Mat(+'(p,qlh,) studied in Section 1.9. Mat(+'(p,q 112,) becomes the Grassmann shell of the superalgebra yl(p, q 102)with respect to the superbracket defined by [ F , , F,)
=
Copyright © 1998 IOP Publishing Ltd
F , F , - (-
l ~ ( F ~ ) JF~, F (F , ~l
(2.1.37)
where F , and F 2 are arbitrary pure supermatrices. The Berezin superalgebra obtained is denoted by yl'+'(p, L ~ I A I~t )is. clear that the set o M a t ' f ' ( p ,qlA,) of c-type supermatrices in Mat(+)(p, ql A,) (see equation (1.9.37~)) forms the super Lie algebra o c j P + ) ( p , q l A , ) with respect to the ordinary Lie bracket. Finally, recall that we introduced in Mut'+'(p,q(A,) the operation 's*' of supercomplex conjugation (1.9.41),which is characterized by property (1.9.44). Therefore, if F , and F , are real c-type supermatrices (see equation (1.9.42b)), then F , F , and hence [ F , F 2 ] are real. As a result, the set OMatS'(p,qlA,) of real c-type supermatrices forms a subalgebra in Oyl'+'(p, q1 A,,) denoted by O q l k + ' ( p , r ~ ( A , . )The . algebra oylc'(p,qlA,) is a real super Lie algebra. To construct the Grassmann shell of the superalgebra sl(p,qlC), we introduce the operation supertrace in the supermatrix algebra Mat'+'(p, q ( A r , ) . Recall that supertrace has been defined for c-type supermatrices only (see equation (1.10.66)).We generalize this definition to the whole Mat'+'(p,q ( A J C ) demanding two basic postulates: 1. str F = ( - 1
+ (:(F)f:\!FM A45
)
for every pure supermatrix F 2 . str ( F , + F,) = str F ,
+ str F,,
for any supermatrices F , and F,.)
i
(2.1.38)
So, the supertrace of a-type supermatrices coincides with the ordinary trace. The operation of supertrace is characterized by the following properties: 1. str (fF)= z str F ,
1
str (Fi) = (str F ) z ,
2 . str i = z ( p - q), 3. str (F")
=
(str F)",
(2.1.39)
where z and Fare an arbitrary supernumber and supermatrix, respectively; str ( F , F l ) , 4. str ( F 1 F 2 )= ( - ly(Fl)c(F?J for arbitrary pure supermatrices F , and F,. Now we consider in M ~ t ' + ~ (qpl A, x ) the subset sl'+'(p, q / A x )of supertraceless supermatrices, str F = 0
V F ~ s l ( pq1, A,).
(2.1.40)
Property 2 in (2.1.38) and property 1 in (2.1.39) show that sl'''(p,qlA,) is a supervector space. Recalling the operation of multiplication by supernumbers in Mut(+'(p,q ( A , ) (see equation (1.9.31)), one finds sl'')(p, qlAJC)to be the Grassmann shell of the Z,-graded vector space sl(p, q1 e).Due to the property 4 in (2.1.39), sl'+'(p, ql A,) becomes the Berezin superalgebra with respect to the superbracket (2.1.37). The subset of c-type supermatrices in sl'+'(p,q I A x ) forms a super Lie algebra denoted by Osl'+)(p, q l A z ) . Finally, the subset of Copyright © 1998 IOP Publishing Ltd
real cr-type supermatrices in Osl"'(p,q~A,) forms a real super Lie algebra denoted by 'slk+'(p, q / A , ) . The Grassmann shell of the orthosymplectic superalgebra osp(p,2ql@)turns out to coincide with the subset of Mat'+'(p,2qjA,)denoted by osp'+)(p,2qlA,) and consisting of all supermatrices under the equation '7.hlPFP,
+ (FST)MP%JN = 0
(2.1.41a)
+ F"q
(2.1.41b)
or, in index notation, qF
=0
where (2.1.42) J being as in equation (2.1.24). The matrix qMN is a supersymmetric real supermatrix in the sense of the definition (1.9.78). The operation of supertransposition for M a t ( + ) ( p ,q I&) is given by equations (1.9.58) (for M a t ' - ' ( p . qlh,) see equation (1.9.59)). Equation (2.1.41) is evidently consistent. Indeed let us extend the operation of supertransposition (1.9.77), defined originally for C-type supermatrices only, to the case of pure arbitrary supermatrices with lower positioned indices: ( H s T ) , ~= ~ ( - ])c(HHczi + e11 + r \ f + 6 1 + c w c s HNM. (2.1.43)
Then, the identities
(GHyT = ( - l)c(G)c(H)HsTGsT VG E Mat'-'(p, ql Am) imply that the supertranspose of equation (2.1.41) is just the same equation. Using the relations (1.9.60) and (1.9.61), one can readily verify that (1) osp(+)(p,2qlA,) is a supervector space and a Berezin superalgebra with respect to the superbracket (2.1.37);(2) with respect to the conjugation (1.9.42), the real even subset 'ospk+)(p,2qlA,) of osp'+)(p,2qlA,) forms a real super Lie algebra. Our last example, very important for the subsequent discussion, is the 'Berezin superalgebra of supervector fields' on a real superspace RP14 parametrized by p real c-number coordinates x m and q real a-number coordinates Ow. We will now use the notation adopted in Section 1-10; in particular, zM = (xm,P ) and aM = (am,a,) = a/azM.By definition, a 'supervector field' on R P 1 4 is a first-order differential operator, K = K M ( z ) d , = Km(x,@am
+ K"x,
@a,
(2.1.45)
with K M ( z )being supersmooth functions on Rp14. The set of all supervector fields on R P l 4 will be denoted by SVF(p,q). A supervector field K is said to Copyright © 1998 IOP Publishing Ltd
Supersymmerry and Superspace
13 1
be c-type (a-type) if its components K ( z ) are pure superfunctions on RPIq with the following Grassmann parities: C-type u-type (2.1.46) c(K"(z)) = cAw & ( K M ( z )=) 1 + (mod2) Supervector fields of definite type are called pure. Every pure supervector K is provided with the Grassmann parity F ( K )defined by E ( K )=
0 1
forc-type K for a-type K .
(2.1.47)
Note that all supervector fields considered in Section 1.1 1, were of c-type. We define in SVF(p, q ) the operations I . addition
K,
+K,
+ KY)d,
= (KAY
2. left multiplication by supernumbers
rK = (aKM)d, 3. right multiplication by supernumbers Kr
(2.1.48)
= ( - p c \ 1( K M C O Z M
4. complex conjugation K
+
E
RMZ,
= ( - l)c(K)c~~(KM)*aM,
Then, all the supervector space axioms prove to be satisfied, and SVF(p, q ) becomes an infinite-dimensional supervector space. It should be pointed out that, due to operations (2.1.48), real C-type supervector fields, K = K , are characterized by real components, ( K M ) *= K M . So, the above definition of complex conjugation is consistent with the notion of real supervector fields introduced in Section 1.11. Let us consider the set of all supersmooth functions on RP1q, denoted by Cr(R"lY). CcL(RP14)is endowed, in an obvious way, with the structure of an infinite-dimensional supervector space. Every supervector field K defines a left linear operator on C"(RP1q) by the rule: @(z) + ( K @ ) ( z )= K M ( z ) z M @ ( z ) V@(Z)E Cm(Rplq).
(2.1.49)
Every c-type supervector field is a C-type linear operator on Cm(Rp14) since it does not change superfunction types. Analogously, every a-type supervector field is a left a-type linear operator on Ccc(RP14).Using equation (1.10.22), one can prove the identity
(KQ)*= ( -
i)c(K)c(WR@*
(2.1.50)
for arbitrary pure supervector field K and superfunction @. Replacing here @ + @* brings this relation into the general form (1.9.40). Copyright © 1998 IOP Publishing Ltd
I 32
1tIctr.s and Metliorls of' S u p e r s j m n e i r j and Supergravity
Given arbitrary pure supervector fields K = K M 8 , and L= L'a,, define their superbracket by the rule [ K , , f j = K L - ( - 1 r(Kk(L)L-K e
we (2.1.51)
which, due to equations (1.10.19, 20) and (2.1.46), is also a pure supervector field, [ K , L ) = ((KL") - ( - 1)"(K)E(L)(LKM))8M.
(2.1.52)
The superbracket (2.1.51) proves to satisfy all the axioms (2.1.30)for a Berezin superalgebra. As a result, SVF(p,4) obtains the structure of a Berezin superalgebra. Then, the subset 'SVF(p, q ) of C-type supervector fields in SVF(p, 4 ) forms a super Lie algebra. Finally, since any pure supervector fields K and L satisfy the identity [ K , L ) = ( - l)"(K)c('L)[L, K)
(2.1.53)
the subset 'SVF,(p, q ) of real C-type supervector fields 'SW(p, q) forms a real super Lie algebra. This super Lie algebra has been studied in Section 1.11. 2.1.5. Representations of (Berezinj superalgebras and super Lie algebras For our discussion of representation theory it seems convenient to start with Berezin superalgebra representations. Let %(A,) be a Berezin superalgebra and Y ( A J be a supervector space. A mapping T : B(A,) --t E n d ( + ) 9 ( A mfrom ) 9?(Am)into the algebra of left linear operators on Y ( A a )is said to be a 'linear representation of the Berezin superalgebra' in Y ( A x )under fulfilment of the conditions: 1. &(T(U))= &(U)
for every pure U E %(A=) 2. T(3u + pu) = stT(u)
+ PT(v)
1
for Vst, P E A, and Vu, v E %(Ax) 3.
(2.1.54)
N u , V I ) = [T(u), T W j for any pure elements U, v E %(A=).
End"'Y(A,) can be provided with a Berezin superalgebra structure with respect to the superbracket: (2.1.55) where F 1and T2are arbitrary left linear operators on Y ( A = ) Thus, . every linear representation T : %(Am)+ E n d ( + ) Y ( A %is) a homomorphism of the Berezin superalgebras. In accordance with the first requirement (2.1.54), every even (odd) element , of %(Ax) maps to a C-type (a-type) linear operator on Y ( A % )respectively. Copyright © 1998 IOP Publishing Ltd
Supersymmetrji und Superspace
1 33
As a result, the q(A,)-representation in Y ( A = )induces a 'linear representation , : 'S(A,) -,'End 2'(Ax), which of the super Lie algebra' '%(A,) in Y ( A x ) T is characterized by the properties:
I
I . T(xu + p v ) = r T ( u ) + PT(V) Vci,
P E Ccand Vu, v E '%(A=)
2. T(Cu,VI)= CT(u),T (dl Vu, V E ' 9 ( A X . ) .
(2.1.56)
For every U E ' Y ( A ~ ) ,the operator T(u) maps the subset '2'(AZ) of even supervectors in Y ( A=) into itself and the subset '2'(A,) of odd supervectors in IP(A,) into itself also. So, while the representation of the Berezin superalgebra 9 ( A x ) acts on the whole supervector space LY(A=), the corresponding representation of the super Lie algebra 'S(A,) acts independently in each of the subspaces '2&) and '2'(A,). This is very important for the following reasons. First, suppose that 2'(A,) has a finite dimension ( p , 4). Then, the subspace 'L?(AJ can be identified with the complex superspace CPlq and the subspace '2'(Am)can be identified with the superspace (see subsection 1.9.2). Therefore, every finite-dimensional representation of the super Lie algebra '%(Am) leads to linear (supermatrix) representations in superspace and superspace is the main subject of our book. Secondly, suppose we have a linear representation T of $(Am) in the supervector space Ccc([wP1q)of supersmooth functions on W I 4 . Then it induces representations of the super Lie algebra '$(A,) in the space 0C"(Rp14)of bosonic fields on RPIq and in the space 'C"(RPIq) of fermionic fields on RPl4. That is why super Lie algebras are of great importance. When 9(Az,) is endowed with an operation of complex conjugation such that the subset 'gR(AZ) of real even elements in g(A,) forms a real super Lie algebra, then every linear representation T : $(A,) -,End 2'(A,) induces a linear representation of the real super Lie algebra 'gR(A,) in 2'(AZ), T : o%R(Az)-,'End 2'(Am),for which the first requirement of (2.1.56) should be substituted by
+
T(EU+ Pv) = NT(u) / ~ T ( v ) , V X , ~ ER,
VU,VE'~R(A,). (2.1.57)
Now we proceed to consideration of superalgebra representations. For the time being, we restrict ourselves to the case of finite-dimensional representations. Let 9 ( C )be a complex superalgebra. A linear mapping T: $(e)-,Mat@, q I C) from $(C)into a 72,-graded matrix algebra is said to be a 'finite-dimensional representation of the superalgebra' $(e) under fulfilment of the following conditions 1. h-(T(a))= x(a)for every pure a E %(C) 2. T ( [ a ,b ) ) = [T(a),T ( b ) }for any pure a, b E %(a=). Copyright © 1998 IOP Publishing Ltd
(2.1.58)
134
l t i ~ w vunci Methods of' Supcrsymmelrj. und Supergrmity
The first requirement means that, for every UE%(C),we have
T ( a )= T(0a)+ T ( ' a )
Therefore, a representation of Y(C) gives us a homomorphism T : a(@) -+ g / ( p , 41 @) of the superalgebras. I t is convenient to identify Mat(p, SIC) with the algebra End 9(@) of all linear operators acting in a Z,-graded vector space Y ( C )of dimension ( p + q ) by the standard rule: every linear operator 92 on U(@) is represented with respect to a given pure basis E , = {E,,,, E,} in 9 ( C ) , m = 1,2,.. . , p and ,u = 1,2, . . . , 4, by the matrix REM@, SIC) defined as follows .#(E,)
= E,R",.
Then the operator .# acts in 9(@) as
X
= x"E,
X ' = .xlME,
-+
a ( x )= X' xIM = R",xN
for every X E 9 ( C ) .After this identification, End 9 ( C )inherits the structure of a Z,-graded associative algebra and becomes a superalgebra with respect to the superbracket (2.1.9). So, every representation T : %(@) -+ Mat(p, ql @) may be understood as a representation T : B(@) -,End U(@) in a Z,-graded vector space. Now we show that any linear representation T of some finite-dimensional superalgebra B(@)in a finite-dimensional &-graded vector space U(@) induces a linear representation 7 of the Grassmann shell %(A,) of the superalgebra 9(C)on the Grassmann shell Y(Am)of the Z,-graded vector space Y(@). First, we construct a one-to-one linear mapping^: End U(@) + End'+'9(A,), preserving the grading and the multiplication. Elements of the supervector space 9 ( A x )will be denoted by an arrow, and they will be written in the form (1.9.274 placing basis supervectors on the left. Then, given a linear operator 9 on U(@), it induces the left linear operator 9 on U ( A , ) defined as follows
2' = EMXtM X',
= RMNXN
for every 2 E U(Am).Hence, the operators G, @ and 9 are characterized by the same (super)matrix R with respect to the basis ( E M }in LY(Amland the basis {E,) in Y ( @If)W . is a pure linear operator on U(Q then 9? is a pure left linear operator on U(A,) of the same type, K(@) = E(&). One can readily = 91&32, V C g l ,92, E End U(@). So, in accordance with check also that
.JO2
Copyright © 1998 IOP Publishing Ltd
Supersjw"try
und Superspuce
135
(2.1.58),we have
"(,
b ) ) = [$(a),
m)}
(2.1.60)
for any pure elements a, be%(@). As the next step, we construct a linear mapping 7:9 ( A m + ) End'+'Y(A,) as follows. Let { e , ) be a pure basis for %(e); every element a E g(C)has the form a = y'e, = y'e,
+ yzex
~'EC
and every element a E $(A,) has the form a = C'e, = ['e,
+ <'el
['EA,.
We postulate $(a) =
['qe,).
Then, using equations (1.9.35b)and (2.1.60), one finds
TCa, b}) = C%4,
$(b)}
for any pure elements a, bE9(A,). Therefore, the correspondence $: %(A,) -,End'+'Y(A,) is a representation ofthe Berezin superalgebra %(A,). 2.1.6. Super Lie groups As is well known, Lie groups and Lie algebras are related by means of the exponential mapping. Given a connected Lie group G and the corresponding Lie algebra Q,, (almost all) group elements can be represented as exponentials of Lie algebra elements, g = expa
a = ['ei
['EC (or
(2.1.61 ) [w)
where ( e i } is a basis for 9,. The components ti,i = 1,2,.. .,dim G, of Lie algebra elements play the role of local coordinates (in a neighbourhood of the identity) of the group manifold. The fact that the union of elements of the form (2.1.61) presents a group is a consequence of the Baker-Hausdorff formula: exp (a)exp (b) = exp
1
[a, b]
1
+ -12[ a ,
[a, b ] ]
1
+[b, [b, a ] ] + . . . 12 (2.1.62)
where dots mean commutators of third order and higher. Now one can ask whether it is possible, using the exponential mapping, to construct some group-like objects starting from superalgebras, Berezin superalgebras or super Lie algebras. The answer is negative in the case of superalgebras or Berezin superalgebras, because the multiplication law in Copyright © 1998 IOP Publishing Ltd
(Berezin) superalgebras is a graded commutator, including both commutators and anticommutators. But there is no graded generalization of the Baker-Hausdorff formula. The situation is different in the case of super Lie algebras, since the multiplication operation is now an ordinary Lie bracket. As a result, one can duplicate the ansatz (2.1.61). Namely, given a super Lie algebra '??(A )(complexor real), we associate formally with every element a, 5(
~ ' E C , (or Rc) < ' E @ ~ (orR,)
+ ('e'
a = ('er = ('ei
the symbol g(t') = exp a.
(2.1.63)
In the set of all symbols, denoted by G, we introduce a multiplication operation using the Baker-Hausdorff formula. The group obtained is said to be a 'super Lie group' (or 'supergroup'), since its elements are parametrized by supernumbers. Super lie groups provide us with an example of 'supermanifolds' which present a generalization of ordinary manifolds. A supermanifold is a space which looks locally like a domain in Wlq.We are not able to give here a detailed treatment of supermanifolds and super Lie groups, but refer the reader to the book by B. De Witt. We content ourselves with giving some examples of super Lie groups. The first example is the super Lie group GL'+)(p,q/Am)of non-singular C-type supermatrices in Mat(+)@,q 1 A,) which was introduced in Section 1.10. Every supermatrix P E G L ( + )( p , q 1 A,) from a small neighbourhood of the unit supermatrix can be represented in the form P = exp F
FEogI(+t(p,q/Am).
It is clear then that the supermatrix multiplication in GL(+'(p,qlAcc)coincides with the definition given above of multiplication in super Lie groups. 'g1(+)(p,q / & ) is the super Lie algebra of the super Lie group G q p , qlAa). A super Lie group which corresponds to the real super Lie algebra 'glF)(p,qlAcc) is the set GLk+)(p,qlAa) of non-singular real C-type supermatrices in M a t ( + ) ( pq, 1 A,). Recall that this super Lie group has been realized in subsection 1.10.3 as the group of invertible linear transformations in the superspace WY. A super Lie group which corresponds to the real super Lie algebra 'slk+)(p, q (A,) is the set SLk+)(p,qlAm)of unimodular real C-type supermatrices in Mat(p, 41A,). Indeed, every supermatrix P expressed in the form P = exp F
FEOslk+)(p,qlA,)
has unit Berezinian, BerP= 1 in accordance with equation (1.9.68) . The super Lie group SLkf)(p,q 1 A,) Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
137
can be realized as the group of unimodular linear transformations of the form (1.10.47) acting in real superspace [WPIY. A super Lie group, corresponding to the real super Lie algebra 'ospk+)(p, 2qlA,) is the set OSPk+)(p,29112,) of unimodular real c-type supermatrices in Mat'+)(p,2q(A,) which satisfy the equation F
~ = ~U]
~
F
(2.1.64)
where the supermatrix q was defined in (2.1.42). The super Lie group OSPk+'(p,29/12,) has a deep geometrical origin. To see this, let us introduce in the real superspace Rp12q with coordinates z M = (xm,P),where m = ,2,. . . ,p and p = 1,2,. . . , 2 q , the non-singular supermetric
+
s2 = xmdm,xn iPJP,,O''= zTqz.
(2.1.65)
Now consider the set of all linear transformations on RP12q of the form z + z' = F z preserving the supermetric (2.1.65).Then every such transformation proves to be given by a supermatrix F under equation (2.1.64). Finally, a super Lie group corresponding to the super Lie algebra OSVF,(p,q) of real e-type supervector fields on RpJy,is the supergroup of general coordinate transformations on [WPIq, which has been studied in Section 1.1 1. To summarize, super Lie algebras and super Lie groups play the same role on superspaces R P l 4 which Lie algebras and Lie groups play on ordinary spaces R".
2.1.7. Unitary representations of real superalgebras Let 5!l be a real superalgebra and 2 be a Hilbert space. A linear mapping T : 9 + End 8from 9 into the algebra End 2 of linear operators on X is said to be a 'unitary representation of the superalgebra' 5!l in 8 under the following conditions:
(2.1.66) for any pure elements a, b e g . The first requirement seems rather unusual but it is necessary to reconcile the main property of Hermitian conjugation (AB)' = B + A + with the superalgebra multiplication. Let us comment further. First, the demand that (T('a))' = - T('a), Voa E '9,is standard. Then every operator exp (T('a))is unitary and the union of all such operators forms a unitary representation of a Lie group related to the Lie algebra '9.Further, for all odd elements 'a, ' b e '9, we have { T ( ' a ) ,T ( ' b ) }= T(Oa) Copyright © 1998 IOP Publishing Ltd
138
Ideas and Methods of Supersymmetry and Superyracity
for some 'a€'%, hence the operator on the right is antiHermitian. So, the demand ( T ( ' a ) ) += -iT('a) is quite correct. Let {e,) be a pure basis for the superalgebra, a = x'el = x i e i + x'e, Lei, ejI = fijke,
x1 E IW
Vu E %
{e,, epj = fi//ei
Lei, e,] = ,L,Pep
(2.1.67)
where the structure constants are real. In accordance with (2.1.66), the representation generators T(e,) are not Hermitian. It is useful to change the real basis ( e , ] to another basis {e,} defined as follows A 1 e I . = -. e i
1 e, = -e,
J1
1
[gi, 6.1 J = -ifijkgk {gg,
[gi, e*,] = -if. I , "6p
(2.1.68)
6,) = f $ i *
Then the representation generators T(6,)are Hermitian. In the parametrization (2.1.68), elements of the superalgebra are a
=
i.xiPi + J.YV,
XI E
R, va E 9.
(2.1.69)
2.2. The Poincare superalgebra 2.2.1. Uniqueness of the N = 1 PoincarP superalgebra Now all the facilities which we require to introduce a basic object of our book - the Poincare superalgebra, which presents an extension of the Poincare algebra 9 - are at our disposal. The main motivation for the appearance of this object in theoretical physics was a very old theorists' dream to find a non-trivial extension (other than the direct sum) of the Poincare algebra, i.e. of the Lie algebra of the space-time symmetry group of any relativistic quantum field theory. The problem has proved to have no affirmative solution in the class of Lie algebras. The well-known theorem of S . Coleman and J. Mandula states that, due to assumptions of the S-matrix approach, the most general Lie group of symmetries in a quantum field theory (whose spectrum contains massive particles) is a direct product of the Poincare group and some internal group,
ll x G
G = G, x [V(l)]"
(2.2.1)
where G I is a semi-simple compact group. The corresponding Lie algebra has the direct sum structure PpoY
Y=%,poY,
(2.2.2)
where Y l is the semi-simple Lie algebra of G I and 3 , is an Abelian algebra. Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
139
Since elements of B commute with elements of 9, the only generators with Lorentz indices present in the symmetry Lie algebra are the Poincare generators {pa,.jab) of space-time translations and Lorentz rotations. After suffering a setback in the class of Lie algebras, it was natural to look for success in the class of superalgebras (in fact, this is the way superalgebras were created). Fortunately, it turned out that it was possible to extend the Poincare algebra by several sets of fermionic generators with spinor indices thus obtaining superalgebras. Let us show that there exists a unique superalgebra extension of the Poincare algebra by four a-type generators (qa,q'), with qz carrying an undotted spinor index and qz a dotted spinor index. Recall, in accordance with the results of subsection 2.1.1, a-type elements of a superalgebra 3 transform in some representation of the Lie algebra '9. Since we have chosen the representations (i,0) and (0,i) to act on the subspaces generated by qa and ij', respectively, these generators must commute with the Poincare generators as follows
where we have used the fact that space-time translations do not act on spinor or vector indices. Now we are going to analyse anticommutation relations in the assumed superalgebra. From equation (2.1.1IC), we can write ( % qp> =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
Supersymmetry aad Superspace
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)
Copyright © 1998 IOP Publishing Ltd
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+ -
Using the matrix realization (2.2.14) of the Poincare superalgebra, it is not difficult to obtain a supermatrix realization of the Berezin superalgebra SP(Az).
Let us endow the supervector space W(A,) with an operation of complex conjugation according to the rule (pa)* E - P a
(Si)*
Gab)* E - j a b
Then every real C-type supervector 1
b", Kah= Copyright © 1998 IOP Publishing Ltd
is of the form
2 E 'Sb(A,)
Ksbjab+ P q , - KbaE R,
-qi.
+ Ehqi E' E Ca.
€5 E*
= (E,)* (2.2.18)
Supersymmetry and Superspuce
145
Due to the (anti)commutation relations (2.2.9),one easily finds that the subset OS./P(A,) of real C-type supervectors forms a subalgebra of the super Lie algebra OS.Y(A.). So oSd,(A,) is a real super Lie algebra. I t is called the '(N = 1 ) super Poincare algebra'. Recalling the spinor notation (1.4.3),the last two terms in (2.2.18)can be rewritten as follows: E'q'
+ E*q+
= Eq
+ q = qE + $
(2.2.19)
where we have used equation (2.2.16).
2.2.5. The super PoincurP group A super Lie group corresponding to the super Poincare algebra is known as the (N = 1 ) 'super Poincare group'. It is denoted by Sn. Every element of SFI is of the form g(b, E, F, K ) = exp = exp
[ (1
i - b ' ~ p ' ~+ @jlp
+ Rifijh,j+ E q +
>I
. (2.2.20)
So, points of the super Poincare group are parametrized by real c-number variables b" and Kab = - Kb" as well as by a-number variables (e2,E') forming a Majorana spinor. Similarly to the Poincare group, supergroup elements g(b) = exp ( - ib"p,)
b" E
R,
(2.2.21)
will be called 'space-time translations' and elements
(i )
g ( K ) = exp -KabjOb
Kab€R,
(2.2.22)
will be called 'Lorentz transformations'. Elements g(c, E ) = exp [i(Eq
+ Gi)]
E* E @,
(2.2.23)
are said to be 'supersymmetry transformations'. The union of all elements (2.2.22),denoted by SO(3, 1 I R,)?,forms a super Lie group, which represents a c-number shell of the Lorentz group SO(3, l)?. Ordinary translations and ordinary Lorentz transformations correspond to soulless parameters in (2.2.21)and (2.2.22).We will refer to SO(3, 1 I RJT as the Lorentz group over R,. It should be noticed that, since U, p] p and U, q] q, the set of elements (2.2.20) with b" E R,, E' E e,, but K a bE R, forms a subgroup of the super Poincare group. Therefore, one may restrict the parameters K a b in (2.2.18)and (2.2.20)to be ordinary real numbers. However, since {q, ij} p,
-
-
-
Copyright © 1998 IOP Publishing Ltd
we have [E,q
+ E , q , E,q + E 2 q 1 = 2(E,oaE, - E2o"C1)pa
and hence exp Ci(c1q + Clii)l exp Ci(E2Q = exp [ i ( ( E l
+ E2ii)l
+ E,)q + ( E l + E 2 ) q + i(~,o"E,- c20"?1)pa)]
(2.2.25)
as a consequence of the Baker-Hausdorff formula, so the parameters b" in (2.2.18) and (2.2.20) should be arbitrary real e-numbers. An invariance in quantum field theory with respect to the super Poincare group is called ' N = 1 (or simple) supersymmetry'. 2.3. Unitary representations of the Poincare superalgebra
We proceed by studying one-particle unitary representations of the Poincare superalgebra. The main goal will be to classify irreducible representations T of S d each of which acts in a Hilbert space Y? of one-particle states. The representation generators will be denoted by = T(p0)
Jab
=
TGab)
Q, = T(q,)
0,= T(q,)
(2-3.1)
where Po and J,, are the Hermitian generators of space-time translations and Lorentz transformations, and 02% and 0,are the supersymmetry generators which satisfy equation (2.2.6). The generators P, and J o b form a representation of the Poincare algebra and the set of all unitary operators
where b" and K R b= -Kb" are real number parameters, gives a unitary representation of the Poincare group. 2.3. I . Positivitj? of energy As is known, the Poincare group has irreducible unitary representations of two types: positive-energy representations, characterized by the condition (1.5.14), and negative-energy representations. Only the positive-energy representations are physically admissible. Thus the negative-energy representations, being possible only in principle, must be discarded. Supersymmetry changes the situation radically. The Poincare superalgebra of the (2.2.9) leads to equation (2.2.8) with k = 1, therefore every state 1") Hilbert (or Fock) space is characterized by a non-negative average energy E'Y = ( Y l ~ I ' y ) 1
=
-((Q:Y
4
IQe:Y) +(Qe:Y l Q ~ Y ) + ( ~ l Y l ~ l Y ) + ( Q ~ Y l ~2* Y 0. ) )
Copyright © 1998 IOP Publishing Ltd
(2.3.2)
Super.y,vmmetr,v und Superspace
147
As a result, all unitary representations of the Poincare superalgebra are
positive-energy representations of the Poincare group. Let us suppose the Poincare superalgebra acts in a Fock space of some field theory, and let Ivac) be a Poincare invariant vacuum state, P,lvac) = 0. Then, due to equation (2.3.2),this state is annihilated by the supersymmetry generators: (2.3.3) Q, I vac) = OhI vac) = 0. So, the vacuum state is a supersymmetrically invariant state. Conversely, having a supersymmetric state, which satisfied equation (2.3.3),it is inevitably a Poincare invariant vacuum state. Every non-supersymmetric state has positive energy. 2.3.2. Ctrsimir operators of the Poincarh superalgebra In order to classify irreducible unitary representations of the Poincare superalgebra, it is worth finding its Casimir operators - that is, polynomials in the S 9 generators commuting with each element of SP. Recall that the Poincare algebra has two Casimir operators: the squared mass operator C , = -POP, and the spin operator C2 = W'VV,',, where VV, is the Pauli-Lubanski vector (1.5.10). The Poincare representations are classified by mass and spin. Due to the supersymmetry algebra (2.2.9),the energy-momentum operator Po commutes with the supersymmetry generators Q, and O, hence the squared mass operator also commutes. So, C , = -PUP, is the Casimir operator of the Poincare superalgebra. However, the spin operator does not commute with the supersymmetry generators. This follows from the identity
where we have rewritten the Pauli-Lubanski vector and the Poincare h (8),hW,, P, = (cf),+P' and generators in the spinor notations ( W X = J,, = +(o"b),pJ,b); in particular, one has
Equation (2.3.4) can be readily obtained from (2.2.11). The fact that the spin operator does not commute with the supersymmetry generators means nothing more than that irreducible representations of the Poincare superalgebra contain particles of different spins. To find a supersymmetric invariant generalization of the spin operator, let us consider the operator 1 z,= w,- -CcQyQ,, 8 Copyright © 1998 IOP Publishing Ltd
Q,]
(2.3.6)
which is a generalization of the Pauli-Lubanski vector. Using equations (2.3.4)and the identity (2.3.7)
[CQmQ513 Q/J = 4Q&5 one finds
[Z,,Q,1
1 2
(2.3.8)
= -Q,P,.
One can also prove the relations (2.3.9) Note that the Pauli-Lubanski vector obeys such relations also (see equation (1.5.11)). In accordance with equations (2.3.8)and (2.3.9),the operator (HOPb- &&Po) commutes with the momentum generators and the supersymmetry generators, (2.3.10) Then, the scalar operator 1
- -(Zap,
2
- ZbP,)(Z"pb - Zbp")
and also, due to (2.3.10),with P, and Q,, commutes with the fourth-order polynomial
c = (Z, P)2 - Z2P2
'&.
As a result,
(2.3.11)
is a Casimir operator of the Poincare superalgebra. It is called the 'superspin operator'. Let us analyse the spectrum of eigenvalues the superspin operator can take. Given an irreducible massive representation of the Poincare superalgebra in a Hilbert space Z , we consider in Z the subspace V , of particle states having a given four-momentum q, (see also subsection 1.5.4). As usual, it is useful to choose the momentum (1.5.26) of a particle at rest. Recall that the Pauli-Lubanski vector transforms V , into itself. Since the supersymmetry generators commute with P, each of Q, and Qa(transforms V , into itself. Therefore, the operator 2, transforms V , into itself. When restricted to V,, the superspin operator (2.3.1 1) takes the form mZ((ZJ2+
+ (GI2)
(2.3.12)
and the operators (l/m)Z, S,, I = 1,2,3, proportional to the space components of Z, satisfy, by virtue of (2.3.9), the commutation relations Copyright © 1998 IOP Publishing Ltd
Supersymmetry und Superspace
149
(1.5.28)of the algebra su(2). Then equations (1.5.29)and (2.3.12)lead to 1 3 Y = 0,- 1 -,... 2' % 2
(Z,P)'- Z2Pz= m 4 Y ( Y + 1)0
(2.3.13)
The quantum number Y is called 'superspin'. The above consideration makes clear that massive representations of the Poincare superalgebra are classified by mass and superspin.
2.3.3. MussiGe irreducible representations Let us fix some mass (m) and superspin ( Y ) . The corresponding irreducible representation T of the Poincare superalgebra provides us with a reducible representation of the Poincare algebra. The main problem now is to decompose T into a direct sum of irreducible Poincare representations. This may be done as follows. In the Hilbert space 2 of particle states we consider subspaces X ( + , , defined by the rule: 2(- and
(2.3.14) 2 ( o ) : { l Y ) € 2 ,Q 2 l Y ) = Q21W= 0)
where Q' = Q"Q,, 0 ' = 0,Q'. Each of the subspaces X ( + ) , and X c o , is invariant with respect to the Poincare group. Indeed, the Poincare transformations U(b,K ) = exp [i(b"P, + (1/2)KUbd,b)]act in 2 by the law: I Y ) + IY') = U(b,K ) l Y ) .
So, if 1") E%(+), Q,IY) = 0, then we deduce from the supersymmetry algebra (2.2.9)that
, Xco,. Analogous arguments are applicable to 2 ( -and It is not difficult to construct projectors for the subspaces (2.3.14).Using the supersymmetry algebra (2.2.9),one readily obtains the identities [QZ,
Q,]= 4P,*Qpil
Q2Q2Q2
[Q2,
Q,]
= -4P,,W
Q2Q,&e2=0
Q2Q,Q2=O
= - 16P2Q2
Q2Q2Q2
=
(2.3.15~) (2.3.15b)
- 16P2Q'.
(2.3.15~)
To prove equations (2.3.156,c), one has simply to notice that any product like Q,,Q,2.. . Q, or Q,,O,?. . . a,", n > 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
151
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 - $), respectively, when Y >O. In the case Y = 0, H c 0 , describes only one Poincare representation, of spin f-. To summarize, the unitary representations of the Poincare superalgebra are classified by mass and superspin. In the case Y # 0, the corresponding
+
Copyright © 1998 IOP Publishing Ltd
irreducible representation describes four particles of spin Y - i, Y , Y , Y + with the same mass 171 # 0. When Y = 0, the representation describes two scalar particles and a spin-$ 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
+ (('%)+~I('%)+W =0
where A , = P,jG?. As a result, one must impose the operational constraints PFa,j0b = PB,@ = 0.
(2.3.23)
Further, recall the identities (2.3.15~1).Then, due to equation (2.3.23), one must also demand (2.3.24) [Q2, Q,] = [ 0 2 , QJ = 0. Finally, for any states 1 Y) E 2,we can write
1
= -Q,[Q&,Q2]1Y)
2
=o
where we have used equation (2.3.24). Therefore, we have PaQ21Y) = 0
VIY)EX.
(2.3.25)
It is necessary to point out that the space 2 must be understood as a space of one-particle states (not a Fock space). Hence if 1") is a non-vanishing Y 0. ) In other words, the state, its average energy must be positive, ( Y ~ E ~ > energy operator E must be invertible. So, equation (2.3.25) forces us to demand the constraints Q2 = 0 2 = 0 (2.3.26) in addition to equations (2.3.23,24). In fact, equation (2.3.24) is now a consequence of constraints (2.3.23) and (2.3.26). Copyright © 1998 IOP Publishing Ltd
Supersymmetry und Superspuce
1 53
By analogy with the massive case, we can introduce Poincare invariant subspaces in 2: (2.3.27) In contrast to the massive case, the subspace &',,,, is now a trivial one, due and ) % ( - ) describes some to constraint (2.3.26). Each of the subspaces 2 ( + massless representation of the Poincare group. The supersymmetry generators onto , % ( - ) and vice versa, transform 2 ( +
-
QZ&'(+)
Q&-)
%(-)
-
%(+)
(2.3.28)
in accordance with the identity PBj,= i{Q%, 0,). To make our discussion complete, we must clarify possible helicity +, ) and i.,- ), which the Poincare representations on %( + ) and eigenvalues i. H,- ) may take.
2.3.5. Superhelicity In the massless case, the relation (2.3.4) is simplified to the form 1 2
1 2
= - -Q&,
[Wiljl,O],
= - -QpP,,
(2.3.29)
as a consequence of equation (2.3.23). Hence, instead of the operator Z, (equation (2.3.6)),it is worth introducing another operator
11,
I
= W, - -(6,),"[Q,,
Q,]
16
(2.3.30)
which commutes with the momentum generators and the supersymmetry generators
C%, PhI
=0
[[I,,
Q,]
= 0.
(2.3.31)
Using equations (1.5.11 ) and (2.3.23,29), one can also prove LUPU= 0
[L,,ILhl
= i&u,,cdLcP.
(2.3.32)
Expressions (2.3.31) and (2.3.32) show that the operator L, being supersymmetric invariant, possesses all the properties of the Pauli-Lubanski vector. Then, applying the same line of argument that was used in subsection 1.5.6 to deduce the massless equation W, = >.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.
+
Copyright © 1998 IOP Publishing Ltd
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)
Supersymmetry and Superspace
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'
Copyright © 1998 IOP Publishing Ltd
=
(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)
Supersymmetry and Superspace
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)
Copyright © 1998 IOP Publishing Ltd
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)
Copyright © 1998 IOP Publishing Ltd
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) '
Copyright © 1998 IOP Publishing Ltd
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
Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
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
Supersymmetry and Superspuce
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
Supersymmetry and Superspace
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?(-)=-
Copyright © 1998 IOP Publishing Ltd
(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.
Copyright © 1998 IOP Publishing Ltd
(2.6.10)
Supersymmetry and Superspace
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.
Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspuce
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 ,,,
,,
, ,,
Copyright © 1998 IOP Publishing Ltd
,
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
Supersymmetry and Superspace
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.
Copyright © 1998 IOP Publishing Ltd
(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,
Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
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)
Copyright © 1998 IOP Publishing Ltd
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
Supersymmetry and Superspuce
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
Therefore, in this approach the determination of transformation laws consists of simple manipulations with the covariant derivatives. Consider the antichiral superfield $(z) conjugate to @(z). It is characterized by the component fields
A(x)= 6 1
&(x) = D&
Copyright © 1998 IOP Publishing Ltd
F(x) = - L D 2 6 l . 4
(2.8.7)
188
I d w s und Methods of' Supersymmetry and Superyruvity
The supersymmetry transformations act on them according to the law 6iqx) = -$(x)
6 $ i ( . ~ )= - 2?$(.~) +2i,5a Z l k 2 i ( x )
(2.8.8)
6F(x) = - icd?$(x).
Rather beautifully, using definition (2.8.4), one can immediately obtain component equations which follow from the massive superfield equations 1-
- -Dz6
4
1 4
- -DzO
+ mO = 0
(2.8.9~)
+ m6 = 0
(2.8.9b)
defining the superspin-0 representation. Namely, taking the space projection of equation (2.8.9)leads to
F + mA = 0
F
+ m A = 0.
(2.8.10)
Then, taking D, from equation ( 2 . 8 . 9 ~and ) D ifrom equation (2.8.9b) and making the space projection gives
i?,&'
+
=O
-idzi$'
+ mGi = 0.
Finally, taking ( - iDz) from equation ( 2 . 8 . 9 ~and ) ((2.8.9b)and making the space projection we obtain
OA+mF=O
OA+mF=o.
bD')
(2.8.11) from equation (2.8.12)
It is seen that the complex scalar field F(x) does not have independent dynamics, since it is expressed in terms of A(x), while the complex scalar field A ( x ) and the Majorana spinor field
satisfy the Klein-Gordon equation
(0- m2)A = 0 and Dirac equation (iy"aQ+ m)Y = 0
respectively. 2.8.2. Chiral tensor superfield of Lorentz type ( n / 2 , 0 ) Consider a chiral tensor superfield O x , 2 2J, ,z ) totally symmetric in its indices. ,
Copyright © 1998 IOP Publishing Ltd
Supersymmetry and Superspace
189
By analogy with the scalar case, we define its component fields according to the rule (2.8.13) ~?l?2...?,@)
1 4
= - -D
2@ul...?,,l.
Obviously, the first and third fields are totally symmetric in their indices. As for the second field, it can be decomposed into two Lorentz irreducible fields, of Lorentz types ((n - 1)/2,0) and ((n + 1)/2,0), as follows
Therefore, a chiral tensor superfield of Lorentz type (n/2,0) contains four irreducible tensor fields of Lorentz types ((n - 1)/2,0), ((n/2,0), (n/2,0) and ((n + 1)/2,0), respectively. Transformation laws of the component fields with respect to the supersymmtery transformations can be obtained in the same fashion as was done in the scalar case. The results are
It is instructive to clarify the component form of the equation DP@/hIC12.. .U , , I
=o
defining a massless superfield. This equation has the following consequences D2@z,.,,2,, = @fi@p?l...?,3 I = @bDzl@,?z.. . ?,, = 0.
Taking the space projection of all the equations gives j q . ,, ?,,
Copyright © 1998 IOP Publishing Ltd
,(4= F?, ...?,AX)= 0
(2.8.16)
I90
Idelis and Methods of Supersymmetry and Superyracity
and 8@A/j,,, , , x , ,
,(XI =
c:PB flpa
I...
(2.8.17)
a.(x).
.,,,
, and F,, ,,,,,, vanish on-shell, and the Therefore, component fields E., rest are on-shell massless fields. The on-shell form of transformation laws (2.8.15) read (2.8.18)
2.8.3. Real scular superfield Consider a real scalar superfield V ( z ) . We define its component fields as follows
E.,
=
-
$&(x) = D,Vl
$,(x) = D,Vl
A ( x ) = VI
1 -D,DZV1 4 1
D(x) = -(D2D2 32
r
& ./
14
= - -DkD2Vj
(2.8.19)
+ D2D2)Vl
Note that the last three component fields here do not coincide with the corresponding ones in the power series expansion (2.0.5)because, for example, we have 1 4
1 - . 4
- -D,D2VI = -2,2$*Vl
1 + -2,kZYVI.
2
The supersymmetry transformations act on the component fields of V ( z ) as follows 6 A ( x ) = -E$(x) - @(x) 6$,(x) = - 2~,F(x)- E' v,,(x) - i&,,A(x)
6F(x)= 4 ( x ) SV,,(x) = 2(EkE.,(x) - E,~,(X)) - 2i(E,C?b,.P(x)
+ ia,,(EICl(x)
+ E413,@(x))
- Wx))
6i.,(x) = - 2~,D(x)- ic,a,V"(x) - 2i&,,~(x)
i 6 D ( x )= - - i ? , , ( ~ ~ r f X ( x+ ) @E.'(x)). 2 Copyright © 1998 IOP Publishing Ltd
(2.8.20)
191
Supersymmetry and Superspace
2.8.4. Linear real scalar supery7eld Finally, let us consider a real scalar superfield restricted by the linearity constraint D 2 V = D2V= 0.
(2.8.21)
In accordance with expressions (2.8.19), this superfield has the following component fields A ( x ) = VI
@,(x) = D,VI
1 V,+(X) = -[I),, 2
$+(x) = D k V l
DJVl. (2.8.22)
Not all of these fields are unconstrained. Indeed, one can readily prove the identity [D2, D2]= -4id"'[D,,
D>J.
This identity and equation (2.8.21) show that the vector field V,(x) is transverse,
dV,(x) = 0.
(2.8.23)
Analogously, every linear tensor superfield contains a constrainted component field. Transformation laws of component fields (2.8.21) follow from expressions (2.8.20): 6A(x) = - E@(X) - @(x) S@,(x)= - &',,(x)
6 v,+(x) =
- 2i(~,d&(x)
- iGkdXkA(x)
(2.8.24)
+ ~ a , p V ( x ) )+ iaX+(@(x)- E@(x)).
It is not difficult to see that P(6Va(x))= 0. To summarize, the reduction of superfields to component fields is done most effectively by means of covariant differentiation supplemented by space projection.
2.9. The superconformal group
We have seen in Section 2.5 that the super Poincart transformations (2.5.3) acting on the complex superspace C4I2leave invariant the surface R414(000) defined by y a - jja = 2i&fB.
(2.9.1)
This surface has been identified with the real superspace R4I4by setting the Copyright © 1998 IOP Publishing Ltd
192
Iiieiis and Metkods of' Supers?w"trql und Supergravity
variables 1 2
xa = -(y"
+ jq
(2.9.2)
to be the c-number coordinates on R4I4. It seems reasonable to ask: do other holomorphic trarisformations of C4I2exist leaving R4I4(8a8)invariant? The answer is yes. Before reading the present section, it is worth recalling the remark given at the end of subsection 1.7.4. 2.9.1. Superconformul transforwtations A holomorphic mapping of C4I2onto itself leaving R4I4(8o8)invariant is said to be 'superconformal'. Clearly, the set of all supercsiilormal transformations forms a group. It is called the superconformal group. All the super Poincare transformations are superconformal. The simplest superconformal transformations are:
Dilu tat ions etAd.
= eAya
eiAd.
8%= eA/2g3
(2.9.3)
Axial or ;I,-rotations
eiRa.
=
e l h .
8" = e - i W p .
(2.9.4)
Here A and i2 are real c-number parameters, d and a denote the generators of the scale and axial transformations, respectively. The above transformations o 8 ) act on ~ ~ l ~ (as~follows:
that is a Majorana spinor, the p,-rotations act on 0 by the rule: eiRa. 0 = e +Qz@ Next, let us consider the antiholomorphic mapping defined on C4I2as (2.9.7)
Copyright © 1998 IOP Publishing Ltd
Supersymmetry und Superspuce
193
This mapping is called the 'superinversion'. Obviously, it represents a superspace analogue of the space-time inversion (1.7.19). One can easily verify that the superinversion moves R414(8ae)into itself. In addition, it coincides with the inverse mapping, (2.9.8)
R2 = 0
Therefore, for every superconformal transformation S , the mapping RSR is superconformal. As a result, taking one of the known superconformal transformations (super Poincare, scale or axial) in the role of S , we may obtain a new superconformal transformation. Recall that the analogous trick has been applied in Section 1.7 in obtaining the special conformal transformations in Minkowski space. First, we choose space-time translations exp(-if" p,) in the role of S and consider the transformations exp ( - if"v,)
= R exp ( -
if"p,)R.
(2.9.9)
Using the relation ( 2 . 5 . 2 ~and ) (2.9.7), one then arrives at Special c,onformul trunsformutions
(2.9.10) generalizing the space-time transformations (1.7.184. As an exercise, we suggest the reader find the restriction of (2.9.10) to R4I4(8o8). Further, let us choose a supersymmetry transformation in the role of S . So we are to evaluate the mappings ei(qzs. + qiSi) = - Rei(q*qz+ l i i j i ) ~ (2.9.1 1) Making use of the relations (2.5.2~)and (2.9.7) gives S-supersymmetry transformations
+ e"q2q2y2. Note, ordinary supersymmetry transformations (2.5.2~)are sometimes called Q-supersymmetric, to distinguish them from the S-supersymmetric ones. Subsequent application of the above trick does not give new superconformal Copyright © 1998 IOP Publishing Ltd
I d i u ~rind Methods of' Supersymmetry and Supergravity
I94
transformations, since one can readily prove that Rei K"bjj,, R = e4 K"*j,h ReiAdR
= ,-iAd
ReihR
=e -i & h ,
(2.9.13)
However, it can be shown that the transformations described turn out to generate the superconformal group. Namely, in a neighbourhood of the identity of the superconformal group, every group element looks like -bapa - f a v a
->I
1 + -Kabjab + Ad + na + + E q + E4 + qs + qs . 2 (2.9.14)
We shall argue this assertion in Chapter 6. In the case of infinitesimal parameters in equation (2.9.14),the corresponding superconformal transformation of C412 reads (2.9.15 ) 0" = 0" E."(y, 0) y'" = y" + E/(y, 0)
+
where ;."(y, 0) = b" j."(y,
0) = E'
+ Kabyb+ Aya + f "y2 - 2ya(f , y ) + 2i0aaE - 20a"8,qyb 1 + -(A 2
- i(ijZb)"yb
- is2)O"
(2.9.16) ~ ~ 0 ~ .
+ f"yb(OoaZb)"- Kz,Op + 2
When restricted to R4I4(0a8)(defined by equations (2.9.1,2)),this transformation acts as follows: E."(x, 0) + e-'"P(x,
8) (2.9.17)
2.9.2. The supersymmetric interval and superconformal transformations We have seen in Section 2.4 that the super Poincare transformations leave invariant the supersymmetric interval
ds2 = wau,
ma = dx"
+ i8a"dB - idOa"8.
(2.9.18)
Let us now consider how superconformal transformations act on this object. Evidently, the dilatations act on ds2 in the manner eiAd:ds2 + e2Ads2 Copyright © 1998 IOP Publishing Ltd
(2.9.19)
Supersymmetry und Superspuce
195
and the ;i,-rotation leave ds2 unchanged. Next, the superinversion (2.9.7) changes CO' according to the rule
where we have introduced the notation .x?*, = xu & i8a"B
As a result, the inversion locally rescales the interval
* ,
R: ds2 -+ ___ ds2. 2 x+:
(2.9.20)
).X ( -
The same is true, owing to identities (2.9.9) and (2.9.1l), for the special conformal and S-supersymmetry transformations. Therefore we come to the remarkable result: the peculiar feature of superconformal transformations in that they, at most, locally rescale the supersymmetric interval.
2.9.3. The siiperconformal algebra The generators of the superconformal group obey the algebra: {q,,
ilk}
= 2P,,
(s,,
= 2v,i
(2.9.21~)
and (2.9.21b)
Copyright © 1998 IOP Publishing Ltd
196
Ideiis
Methods of' Supersymmerrj?und Supergruvity
(itid
conjugation (considering pa, vu, d, a as Hermitian operators and setting tixi,)+= j,,, (qu)+= q, and (s,)' = S,). The above (anti)commutation relations define the superconformal algebra. To derive (2.9.21), one can consider a simple representation of the superconformal group acting on the space of scalar superfields by the rule y: U @ )+ U,(Z) = U ( g -
(2.9.22)
'Z)
for every superfield U and every element g of the superconformal group. This representation is characterized by the super Poincar6 generators (2.4.37) (with M X P= &lip = O), the dilatation generator
D
1
= - {(xf+
2
+ xP-
,)do
+ O"2, + B%??}
(2.9.23~)
the axial generator 1 A = -(02c?' - BkZ,) 2
(2.9.23b)
the special conformal generators
(2.9.23~) and the S-supersymmetry generators
-
S , = - ix, - ,P,B~C?,,~
+ x(
+
,P,dp - 2i823,.
(2.9.23d)
Other superfield representations of the superconformal group will be discussed in Chapter 6. In conclusion, we would like to describe one of the possible explanations of the symmetry between expressions in the 1.h.s.and r.h.s. of (2.9.21).Consider the transformations (2.5.4). Together with transformation (2.5.2u), where b" is complex, they define the action on C4I2of the complex shell of the super Poincare group. Let us introduce the holomorphic mappings exp (if%,) exp (ig's,)
= R exp (i&ij')R
R exp (if"p,)R exp (iK&
=
R exp (iKZq,)R.
(2.9.24)
Now, the transformations (2.5.2~) and (2.9.3,4), where b" and A, S2 are complex, and the transformations (2.5.4) and (2.9.24) define the action on C4I2 of the Copyright © 1998 IOP Publishing Ltd
Supersymmetry und Superspuce
I97
complex shell of the superconformal group. It is a simple exercise to check that R exp (iAd)R = exp ( - iad)
R exp (iRa)R = exp ( - ina)
R exp (iN’j,,{)R = exp ( i W q i , j ) .
(2.9.25)
The relations (2.9.24,25) make it possible to reconstruct the expression on the r.h.s. of (2.9.21) starting from the corresponding ones on the 1.h.s.
Copyright © 1998 IOP Publishing Ltd
3 Field Theory in Superspace Gaily bedight, A gallant knight, In sunshine and in shadow, Had journeyed long, Singing a song. In search of Eldorado. Edgar Allan Poe: Eldorudo
3.1. Supersymmetric field theory We proceed to a systematic study of supersymmetric field theories. By definition, a field theory is said to be ‘supersymmetric’ if its symmetry group coincides with the super Poincare group or includes this supergroup as a subgroup. The family of all supersymmetric field theories forms a subclass in the class of all relativistic (or Poincare invariant) field theories. Our primary goal is to define requirements on a field theory in order for it to be supersymmetric. Then, we are to clarify dynamical properties of supersymmetric field theories, both at the classical and quantum levels. This chapter is devoted to consideration of classical aspects. Quantum theory will be discussed in the next chapter. 3.1.1. Quick ret‘iew ofJield theory To begin with, it is worth restoring in mind basic principles of classical field theory; for a more detailed treatment see, for example, the lectures of B. De Witt*. As is well known, any dynamical system is determined by specifying *B.S. De Witt, The Space-time Approach to Quantum Field Theory, in: Relutiuity, Groups und Topology, eds B.S. De Witt and R. Stora (Elsevier Science Publishers B.V., 1984) 198 Copyright © 1998 IOP Publishing Ltd
199
Field Theory in Superspace
a ‘space of dynamical variables’ (or ‘space of histories’) @ and an ‘action functional’ S[cp]. Each point cp = {q’(x)} of @ is said to be a ‘field history’. Here cp’(x) are smooth bosonic and fermionic fields on a space-time, where index ‘ i ’ (labelling field statistics types and tensor types) takes a number of values fixed for the given system. Fields cpi forming histories possess an arbitrary behaviour in finite regions of the space-time but obey certain boundary conditions at infinity, appropriate for the system. The action functional S[cp], which is a mapping
S : @ + R,
(3.1.1)
determines the dynamical field equations of the system (3.1.2)
where left functional (or variational) derivatives are defined as follows
with 6qi(x) being arbitrary field variations. In obtaining the dynamical equations, the following identity (3.1.4)
is often helpful. Every solution qo = { q b ( x ) > of the dynamical equations is said to be a ‘dynamical field history’. The set of all dynamical histories forms a subspace in 0,called the ‘dynamical subspace’ (or ‘mass shell surface’) and is denoted by (Do. Throughout this book we adopt the following convention with regard to the global structure of 0:for every history cp E @, there exists cpo E O0such that the displacements Acpi(x) = cp’(x) - cp’,(x)
have compact supports in the space-time. In the case of a dynamical system in Minkowski space, the Poincare group is assumed to act on the space of histories @ by means of some transformations cp’(x)
--*
cp”(x) = cpL(X)
(3.1.5)
defined for every group element g E l-I. The dynamical system is said to be a ‘relativistic field theory’ if the dynamical subspace a0is a Poincare invariant surface in @, i,
SCql = 0 * ir SC40ql = 0.
(3.1.6)
This requirement is satisfied automatically when the action functional is chosen to be a scalar with respect to the Poincare group
XcpI = SCcp,I Copyright © 1998 IOP Publishing Ltd
(3.1.7)
for every field history c p ~ @and every group element y E n . One can look on equation (3.1.7)as a postulate of relativistic field theory. More generally, having a field theory with some invariance group (rigid or gauge), the action functional should remain unchanged under all transformations from the invariance group. One more assumption of relativistic field theory is that dynamical variables cp‘(.u) may be chosen in such a way that Poincare transformations (3.1.5) are linear and homogeneous. As a result, cp’(x) are tensor fields on Minkowski space, with the Poincare transformation law (1.5.13). The field theories usually considered are local ones. A field theory is said to be ‘local’ if the dynamical equations involve a finite number of time derivatives (and space derivatives, as a consequence of the Poincare covariance). In local field theory the action functional has the form
l
XP1 = d4xU(o,a,cp,. . ., d,,Z,> ‘ . d,cp) *
(3.1.8)
where the integral is performed over Minkowski space. The integrand L is called the ‘Lagrangian’ (or the ‘Lagrange function’). To guarantee equation (3.1.7), it is sufficient to choose the Lagrangian to be a scalar field, L,(x) = L(g- . x)
(3.1.9)
with respect to the Poincare group. Explicitly, since the Poincare transformations preserve the space-time volume, (3.1.10) we have S[qJ =
s
s
d4xL(g-’.x) = d4x’L(x’) S [ q ] .
In practice, one never needs to know the concrete values of S [ q ] on fixed histories, but only its functional structure. In this respect, a number of formal manipulations with the action functional, like integration by parts for instance, are available. These operations are justified by adding suitable boundary terms to S [ q ] or by imposing convenient restrictions on the dynamical variables, as will always be assumed below. The most popular field theories are those in which the dynamical equations are at most second order in time derivatives. In this case the action functional can be represented in the form
s
SCSOI = d4xL(cp,d,v) modulo boundary terms. Copyright © 1998 IOP Publishing Ltd
(3.1.11)
Field Theory in Superspace
201
3.1.2. The space of superfield histories; the action superfunctional
In a supersymmetric field theory the space of field histories is a transformation space of the super Poincare group. So it is worth finding a systematic way of obtaining representations of the super Poincare group on functional spaces. At present, the only known systematic and most elegant way is based on the use of superspace and superfields. Indeed, we have seen that tensor superfields provide us with representations of the super Poincare group. What is more, every unitary representation of the super Poincare group admits a superfield realization. Furthermore, each superfield is determined in terms of its component fields and they automatically form a multiplet with respect to the super Poincare group. Therefore, every space of tensor superfields leads to some space of tensor fields (‘component field space’) with defined action of the super Poincare group. Conversely, the question arises: does any space of fields with defined action of the super Poincark group admit a structure of component field space for some space of superfields? We do not know the answer to this question. But all known N = 1 supersymmetric field theories admit superfields realizations, and we restrict our consideration to the case of such theories. In each theory from this class, there are two equivalent realizations of the space of dynamical variables: as the space of field histories @ and the ‘space of superfield histories’ which will be denoted by V. Any point of V presents a set of tensor superfields U = ( ~ ‘ ( z )on } R4I4and the totality of all their component fields gives a field history ( q ’ ( x ) } from @. In other words, Q, is the component field space for V.Since the spaces Q, and V are in one-to-one correspondence, the action functional S [ q ] can be re-expressed as a functional on the space of superfield histories, i.e. as a mapping
s:v + R,
(3.1.12)
which will be denoted by S [ u ] . It is useful to call S [ u ] the ‘action superfunctional’ (more generally, any mapping from a space of superfunctions into the Grassmann algebra Am is said to be a ‘superfunctional’). If v is a superfield history and q is the corresponding field history, then (3.1.13) The ‘super Poincare invariance’ means that the action superfunctional should remain unchanged under the super Poincare transformations (2.4.29), (3.1. 14) for every supergroup element g E SH. The dynamical subspace Q 0 consists of field histories corresponding to stationary values of the action functional. When qo is a dynamical history, we have S[q,
+ 6q] - S[qo] = 0
Copyright © 1998 IOP Publishing Ltd
Vdq’(x).
(3.1.1 5 )
202
Ideus und Methods of Supersymmetry and Supergruuity
Let us denote by V, the subset in V,for which Q0 is the component field space. The set V, is the superfield version of the dynamical subspace. Any U, = { v’,(z)} E V, will be called a ‘dynamical superfield history’. Due to one-to-one correspondence between Q, and V,equation (3.1.15) leads to
S[u,
+ 6v] - S[vO] = 0
V6v’(z)
(3.1.16)
for every dynamical superfield history u,(z). So, the action superfunctional S [ u ] takes stationary values at each point of V,. We see that the dynamical beltuuiour of the superfield system is determined by the stationary action principle. To rewrite equation (3.1.16) in a differential form, it is worth discussion superfield variational derivatives.
3,1.3. Integration over R4I4and superfunctional derivatives Now is the time to resort to the integration theory over RPl4 developed in Section 1.10. We will be mainly interested in integrals over R4I4of the general
form
s sss s s
d8z u(z) = d4x d28 d2B v(x, 8,8)
(3.1.17)
where u(x, 8,8) is some superfield. The integrals over a-number variables 8* and Bj, are defined as follows d?P Bg = 6’j
de, 9” = 6,P
(3.1.18)
and the multiple measures are
(3.1.19) instead of equation (1.10.41), such that
s
s s s
d2ee2 = 1
d28G2= 1.
(3.1.20)
The difference in definition (by a constant) of multiple integrals (1.10.41) and (3.1.17,19) is introduced to have the property dszV(z) = d4xD(x)
(3.1.21)
where D(x) is the highest component field of V ( z )in the 9, &expansion (2.4.21). Therefore, if v(z) is a real superfield then integral fdszo(z) is a real Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
203
supernumber. More generally, we have (fd8zu(z))* = jd8zu*(z).
(3.1.22)
Superfields arising as integrands will always be assumed to vanish when the bodies of x m go to ( f CO),which implies, together with equation (1.10.3IC), the relation
s
d8z(aAu(z))= 0
(3.1.23)
where are the partial derivatives defined in equation (2.4.25).This result presents the superspace rule for integration by parts. It can be rewritten in terms of the covariant derivatives (2.5.25):
s sss
d8z(DAo(z))= 0
(3.1.24)
which follows from
d8zD,t' = d8zd,u
+i
d8z(o'B),i3,0 =
Recall that integration in a-number variables is equivalent to differentiation. Then, due to equation (3.1.20),one obtains
(3.1.25) By analogy with the derivation of equation (3.1.24)from equation (3.1.23), the final relations can be rewritten in terms of the spinor covariant derivatives:
and
(3.1.26b) where symbol 'I' means space projection (see Section 2.8). Introduce the &function on Iw4l4:
s*(~) = s4(x)~(e)s2(0)
s2(e)
Copyright © 1998 IOP Publishing Ltd
= 02
~ ( 8=)8 2 .
(3.1.27)
204
Ideas and Methods of Supersymmetry und Supergravity
I t is characterized by the properties:
s
dsz'6(Z - z')u(z') = U(Z)
68(z - z')o(z') = 68(z - z')u(z) P(Z) = S E ( -z)
(68(z))2= 0
(3.1.28)
6"O) = 0
for an arbitrary superfield v(z) (compare with equations (1.10.38,39)). After the given excursion into integration theory, we are in a position to introduce variational derivatives associated with different superfield types. Let us consider a dynamical system with the space of histories V being the set of all real scalar superfields V(z), V = E under some boundary conditions. For any superfield VEV, we denote by {cp'(x)} the set of component fields of V(z). Using the supernumber norm (1.9.Q V can be turned into a metric space with the distance function p( V , , V 2 )between two arbitrary superfields V l , V , E V defined as follows
are obvious. The mapping (3.1.12)defining the action superfunctional will be assumed to be continuous. Then, under reasonble assumptions about the form of SCV], two values of the action values S [ V ] and S [ V + S V ] where SV(z) is an infinitesimal displacement, p ( V , V + S V ) -,0, are related by the rule 6S[V] = S[V+ SV] - S[V] =
+ O((6V)z).(3.1.30)
The superfield SS[ V ] / SV ( z )will be called the 'superfunctional derivative' of S [ V ] at V [ z ] . In complete analogy with ordinary variational analysis, one can prove that
s
d*ZV(z)'€'(z) = 0
V V ( z )=
P(z)3 Y(z) = 0.
(3.1.31)
Here V can be restricted to have compact support in the space-time. Therefore, due to the arbitrariness of the superfield variation SV(z)in (3.1.30), Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
205
the stationary action principle (3.1.16) leads to the equation
SS[ V ] / SV ( 2 )= 0
(3.1.32)
which is the ‘dynamical superfield equation’ of the system. For calculating the superfunctional derivative of different possible actions SCV], it is useful first to do this in the case of a special superfunctional defined by
F: v
+ R,
F(V)= V(z/)
where z’ is a fixed point of R4I4. Making use of the superspace 6-function (3.1.27) gives
s
6F( V ) = 6 V(Z’)= dsz6 V(Z)S(Z- 2’). Therefore, we have the relation
6 V(z’) 6V ( 4
-= P ( z - z‘)
(3.1.33)
which can be used further in practical calculations as a formal rule. When a dynamical system is described by p real scalar superfields V’(z), V’ = P’, completely arbitrary modulo certain boundry conditions, equation (3.1.30) should be substituted by
s
S S [ V ] = d8z6V’(z)-
6SL.V) dV‘(2)
(3.1.34)
and, instead of equation (3.1.33), we will have
(3.1.35) When the total number of V’ is even, it is sometimes useful to copbine the superfields uGder considerations into complex scalar superfields V‘ and their conjugate P’(z). Then equation (3.1.35) tales the form SV(Z’) 6 B(2’) -SP(Z)
SV(2) -
*
6 V ( 2 ’ ) 6 P(2’) _-- 6*(2 - z’) SV(2) S V ( Z )
--
(3.1.36)
where indices are suppressed. Now consider a dynamical system, for which the space of histories V coincides with the set of all possible pairs (@, where Wz)is an arbitrary chiral scalar superfield, Di@= 0, modulo boundary conditions. How can one represent an infinitesimal variation of the action superfunctional S[@,
a),
a]?
Copyright © 1998 IOP Publishing Ltd
206
ldcus trnd Mct1iod.y of Supersymmetry and Supergravity
Recall, each chiral superfield @(z) is of the form @(z) = eiJY6(X ,e)
2 = 8aa8d,.
(3.1.37)
Hence @(z) is determined by superfield &(z, 0) which represents a superfunction on superspace rW: x C;.Therefore, one can look on the action superfunctional as a superfunctional of &(x, e) and 8(x, 8) = (Nx,e))*, S[@,
a] = S[$,i].
(3.1.38)
Obviously, &(x, 8) is an arbitrary (modulo the boundary conditions) bosonic superfunction on 53: x C,'. Then we can write
6S[@,
a] = S[@ + 60, @ + sa] - S [ @ , a]
where we have introduced the notation: d6z = d4x d 2 8
d6Z = d4x d28.
Note that SS/S&(x, e) is a superfunction on R I x C,'. Furthermore, due to equation (3.1.37), the following identity
(3.1.40)
(3.1.41) holds if the component fields of & vanish at infinity. Therefore, one can rewrite equation (3.1.39) in the form
(3.1.43) Chiral superfield SS/S@(z)is called the 'superfunctional derivative' of S [ @ , $1 at @(z). By analogy with equation (3.1.31) one readily proves that
s
d6z &(x, e)9(x,e) = o
v&(x,
e) a 9 ( x , e) = 0.
(3.1.44)
Then the stationary action principle (3.1.16) and equation (3.1.42) give the Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
207
dynamical superfield equations:
(3.1.45) Since the action superfunctional is real, these equations are conjugate to each other. For calculating the superfunctional derivatives, it is useful to operate with the identities: 1 -= - - D2S8(z - z’) 6@(z) 4 S@(z’)
SG(Z’)
E
6 + ( ~z’),
1
- -D2a8(z - z’) G S-(Z, z’) 6C-m 4
-=
(3.1.46)
6@(z’)- 6 a q Z ’ ) ---=o.
60(z)
6qZ)
One can readily prove these identities by considering the superfunctional F[@,G] = @(z’) with Z‘ being a fixed point of
s
[w4I4.
Its variation is represented as
GF[@, a] = 6@(~‘) = d8zd8(Z- z‘)SO(Z)= -
d6ZD2(d8(Z- z‘)~@(z))
= ld6z6@(z)6+(z,z’)
where equation (3.1.26~) has been used. The 6 + ( z ,2’) will be called the ‘chiral delta-function’. We list its basic properties:
Djr6+(z,z‘) = 0
s
S+(z, z‘) = 6+(z‘,z )
d6z‘6+(z, z’)@(z’) = @(z)
(6+(z,z’))2 = 0
(3.1.47)
S+(z, z ) = 0
where @(z) is a chiral superfield. The 6 - ( z , z ’ ) is called the ‘antichiral delta-function’. Conjugating the above identities, one obtains the basic properties of 6 - (z,z’). Equations (3.1.33) and (3.1.46) show that there is no universal form for superfunctional derivatives, in contrast to the space-time case, where we had equation (3.1.4). The point is that different superfields are in fact, defined, on different superspaces (real superfield on [w4I4, chiral superfield on rW; x C:). We have discussed two possible realizations of dynamical variables: in terms of real scalar superfields and in terms of (anti)chiral scalar superfields. Copyright © 1998 IOP Publishing Ltd
208
Ideas und Methods of‘ Supersymmetry and Superyruuity
Generally, throughout this book, we will consider only supersymmetric field theories described by unconstrained tensor superfields and unconstrained tensor (anti)chiral superfields. This means that the superfields variables V’(z), parametrizing the space of histories V,may take arbitrary values modulo the boundary conditions and modulo the constraints of chirality and antichirality, which superfields d ( z ) satisfy. For every such theory the stationary action principle (3.1.16) is equivalent to the dynamical superfield equations SS[U]/SU‘(Z)
=0
(3.1.48)
and the superfunctional derivatives are calculated by applying identities like (3.1.33), (3.1.36) or (3.1.46),depending on superfield types.
3.1.4. Local supersymmetric jieid theories We are going to describe a wide family of local supersymmetric field theories. It is useful to start with two simple theorems.
Theorem 1. All the super Poincare transformations on R4I4 have unit Berezinian, zA,z’A=g.zA
gESn
aBer -z’~
)
= 1.
(3.1.49)
Proof. It is sufficient to prove the theorem for particular super Poincare transformations (2.4.10-12). In the cases of translations (2.4.10) and supersymmetry transformations (2.4.12),the statement is evidently satisfied. In the case of Lorentz transformations (2.4.1l), the statement is also satisfied since matrices (eK)aband (eK),p are unimodular. Theorem 2. Let Y(z) be a scalar superfield, and Y,(z) be a chiral scalar superfield, DhYc= 0. Then the integrals
s
I = d8zY(z)
I
I, = d6zYc(z)
(3.1.50)
are invariant under the super Poincare transformations.
Proof. In accordance with equation (2.4.34),a supergroup element y E Sll moves dp(z) into Y’(z) = Y(g-
z)
and hence
I
S
I’ = d8zY’(z) = d8zY(g- .z). Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspuce Introduce the new integration variables
209
("i..).
d8z = d'z' Ber-
i3ZA
Making use of (3.1.49) gives I' = I. To prove the second part of the theorem, we note that Yc(z) can be represented in the form Ip,(z) = - 4-iD2U(z)
(3.1.51)
where U ( z ) is a scalar superfield, see also subsection 2.5.6. Then, due to equation (3.1.26~1,we have
s
I , = d8zU(Z). Therefore, the second part of the theorem is reduced to the first part, and the proof is complete.
Remurk. In accordance with equation (3.1.26a), we have
s
d'zL?c(z) = 0
D&YC= 0.
(3.1.52)
Now, consider a supersymmetric field theory with an action superfunctional of the general form
s
S [ u ] = d'zY(u,DAu, ...,D A ,... DA,V)
Y = 9
DdrYc=O
(3.1.53)
where Y and Y care supersmooth functions of their arguments; the symbol 'c.c.' means complex conjugation. The reality condition 9 = 8 is imposed in order that S [ u ] be real. To guarantee the requirement of super Poincare invariance (3.1.14), L? and Y cshould be scalar superfields constructed in terms of tensor superfields ~ ' ( z )D,u'(z), , D,D,u'(z), . . . , The function 9 will be called the 'super Lagrangian' and Y cwill be called the 'chiral super Lagrangian'. Using the technique of reduction from superfields to component fields (see Section 2.8) one can readily see that S[u] leads to a local field theory at the component level, with the action functional
s
S [ q ] = d4xL(q,i3,(p,. . ., a,, 1
1
... d,,(p)
1i-l
L = -{D2 3 2 " (D42 ' Y - D 2 Y c - - D 2 P C Copyright © 1998 IOP Publishing Ltd
(3.1.54a)
2 10
Ideas uno' Methods of Supersymmetry und Superyruvity
(see also equations (3.1.26)).Up to total derivative terms, the Lagrangian can be represented in two different forms:
(3.1.54b) or
(3.1S4c) Any action superfunctional of the form (3.1.53) will be called 'local'. We have taken the super Lagrangians to be dependent on covariant derivatives D, of U' but not on partial derivatives aA of d.The reason for this is that the spinor partial derivatives are not tensor operations; when ~ ' ( zare ) tensor superfields, then dlu'(z) or ahv'(z) are not. It should be pointed out that in Minkowski space partial derivatives 3, are automatically covariant ones. In accordance with (3.1.26), one can rewrite the action superfunctional (3.1.53) as follows
s
S[U] = d6zPc(z)+
C.C.
(3.1.55)
1Pc= 44, - -D244.
8
Evidently, this action superfunctional is local. As a result, one can work with chiral super Lagrangians only. But a reverse transformation, being possible in principle, may result in a non-local superfunctional. We can represent 9, in the form (3.1.511, to obtain
s
S[V] = d 8 z P
2 = L+
U
+ D.
(3.1.56)
As a rule, this superfunctional is non-local. For example, consider the chiral super Lagrangian m 2
6Pc(O)= -O2
where m is a constant, and O(z)is a chiral scalar superfield. Using identities (2.5.30e) and (2.5.33), we have
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
21 1
This leads to (3.1.57)
and the obtained superfunctional is non-local. In conclusion, we must answer the following question: what restrictions on the action superfunctional can we impose in order to make the dynamical field equations at most second order in time derivatives? Recall that the space-time derivatives can be expressed by the rule
2 = “
1
- -(EaYZ{DZ,
Dk).
4
Therefore, the dynamical superfield equations should be at most of fourth order in the spinor covariant derivatives. Then, the action superfunctional can be represented in the general form
s{
S[C] = dszU(t., D,u, D,DBv)
+
sd6zU,(o. D,c, D,DBu, D,D,D,u)
I
+ C.C.
(3*1.58)
3.1.5. Mass dimensions As is known, the action functional is dimensionless (in units of mass). The same is true for the action superfunctional. We wish to find mass dimensions of super Lagrangians. By virtue of equations (2.4.42)and (3.1.20) one obtains
[d20] = [d’e] = 1. Since [d4x] = -4, we have [d8z] = -2
[d6z] = -3.
(3.1.59)
hence, the dimensions of super Lagrangians are
[9] =2
3.1.6. Chiral representation
[U,] = 3.
(3.1.60)
When investigating a dynamical system, it very often proves useful to consider different pictures (or representations) for dynamical variables. For example, the two most popular quantum mechanical pictures are the Schrodinger representation and the Heisenberg representation. In field theory one may choose the dynamical variables to be fields in Minkowski space or, after taking Fourier transforms, in momentum space. In the case of supersymmetric Copyright © 1998 IOP Publishing Ltd
field theory, there is a representation for dynamical variables - the 'chiral representation'-which does not have an analogue in ordinary field theory. This representation is characterized by the requirement that, for every chiral superfield 0,its chiral transform 0''' coincides with the superfield 61defined by equation (3.1.37). The chiral representation is introduced as follows. Consider a dynamical system with the action superfunctional S [ v ] being of the form (3.1.53). One changes every superfield variable V ' ( z ) to L.(C)'(:)
= e-I"v'(z)
=
eaq7~,
(3.1.61a )
and every differential operator 9 to gco = e - l "
/c e
(3.1.61b)
IH .
Obviously, the super Lagrangians can be represented as Y(L., D,c,. . . ) =
f
~ ( ~ ?(DcA( )O t i ( C ) ,. . .
Y,(c,D,o,.. . ) = e'fYc(c(C) > D )A '(it)'(
,...)
E ei f l y ( C ) ( , ( C ) ,
DA (C+,,(C), . , . )
eiw~E(C)(U(C),DA(C)u(C) ,...).
Therefore, the action superfunctional does not change (modulo total derivative terms), In accordance with equations (2.5.19,21), the chiral transformed covariant derivatives are
Df) =
-zk
D',") = e-2iXdze2iX= d,
+ 2iB'dEb.
(3.1.62)
Therefore, in this representation each chiral superfield depends only on xa and Ox,
D,o, = o
-
a y z )= 6 ( ~ e).,
(3.1.63)
Further, recalling the explicit form of the super Poincare generators (2.4.37), their chiral transforms are Pa' = Pa
aic)= i8,
JL:) = Jab Qj( (C) - -ie-Z'X;?.e*'" = -i;?& - 2 p d , k
(3.1.64)
It is clear that all the super Poincare transformations preserve the &independence of chiral superfields. The chiral representation is best suited for operating with chiral superfields. For other superfield types, it leads to complications. For example, the operation of superfield complex conjugation in this representation is as follows v ( C ) = e 2 i J y p(0 (3.1.65)
where V ( z )is an arbitrary superfield and P(z) is its conjugate. In particular, Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
21 3
if V ( z )is a real superfield, P = V , then -
v ( C ) = e2iX jAC).
We see that the chiral representation modifies the reality condition. On these grounds, the previously used representation will be called ‘real’, to distinguish it from the chiral representation. One can easily see that e - ~ H -iN’
e
h8(z - z’) = P ( z - 2’).
Therefore, the chiral transform of S+(z, z’) is given, using equation (3.1.62), in the form
z’) = 64(z - z y z ( e - e’).
(3.1.66)
By analogy with the chiral representation, one can introduce an ‘antichiral representation’ best suited to describing antichiral superfields. It is defined by replacing the operator A? by ( - A? ) in equations (3.1.61).
3.2. Wess-Zumino model We now review the most popular supersymmetric field theories. To maintain the tradition, we start with dynamical systems described by (anti)chiral superfields-supersymmetric analogues of scalar fields.
3.2.1. Massive chiral scalar superfield model As has been shown in Section 2.6, the massive superspin-0 representation can be realized in terms of a chiral scalar superfield O(z) and its conjugate @(z)using the equations 1 -2--DcP+mO=O 4
1 --D20 4
+ma
=0
(3.2.1)
with m being the mass. We wish to find a dynamical system for which equations (3.2.1) are the dynamical superfield equations. The above equations are linear in O and hence the action superfunctional SCO, must be quadratic in the superfields under consideration. Recall that the variational derivatives S@(z‘)/SO(z) and h@(z’)/6@(z)involve the covariant derivatives (see equations (3.1.46)).Therefore it seems reasonble that the super Lagrangians will depend on O and (5 only. Then there are three possible
a]
Copyright © 1998 IOP Publishing Ltd
a,
structures I, =
5
daz@@
I, =
s
d6z02
s
l3 = d6Za2
which may contribute to S [ @ , @I.Two other structures J d8z Q2 and d8z G 2 vanish identically, in accordance with equation (3.1 S2). Making use of equations (3.1.46) gives
S
s
d6z‘@(z’) S+(Z, z’) = 2 d6z’ @(z’)6+(z’,Z )
=2
s
d’z’ @(z’) d8(z’ - Z ) = 2@(z) --
6@(4
- 0.
It is instructive also to reobtain 61,/6@(z) in the chiral representation. One finds, after using equation (3.1.66), ___ =
S@‘C’(Z)
s s
2 d6Z‘@“’(z’) ~‘,C’(Z,z’)
=2
d4x’d28,@(‘)(x’, 0’) d4(x - x’) a2(8 - e’) = 2@(”(z).
The above identities show that the action superfunctional is of the form S[@,
a] =
s
TI
d8z&D+ - d6zQ2 + E d6Za2 2 s
(3.2.2)
Here the first term is called the ‘kinetic term’ and the rest are said to be the ‘mass terms’. Obviously, the action is invariant under super Poincare transformations. Representing @(z)= e’.”@(x,e) and a ( z ) = e-’.*@(x, 8) and integrating by parts, S [ @ , G] can be rewritten in the form
i
S[@, @] = dBz6(X,6)e2’.“@(x,e)
Copyright © 1998 IOP Publishing Ltd
+ (3.2.3)
Field Theory in Superspace
21 5
One can readily find the mass dimension of @ and 6.Explicitly, equations (3.1.59,60)lead to (3.2.4) [@(z)]= [G(z)] = I .
3.2.2. Mussless ckiral scalar superfield model A massless model is obtained by setting the mass parameter in (3.2.2) to be vanishing. The action superfunctional reads
s :s
S[@,@] = d8zG@= - d8z(@+
a)’
(3.2.5)
and the dynamical equations are 1-
- -D2$ 4
1
- -D’@ = 0.
=0
4
(3.2.6)
These equations determine on-shell massless (anti)chiral scalar superfields (see subsection 2.7.2), and the corresponding superhelicities are =0
ti(@)
ti(@)
=
- lj2.
(3.2.7)
3.2.3. Wess-Zumino model We now consider the model of interacting (anti)chiral scalar superfields, proposed by J. Wess and B. Zumino, with the action S[@,
a] =
s s
s
d8zG@+ d6zYc(@)+ d6Z8,(6) D&2?c(Q(z)) = 0.
(3.2.8)
Here LYCis a holomorphic function of an ordinary complex variable, called the ‘chiral superpotential’. Clearly, Yp,(z)= Yc(@(z)) is a chiral scalar superfield, therefore the action superfunctional is super Poincare invariant. The dynamical superfield equations have the form 1 -
- -D’G 4
+ 9:(@) =0
1 4
- -D2@
+ 8b(G) = 0.
(3.2.9)
We will refer to the above theory as the ‘general Wess-Zumino model’. At the classical level, there is no restriction on the form of YC(@). The situation is different at the quantum level. As will be shown in Chapter 4, the theory with classical action (3.2.8) is renormalizable if the chiral superpotential is at most a third-order polynomial,
m E. 9,(@) = p@ + -0’+ -@3 2 3! where
,U,
(3.2.10)
m and 3. are coupling constants. Note that the first term in Yc(@)
Copyright © 1998 IOP Publishing Ltd
can be removed by the superfield redefinition Q, -+ constant. Then. one obtains the action
Q,
+ (,where ( is a suitable
This theory will be called the 'standard Wess-Zumino model'.
3.2.4. Wess-Zunzino model in componeni ,form Let us rewrite the action superfunctional (3.2.8) in terms of the component fields of Q, and 0.We shall follow the prescription (3.1.54~)to obtain S[go]. Recalling definition (2.8.4) of the component fields, we have 16
1 16
+ 81
1 + --Q,/D2D2@,( 16
= - ( D 2 Q , ) l D Z ~ ! -(DW)lD2Dz@\
+ A(x)OA(x)
1
= F(x)F(x)- - Iy(X)d,$(X)
2
the next contribution is
1 4
- 'Dzpc(@)I= F(x)P;(A(x))- - $&(x)lp(x)P:(A(x)).
4
Then, after introducing the notation V ( A ) = 9&4) Copyright © 1998 IOP Publishing Ltd
(3.2.12)
217
Field Theory in Superspace the action functional reads
(3.2.13) where cp = { A , A, $ E , p,F , F } represents the complete set of field variables. It is instructive to rewrite the action in four-component spinor notation. Introducing the Majorana spinor (3.2.14) and recalling expressions for y-matrices (see Section 1.4), one obtains
Not all of the component fields have non-trivial dynamics. The first dynamical equation (3.2.9)is equivalent to three component equations
F
+ V ( A )= 0
i(oa8a$))l- y ' ( A ) t j a= 0 1 O A + V ' ( A ) F - - V " ( A ) $ $ = 0. 4
(3.2.16~) (3.2.16b) (3.2.164
Therefore, on-shell F(x) is expressed algebraically in terms of A@); F(x) does not have an independent dynamics. On these grounds, F(x) and F(x) are said to be 'auxiliary fields'.
3.2.5. Auxiliary fields Now, we would like to describe what the notion 'auxiliary fields' means in general. Let cp = (@,@}be field variables of a field theory with an action functional S[cp] = S [ @ , @I.Suppose that the dynamical equations for fields ip are such that: (1) they do not involve time derivatives of ip; (2) they can be uniquely resolved by expressing ip in terms of the remaining fields, SS[cp]/S@ = 0
@ = f(@)= ip,o.
Under these assumptions, @ are said to be auxiliary fields and @ are said to be physical fields. Copyright © 1998 IOP Publishing Ltd
21 8
Ideas and Methods qf Supersymmetry and Superyraoity
Instead of working with the total set of fields, one may eliminate from the very beginning the auxiliary fields, resulting in the 'physical' action
SC41 = SE@,$1 I @ =I(@) Then, two actions S [ q ] and S [ @ ] lead to equivalent dynamical equations in @:
therefore
However, very often it is convenient to keep the auxiliary fields, since their elimination may produce complications such as the loss of explicit covariance or the non-locality of the 'physical' action and so on. In particular, linear symmetry transformations may turn into nonlinear ones after elimination of the auxiliary fields. If S [ q ] possesses an invariance under linear transformations of the form
a@= R A "4 + R A"4
= R"
" 4 + R"
"@
then S [ @ ] is invariant under transformations d@ = R A " @
+ R " "f(4).
which are nonlinear, as a rule. The Wess-Zumino model provides us with an example. 3.2.6. Wess-Zumino model after auxiliary field elimination Substituting equation (3.2.16a) into (3.2.15) expresses the Wess-Zumino model in terms of physical fields @ = { A , A, t,bm, p> as follows
where U ( A ,A) = lV(A)12
(3.2.18)
is the potential of scalar fields. Note, the scalar potential is positive semidefinite, which is a direct consequence of the energy positivity in supersymmetric theories (see subsection 2.3.1). Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
219
In the case of the standard Wess-Zumino model (3.2.1l), we have "+/(A)= mA
+ -21) A 2
and the action (3.2.17) takes the form
(3.2.19)
where we have supposed that the coupling constant E. is real. The resulting action describes cubic and quartic self-interaction of the scalar fields and Y ukawa coupling between the scalar fields and the Majorana spinor field. Now, let us discuss supersymmetric properties of the Wess-Zumino model, before and after elimination of the auxiliary fields. Since the action superfunctional (3.2.8) is super Poincare invariant, it does not change under infinitesimal supersymmetry transformations
de@ = i(EQ +
a)@ 6$
= i(EQ
+ q)a.
At the component level, they are given by equations (2.8.6,8) (these transformations leave invariant the action functional (3.2.13)).The equation (2.8.6) defines a linear field representation of the supersymmetry algebra. In particular, commuting two supersymmetry transformations gives a space-time translation,
b" = 2i(~,a"Z, - E,&,)
(3.2.20)
in accordance with equation (2.2.24). As for the action functional (3.2.17),it is invariant under the supersymmetry transformations
which follow from equations (2.8.6,8) by setting F = - P ( A )and F = - $'-(A), We see that the spinor transformation laws become nonlinear after elimination of the auxiliary fields. This, however, is not the whole story. Taking the commutator of two supersymmetry transformations d,, and a,, Copyright © 1998 IOP Publishing Ltd
220
Ideus und Methods of' Supersymmetry and Supergravity
applied to A and $%. one finds (3.2.22)
The second relation indicates that the supersymmetry algebra becomes broken when the auxiliary fields are eliminated! The supersymmetry algebra closes only modulo the dynamical equations of the spinor fields. To summarize, auxiliary fields are needed to make the supersymmetry transformations linear as well as to make possible the very existence of the super Poincare algebra off-shell. Without use of auxiliary fields, the super Poincare algebra cannot in general be realized on the space of field histories but only on the mass shell surface. It should be pointed out that the problems just discussed arise only when working with ordinary (component)fields. Living in superspace and operating with superfields, there is no need to worry about supersymmetry. Superspace makes supersymmetry manifest.
3.2.7. Generalization of the model The Wess-Zumino model can be easily generalized to describe the dynamics of n chiral scalar superfields 0'and their conjugates @,@(z) = (@(z))*. An action superfunctional is taken to be
d6zY,(O', 0 2 ,,,. ,W)+ C.C. where 9, is a holomorphic function of n complex variables. The action leads to the following equations of motion for ai
Turning from the superfields to the component fields $;(x) = DzQi1
A ' ( x ) = @'I
1 4
F ' ( x )= - - D2@
(3.2.25)
the action reduces to the form
+
1 d26P(A) . . F'-Y,(A) - - A $ ' I c / J 4 aA'aAJ
[
Copyright © 1998 IOP Publishing Ltd
11
+ C.C.
.
(3.2.26)
Field Theory in Superspuce
221
Ifone eliminates the auxiliary fields, the purely scalar part of the action reads S [ A , X] =
-
J
d4x{daZd,Ai
+ U ( A , A)} (3.2.27)
The scalar potential is positive semidefinite. From the expression (3.2.27) it is seen that not every scalar field theory of the general form
'S
S [ q ] = - - d4xd"cp'a,cpz - [d4xxL'(q)
2
(3.2.28)
where cp' are real scalars can be supersymmetrized. An admissible model is one with an even number of real fields which can be combined into complex ones in such a way that the scalar potential has the structure (3.2.27).
3.3. Supersymmetric nonlinear sigma-models In this section we endeavour to clarify the conditions under which a nonlinear scalar field theory of the general form
'S
SCcpI = -- d4xgz,(d 2
au(p'aCr(pJ
(3.3.1)
generalizing the kinetic term in equation (3.2.28), can be extended to a supersymmetric field theory. First, it is worth saying some words about models (3.3.1) known as nonlinear o-models.
3.3.I . Four-dimensional o-models Nonlinear o-models naturally arise in a geometrical framework as follows. Let A"be an n-dimensional Riemann manifold with metric g,,(q), where q' are local coordinates on A".The o-model space of field histories coincides with the set of all smooth mappings cp: R4 + A"
(3.3.2)
from Minkowski space into An('target space'). So, fields are coordinates on A"depending on points of Minkowski space. An action functional is required to be invariant under the Poincare transformations on Minkowski space and Copyright © 1998 IOP Publishing Ltd
222
I d u s und Methods of Supersymmeiry and Supergruvity
the general coordinate transformations
v'
+
cp" = f ' ( c p )
(3.3.3)
of the target manifold. Evidently, the choice (3.3.1) is consistent with these requirements. Of primary importance for us will be a-models on complex manifolds. Recall, a manifold ,H2" is said to be a complex n-dimensional manifold if local coordinates q', I = 1,2,.. . ,2n, on the manifold can be chosen in such a way that, after introducing complex local coordinates ui and U!, ,
1
q' = -(U'
2
1 . . qifn = -(U' - U!) 2i
+. U!).
-
i, i = 1, ..., n
(3.3.4)
the transition functions are holomorphic, u'i
=f y u )
(3.3.5)
in any non-empty overlap of two arbitrary charts. Given a positive metric ds2 = g f J ( q )dq' dqJ on the manifold, in complex coordinates it takes the form ds2 = 2gijdu' dUj + gij dui d d 9.. = (g..)* J!
+ gij dUi d d
9.. !! = g.. !! = (Sij)*.
(3.3.6)
The metric is said to be Hermitian, if it is of the form ds2 = 2giJu, U) dui dU!.
(3.3.7)
Under the coordinate transformations ( 3 . 3 3 , the components of Hermitian metric change according to the rule (3.3.8) In the case of a complex target manifold with some Hermitian metric, the action (3.3.1) is rewritten as
s
S[U, U] = - d4xgij(u,U)a"Ejd,~'.
(3.3.9)
One can look upon the actions (3.3.1) and (3.3.9) as nonlinear analogues of the kinetic terms in (3.2.28) and (3.2.27), respectively. 3.3.2. Supersymmetric a-models We are now going to generalize the superfield kinetic term in (3.2.23) to nonlinear case. Demanding the super Lagrangian to be independent of the Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace covariant derivatives of (anti)chiral superfields @ and form of the action superfunctional reads
223
ai,the most general
s
S[@, 61 = d8zK(@',@)
(3.3.10)
where K is a real smooth function of n complex variables and their conjugates. Since the integral over R4I4 of a chiral superfield vanishes (see equation 3.1.52)), K(@,@) is determined only modulo the transformations
+
K(@,@)-,K ( @ , @ ) A(@)
+ A($)
(3.3.1 1)
with A being an arbitrary holomorphic function of n complex variables. The action (3.3.10), which has been suggested by B. Zumino, defines a 'supersymmetric nonlinear a-model'. Evidently, the action is super Poincare invariant. Furthermore, the chirality-preserving reparametrizations @i + @ i
@!
=f'(@)
+ @'!
=f@)
(3.3.12)
leave the action invariant provided K(@,@) changes as follows K'(W, 5') = K(@,@)
(3.3.13)
modulo a transformation (3.3.11). Let us investigate the component structure of the theory (3.3.10). In accordance with the prescription (3.1.54c),the action functional is represented as
's
s
S = - d4xD2D2K(@,@)I = d4xL. 16
Defining the component fields of @' by the rule (3.2.25),a simple calculation leads to
(3.3.14)
where we have introduced the notation
-
Ki, ... i,,!,.. . !', =
a p + q K ( A ,2
)
a A i l , , . aAbaAi1.. . dAle
By comparing the obtained supersymmetric action with the nonlinear a-model action (3.3.9), it is seen that the dynamics of scalar fields A' and A! Copyright © 1998 IOP Publishing Ltd
224
Iric~trsonti Metliotis of Supersymmetry and Supergravity
is determined by a Hermitian metric of the special type y. = a 2 K ( A ,2). ?
(3.3.15)
?A'dA! Every such metric induces an interesting geometry known to mathematicians as Kahler geometry.
3.3.3. Kuii ler munifolds Let *// be a complex manifold with Hermitian metric gifu,U). Using the metric one can construct the second-order differential form
(3.3.1 6) which is called the 'Kahler form'. The exterior differential ofw is written as
The Hermitian metric is said to be 'Kahlerian' if the corresponding Kahler form is closed, dw = 0.
(3.3.17)
Then, A'is called a 'Kahler manifold'. The above requirement is equivalent to the equations (3.3.18) These imply that the metric can be locally represented in the form (3.1.15). The corresponding function K(u, U) is called a 'Kahler potential'. It should be pointed that K(u, U) is defined locally and only modulo the transformations K(u, a) -+ K(u, U)
+ A(u) + A(U).
(3.3.19)
Let us calculate the Christoffel symbols and the curvature tensor of a Kahler manifold. For this purpose we introduce 'real' coordinates cp' = { ui, U!} labelled by capital Latin letters. In these coordinates the metric and its inverse are (3.3.20) Here g! J. = yJ!'. g!j = yj! and y'iy !k = SL, g!'q.Ji= Si., In accordance with equation (1,6.12), the Christoffel symbols are defined by the rule
Copyright © 1998 IOP Publishing Ltd
Fiekd Theory in Superspuce
225
Making use of the explicit form of the metric (3.3.20)and equations (3.3.18) gives
(3.3.21) where equation (3.3.6) has been used. O n the same grounds, the other components prove to be vanishing,
r;!,= rikj _ _ = r!.J k = r!
jk
= 0.
(3.3.22)
In accordance with equation (1.6.14), the curvature tensor is a l j K L
= d&J
- d$-'KJ
+ PKErELJ -
rILErEKJ.
Looking at (3.3.22),one reads off =O
g\KL
.%'jKL
=O
(3.3.23)
hence .%'ijKL = g i ! @ j K L = 0 and W , j K L = g i i g ) K L = 0. Due to the properties of the curvature tensor, we also have g K L= i j0 and aKtij = 0. Therefore, the only non-vanishing components are: &?ijk!, g i j k gl l,j k ! and gij k l , which are related to g$!, gik1, &?'jk! and wijkl, respectidy. A straightforward calculation gives
(3.3.24)
Expressions (3.3.21,24) lead to
Then, taking into account equation (3.3.15), one obtains the result = Kikj!
aijk!
- g""iknKmj!
(3.3.25)
where we have used the notation adoted in the previous subsection. Now, it is obvious that 2ijk!
= *%kji! = gi!kJ:.
(3.3.26)
Relations (3.3.23,25) determine the curvature tensor of a Kahler manifold. Copyright © 1998 IOP Publishing Ltd
226
Iciecis cind Methods of' Supersymmetry and Supergravity
3.3.4. Kaiiler geometry and supersymmetric o-models Above it has been shown that in supersymmetric a-models, scalar fields couple via the Kahlerian metric and the Kahler potential coincides with the super Lagrangian in (3.3.10). So, every supersymmetric a-model defines some Kahler geometry. Remarkably, the converse is also true. Namely, with a given Kahler manifold one can associate a supersymmetric o-model. However, in this situation (anti)chiral superfields occurring in the action (3.3.10) take their values not in the Kahler manifold but in a supermanifold, in which the Kahler manifold is embedded. First, let us show how to embed a real n-dimensional manifold A" in a real supermanifold of dimension (n,0) (a supermanifold of dimension (p, q) looks locally like a domain of RPIq). The basic observation is that every smooth function of n ordinary real variables f(q'), defined on an open set VE R", can be extended to a supersmooth function of n real c-number variables f(o'),defined on the domain P = n:-'(V) c R:, where n: R: + R" is the map projecting any point (a')E Rg into its body,
n:((w'))=
= cp I .
The above mentioned supermanifold is a fibre bundle J,@ over base manifold A nwith fibre F = ;ER", and n: &"-+A"being the projection mapping. It is endowed with the structure of a supermanifold as follows. Choose an arbitrary chart ( U , q ) in A",where U is an open set of Anand q: U -, V c R" is a homeomorphism defining local coordinates q' on U . Next, we extend ( U , q ) to a chart (8,w ) in A",where 0 = l l - ' ( U ) and w: 8 -, P= n:-'(V) t R: is a homeomorphism, introducing local c-number coordinates a*on 0, which are taken to satisfy the relation q 0 ll = n: 0 a. If two charts U and U' have non-empty overlap with the transition functions being of the form (3.3.3), then their transition superfunctions for the corresponding charts U and U' are taken to be the supersmooth extensions of the function f'(q),
n:-'(d,
0 ' + "I
- f '(4. -
(3.3.27)
As a result, A@ turns out to be a supermanifold. Furthermore, given a manifold A2"endowed with the structure of a complex manifold, the corresponding supermanifold &2n also admits a as in equations (3.3.4,5) complex structure. Explicitly, if one parametrizes A2" and introduces local complex coordinates Qiand W on A2"
then the transition superfunctions prove to be (anti)holomorphic, qyi = f'(@) Copyright © 1998 IOP Publishing Ltd
(pi =
3@),
Field Theory in Superspuce
227
possesses a Kahlerian metric, it can be extended to a Kahierian Finally, if. /lzn metric on . /tzn. Now, we are in a position to give the geometrical interpretation of i' supersymmetric nonlinear a-models. Let dl be a Kahler manifold and d be the corresponding supermanifold. We introduce into consideration the family of superholomorphic mappings of the form cp:
c y +2
(3.3.29)
In local complex coordinates on ? every l such mapping looks like I
@(y, 0) = A'(y)
+ O'$:(y) + O2Fi(J>)
d@'/dy" = d@'/@
0.
(3.3.30)
When restricted from to the surface R414(00B) in C:12, on which ya = .Y" + i080, ai becomes a chiral superfield. So, we recover a-model superfield variables. Therefore, one can take the following point of view. The set of all superholomorphic mappings (3.3.29) is identified with the a-model space of superfield histories. The action superfunctional is taken in the form (3.3.10), where K ( 0 , is a Kahler potential related to the Kahlerian metric on by the rule
a)
g,,p, a)= a 2 ~ ( @ a)/aa+a@. ,
Note that K ( @ , a ) is locally defined on 2?. But the action (3.3.10) is well defined globally on R4I4,since the inherent arbitrariness (3.3.1 1) in the choice of the super Lagrangian coincides with the arbitrariness (3.3.19) in the choice of the Kahler potential. In conclusion, let us investigate the action of the transformations (3.3.12) (coordinate transformations on 2)on the component fields of @,'(z).One readily finds A'' = f ' ( A )
(3.3.31)
It is seen that $: transforms as a vector field on .a.The transformation law for F' is not covariant. Redefining F' by the rule (3.3.32) where
rijkare
the Christoffel symbols (3.3.21), one obtains the vector
Copyright © 1998 IOP Publishing Ltd
228
I0r~cr.su i i d Mt>iliodsof' Supersymmetry and Superyruvity
transformation law
(3.3.33) After the redefinition (3.3.32),the Lagrangian (3.3.14)can be rewritten in the form L = -qi,
i
?A!S,Ai - .F!.Fi+ - $
p -.
1
4
g
V,$!
)
1 + -9!ilk!$i$k$i$!
(3.3.34)
16
Here .JRiik! is the curvature tensor (3.3.25),and V, denote the target space covariant derivatives,
v& = &$; + rijL(?,N)$; va$i = ?,(ti + Ti.!! 2 &J:. a
We see that the supersymmetric o-model Lagrangian is expressible in geometrical terms.
3.4. Vector multiplet models 3.4.1. Mussive vector multiplet model
In Section 2.6 it was shown that the superspin-$ representation of the Poincare superalgebra can be described in terms of a real scalar superfield subject to the Klein-Gordon equation and a supplementary condition. Now, we present the supersymmetric model of an unconstrained real scalar in which the Klein-Gordon equation and the superfield V(z), V = supplementary condition arise as consequences of the dynamical equation. Let us consider the dynamical system with action superfunctional
s
's
S [ V ] = - d8zVD'D2D,V+ m2 d8zV2. 8
(3.4.1)
The kinetic term is real due to the identity (2.5.30d).Integrating by parts and making use of equation (3.1.26a), one can represent the action in the form
's
S [ V ] = - d6zW"W, 2
+ m2
s
d8zV2
(3.4.2)
= 0.
(3.4.3)
where we have introduced the superfield
w
"
1 4
= --D~D,V
Copyright © 1998 IOP Publishing Ltd
D,w,
Field Theory in Superspuce
229
Due to equation (2.5.30d),this chiral spinor superfield satisfies the reality constraint D"W, = D,m/".
(3.4.4)
Let us also point out the identity
(3.4.5)
To find the dynamical equation, one uses the relation (3.1.33).The result is 1
-D1D2D,V+ m 2 V = 0. 8
(3.4.6)
Acting by the operatorsDZorD2 on the left side and using equations (2.5.30f), one obtains D2V= D 2 V = 0.
(3.4.7)
Therefore, on-shell the superfield V is linear. Then, by virtue of equation (2.5.30e),the equation of motion is equivalent to
(0- m2)V= 0.
(3.4.8)
These are the equations (3.4.7,8),which determine the irreducible superspin-i representation.
Remark. Mass dimensions of V ( z )and W,(z) are [VI = 0
[W,] = 3/2.
Remark. The action superfunctional has been taken in the form (3.4.1)in order that the dynamical subspace was given by equations (3.4.7,8)as well as to have a standard mass term for the component vector field of V(z),
On-shell, only four component fields of V ( z )are non-vanishing (equations (2.8.22)). The multiplet of fields includes a transverse vector field. On these grounds, the theory under consideration is called the massive vector multiplet model. The superspin-i representation describes four particles of spins 0, i,$ and 1 . It should be pointed out that there are three super Poincark representations, of superspins i, 1 and 3,including vector particles. But only the former representation describes particles of spins not higher than one. Copyright © 1998 IOP Publishing Ltd
3.4.2. Massless vector multiplet model Let us set the mass parameter in equation (3.4.1) to zero. The resultant action (3.4.9)
turns out to be invariant under the following gauge transformations 1
V + V ' = V + -(A - A)
2
DJ
(3.4.10)
=0
where A(z) is an arbitrary chiral superfield. Making use of equation (2.5.30~) gives
w;= w,+ ~8 D Z D J= w,.
mi
We see that (anti)chiral superfields and W , are gauge invariant objects. W +and W, will be called 'superfield strengths'. The dynamical equation can be written in two equivalent forms D"W, = O o D h W * = 0
(3.4.11)
due to the identity (3.4.4). Now, it is time to recall the results of Section 2.7. Obviously, the above equations state that W, and Wlhare on-shell massless superfields and their superhelicities are K(W,) = 1/2
K(Wu) =
-1.
(3.4.12)
Therefore, W, describes two massless particles of helicities W +describes particles of helicities - 1 and -+.
4 and
1, while
3.4.3. Wess-Zumino gauge It is instructive to investigate the component structure of the theory. First, let us discuss gauge transformations. Since A is chiral, we can write A(X,
e, 8) = eiseoaO(u(x) + e,p,(X) + e 2 f ( x ) ) .
(3.4.13)
Here u(x) and f ( x )are arbitrary complex scalar fields and p,(x) is an arbitrary undotted spinor field. Then the transformation law (3.4.10) takes the form
6 V(X,
e, 8) = -i @(X) 2
-
1
- - exp&) 2
i
- -e 2 f ( x )
2
+ -2i B$(X) (3.4.14)
where dots mean terms involving partial derivatives and at least third order in 8, 8. Obviously, the component fields A , $, and F of our superfield (see Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
231
equation (2.8.19)) have arbitrary displacements, hence they can be gauged away. Then V is reduced to V = eoa8va
+ 8 2 m , + e*giR +
0 2 8 2 ~ ~
(3.4.15)
This gauge fixing, known as the 'Wess-Zumino gauge', can also be defined as follows VI = D,Vl = D2V1 = 0.
(3.4.164
Under these requirements, the component fields in the expansion (3.4.15)are given by the same relations as in equation (2.8.19), 1 V,, = 2 [D,,
DJ VI
I,,
14
= - - D 2D,Vl
(3.4.16b) Imposition of the Wess-Zumino gauge does not completely fix the invariance (3.4.10). The transformations preserving the above gauge correspond to the choice
w
A ( ~e,, 8) = e i ~ ~ t l ~ i t J
5 =T
(3.4.17)
with <(x) being a real function, and their explicit form is
+ a,(
i; = E,,
V : = I/,
D' = D .
(3.4.18)
Obviously, the gauge transformation of V , is the same as in Maxwell electrodynamics. As for the component fields of the superfield strength W,, they can be easily determined using equations (3.4.16). One finds W,I = E",
1 D"W,I = --{D2,D2}V/ = - 4 0 8
1 4
- -D2W,I
(3.4.20)
1 2
= -D,DPWp/ = -id,,P
where we have introduced the vector field strength Fa, = a,
- abV, (3.4.21)
Copyright © 1998 IOP Publishing Ltd
Therefore, the expansion of W , as a power series in 8 reads W,(.u, 0) = i.,
+ 200, + 2iF,lj6P - i?,i%12.
(3.4.22)
The Wess-Zumino gauge is especially helpful, due to the gauge invariance of S [ VI. for calculating the action functional. Following the prescription (3.1.54)gives
Making use of equations (3.4.20) leads to
I t only remains to recall the identities
j n
n
d4xF"PFZp= d 4 x P f iFi+ -
(3.4.23)
Then, the action functional takes the final form FnbFnb - ij.o"d,Z
+ 20'
(3.4.24)
Beautifully, the first term coincides with the action of Maxwell electrodynamics.
3.4.4. Supersymmetry transformations The action (3.4.9) is invariant under the supersymmetry transformations 6,V = i(EQ + @)V
(3.4.25)
having the form (2.8.20) at the component level. Every such transformation breaks down the Wess-Zumino gauge. Indeed, if I/ is under the constraints (3.4.16), then making a supersymmetry transformation generates nonvanishing component fields and F ,
IC/;
= -?"V,.,
F' = - E 2
in accordance with equation (2.8.20). So, the Wess-Zumino gauge is not supersymmetric. Of course, one can restore the Wess-Zumino gauge by accompanying the Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
233
variation 6,V with the €-dependent gauge transformation
6,,(,)V= L(A(€) - A(€)) 2
+
A(€) = eifhvOc’(, ( 2 i W ? ~ ~2i023).
(3.4.26)
The resultant supersymmetry transformation & V = i(EQ
+ $)V+
i 2
-(A(€) - A(€))
(3.4.27)
has not, however, the universal form (3.4.25).In other words, imposing the Wess-Zumino gauge modifies the superfield supersymmetry generators. As a result, the Wess-Zumino gauge, being quite useful for component calculations, does not fit into covariant (explicitly supersymmetric) superfield calculations. The supersymmetry transformation (3.4.27)acts on the component fields (3.4.166)in accordance with the rule
& V , = 6, V , = E ~ , X+ io,? &,E,.
1 8
= 6,& - -D,D2A(€)(=
-2iF,,&B
- 26,D
(3.4.28)
where we have used the relation (2.8.20).The last two expressions can also be obtained by applying results of subsection 2.8.2 to the chiral spinor superfield W,.
3.4.5. Super Lorentz gauge As we have seen, the Wess-Zumino gauge is not supersymmetric. One would like to have a covariant gauge fixing condition - that is, one commuting with the supersymmetry transformations (3.4.25). A useful gauge choice consists in imposing the linearity constraint (3.4.7), which restricts the component vector fields of V ( z )to be transverse (see subsection 2.8.4). On these grounds, this gauge is known as the super Lorentz gauge.
3.4.6. Massive vector multiplet model revisited
The massive vector multiplet model defined by action (3.4.1)possesses a hidden point: several component fields have non-canonical dimensions. Therefore, Copyright © 1998 IOP Publishing Ltd
Ideas and Methods of Supersymmetry and Supergravity
234
we should carefully define the component fields of V in order to avoid higher derivatives at the component level (in any case, such potential higher derivatives are not dangerous and can be eliminated by a field redefinition). Instead of using the definition (2.8.19), now it is more reasonable to introduce the component fields of V by the rule
1A ~ ( x= ) --D~D,VI
4
1
D ( X )= - D ~ D ~ D , v I .
(3.4.29)
16
With such a definition the component fields of W, remain unchanged, see relation (3.4.20). Hence, we can use our old result (3.4.24) establishing the component form of the first term in action (3.4.2). As to the mass term in action (3.4.2), its component structure is
m2
[d8z V 2 [d4 [ =
x
-
1 2
1
?au Sa, B + 2m B D $- 2GG - -m2
-iXaua,j
- m(hX
+ ii)} .
Vu Vu
(3 A.30)
It is seen that the, superfield mass term contains the kinetic terms of the component fields B and x . The component Lagrangian of the massive vector multiplet reads 1 1 1 1 L = - -FUbFub - -m2VuV, - -aUBauB- -m2B2 4 2 2 2 - i ~ d ' a , i - iXd"',j - m(AX i j >
+
1
+ 2GG + 2 ( 0 + -mB)2 2
(3.4.31)
Eliminating the auxiliary fields, the Lagrangian turns into 1 1 1 1 L = - - F U b F u b - -m2V"V, - -aUBauB- -m2B2 42 2 2 - \kD(iyUau m)\kD .
+
(3.4.32)
Here we have introduced the Dirac spinor (3.4.33)
The massive vector multiplet model can be reformulated as a gauge theory such that all component fields have canonical dimensions in a Wess-Zumino gauge. Such a formulation is based on a superfield generalization of the Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
235
Stueckelberg approach to the gauge covariant description of massive vector particles. We introduce a compensating chiral scalar superfield 0 ( z ) and its conjugate & ( z ) by making the replacement V - V + - ( @ - 6i ) 2m
(3.4.34)
in action (3.4.2). The resulting action superfunctional S [ V . 0,$1 =
/ /
1
+ m 2 d8z V2 d8z V ( O - 6) + 5 1 d8z 60
d6z W' W,
2 + im
1
(3.4.35)
is invariant under the gauge transformations 1
-
S V = -(A - A ) 2
(3.4.36)
S0 = m A
with A an arbitrary chiral scalar superfield. The gauge freedom can be used to completely gauge away 0.In the gauge
(3.4.37)
0=0
we returri to the original model (3.4.2). Instead of imposing such a gauge fixing, however, we are now able to choose the Wess-Zumino gauge (3.4.15). Then, the residual gauge invariance given by equation (3.4.17) can be further used to eliminate the real part of 01.As a consequence, it is in our power to impose the conditions VI = D,VI = D2VI = 0
0 1
+ $1
=0
(3.4.38)
which completely fix the gauge freedom. Gauge fixing condition (3.4.38) is most useful for passing to components. From action (3.4.35) one readily obtains the component Lagrangian 1 1 1 1 L = - -FUhFuh - -m2VUVu- -a'AauA - -m2A2 4 2 2 2 I i - iAa'aui - -$a'au$ - -m(A.$ - i$) 4 2 1 + L2 F F -k 2(D - -mA)2 2
(3.4.39))
where A = 2 = -i01. The component Lagrangians (3.4.31) and (3.4.39) are seen to coincide if we identify B = -A,,G = + F and x = ;$.
Copyright © 1998 IOP Publishing Ltd
236
Iciecis cincl Methods of Supersymmetry und Superyruvity
3.5. Supersymmetric Yang-Mills theories
The main goal of the present section is the construction of Yang-Mills theories possessing supersymmetry. Our strategy consists of finding superfield theories having the Yang-Mills form at the component level. The role of matter will be played by (anti)chiral scalar superfields since their component content is scalar and spinor fields only. The role of gauge object will be played by a multiplet of Lie-algebra-valued real scalar superfields V'(z) transforming by 6V' = :(A1 - A') 2
+ O(V)
DhA1 = 0
(3.5.1)
with A'@) being arbitrary chiral superfield parameters. This transformation law is reasonable on two grounds: (1) when reduced to the linearized level, it describes a set of free vector multiplets (see equation (3.4.10); (2) it admits the Wess-Zumino gauge
+ 826ui,' + 6*8$' + 6282D'.
V' = 80"8V,'
(3.5.2)
To start with, we consider an Abelian gauge model.
3.5.1. Supersymmetric scalar electrodynamics As is well known, an Abelian gauge vector field V,(x), transforming by the law SV,(x) = a,((x), arises as a compensating field when trying to make local the rigid phase invariance
A(x) -,e'*A(x) in the theory of a complex scalar field A(x) with the action
s
S = - d4xdmAamA. A gauge real scalar superfield V ( z ) ,transforming by the law (3.4.10), can be introduced in a similar fashion. Consider the massless chiral scalar superfield model (3.2.5). The action is invariant under the rigid U(1)-transformations
~ ( z+) ei%(z)
R = 0.
(3.5.3)
In trying to localize these transformations, one is faced with the following problem: the choice of R(z) as a real superfield is inconsistent with the chirality of 0, Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
231
n(z)= n ( z )=-Db(eiR@)# 0. Therefore, if one wishes to have a local transformation, the parameter should be chiral
0 --+ W = eiA@
DJ
= 0.
(3.5.4)
But this breaks the invariance of the action: iij@
-,a e i ( A - A)@
Following the ideology of gauge field theory, we introduce a compensating real scalar superfield V , with the transformation law (3.4.10), and change the action to sd8z@eZv@.
(3.5.5)
Obviously, this superfunctional is gauge invariant. Note that the transformation law (3.5.3) corresponds to unit U(1) charge. In the case of a chiral scalar superfield, having U(1) charge q, the above law reads @(z)--t e'q*Wz).
(3.5.6)
This leads to a slight modification of the gauge invariant action (3.5.5) consisting in the insertion of q into the exponential, resulting in
S[@,
a; V ] =
s
d8z@e2qv@.
(3.5.7)
The action is invariant under the gauge transformations @ = ei4A@ e2qV'
=
DkA= 0
eiqAe2qVe-iqA
(3.5.8)
Adding the free vector multiplet action (3.4.9),the total gauge invariant action takes the form
s
'S
S = d8z@e2qv@+ - d6zW aW,. 2
(3.5.9)
Unexpectedly, the action turns out to be non-polynomial. However, in the Wess-Zumino gauge (3.4.15), where V 3 = 0, the exponential terminates. Let us determine the component form of the action (3.5.9) in the Wess-Zumino gauge. Using equations (3.4.16) and defining the component fields of 0 by rule (2.8.4), one obtains the Lagrangian Copyright © 1998 IOP Publishing Ltd
238
Ideas L=
{
iind
- (2"
Methods of Supersymmetry and Superyruviiy
+ iqk")A(Z,
- iqV,)A
i + 2qAAD - -$a"(a, + iqV,)$ 2
After eliminating the auxiliary fields F and D, this takes the form i L= - (8" iqVm)3(2m- iqV,)A - Lq2(AA)2- -$a"(a, + iqV,)$ 2 2 (3.5.11)
+
One can look upon this model as a supersymmetric generalization of scalar electrodynamics. The Lagrangian (3.5.1 1) is invariant under the gauge transformations A' = ek5A $; = e'qC$% (3.5.12) i.j, = &. V i = V , + 2,t Using the spinor component field of 0,we can construct three four-component spinor objects: (1) the Majorana spinor field (3.5.134 (2) the left Weyl spinor
YL = (
(3.5.13b)
Z X )
(3) the right Weyl spinor (3.5.13~)
In accordance with equation (3.5.12), IM, I, and IR transform as follows y 'M -- eiq5zyrM yrL -- ei~SyrL y'R
--
e-'YSyr
R*
(3.5.14~) (3.5.14b) (3.5.14c)
Correspondingly, the gauge invariant spinor kinetic term in equations (3.5.10, 11) can be rewritten as Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
1
- -$oo((a, 2
+ iqV,)$
1:
1 -
=
- -yLy"(d, - i q V , V L 2 1-
- -Y,y'((d,
2
+ iqV,)Y,
239
(3.5.15b) (3.5.15~)
Therefore equation (3.5.12) describes local chiral or y,-transformations. We see that the theory under consideration describes chiral fermions at the component level. 3.5.2. Supersymmetric spinor electrodynamics To obtain a Dirac spinor field in component form, let us consider two chiral scalar superfields @+ and 0-with U ( ] )charges 4 and -4, respectively. Their spinor component fields
D,@+I transform according to the laws $+1=
$'+1= eiqt$+z
= D1@-
$-a
$'-%
= e-iqC+
1
in accordance with equation (3.5.12). Then, the Dirac spinor (3.5.16 ) is characterized by the transformation law D --
e k t yD
(3.5.17)
inherent to spinor electrodynamics. The most general (renormalizable) gauge invariant action for @+ and @-, coupled to a gauge superfield I/, reads SsED = (dsz{G+e2'7v@++
+m
s
- II
s
d6z@+@-+ m d6FG+@-+
(3.5.18)
Note that a mass term is forbidden in the case of a single chiral scalar superfield with non-zero U(1) charge. In components, the above action is SSED = jd4XLSED
Copyright © 1998 IOP Publishing Ltd
(3.5.19)
240
Ideas und Meihods of' Supersymmetry and Superpacity
where we have introduced the Majorana spinor
Obviously, the model obtained represents a supersymmetric extension of spinor electrodynamics. In conclusion, we rewrite the action (3.5.18) in a different form. We unify a+ and 0-into a two-component column
(3.5.20) and make the redefinition
(3.5.21) After this, the action takes the form
s
SsED= d8zie2qVo2 x + (Sd%(
5 x T +~
where o2 is the second Pauli matrix. The transformation law of
x-
= eiqAo~
x
is
(3.5.23)
In summary, we have constructed the supersymmetric generalization of spinor electrodynamics. 3.5.3. Non-Abelian gauge superfeld We now generalize the above results to the non-Abelian case. Let G be a compact connected Lie group (in general, G is the product of a finite number of simple and U(1) group factors). Given a finite-dimensional unitary G-representation, we denote by where I = 1,2,.. . ,dim G, its Hermitian generators: Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace [T', T J ]= if I J K T K
(TI)' = TI
241 (3.5.24)
and the structure constants are assumed to be totally antisymmetric. Any operator of the representation is written in the form
since every compact connected Lie group is covered by the exponential mapping. Consider a system of free massless chiral scalar superfields @'(z).The action S
=
J
S
ai= (@')*
dszai@' = ds&@
is invariant under the rigid transformations = eiq'T'@
R,.
As in the Abelian case, when attempting to localize these transformations, the parameters are to be taken chiral,
In addition, it is necessary to modify the action by introducing a compensating multiplet of real scalar superfields V1(z)transforming by the rule e2v' = eiA e 2 V e -iA
The gauge-invariant action is
s
I/= I/+= I/'TI.
s
S = d*zGi(e2v)ijW'= daz@e2'@.
(3.5.26)
(3.5.27)
Lie-algebra-valued Hermitian superfield V with the transformation law (3.5.26) is said to be a 'gauge superfield' (or 'Yang-Mills superfield'). The transformations (3.5.25,26)will be called 'supergauge' transformations. The equation (3.5.26) shows that I/' is Hermitian, (V)' = V'. Next, due to equation (3.5.26) and the Baker-Hausdorff formula (2.1.62), V' has the form V' = V'IT'. To see the explicit connection between V" and V', let us consider infinitesimal gauge transformations (3.5.26). 3.5.4. Infinitesimal gauge transformations In the infinitesimal case, equation (3.5.26) takes the form & 2 V = e2(V+S V ) - e2V = iAe2V - ie2vA.
(3.5.28)
Our goal is to determine 6V in terms of V , A and A, For this purpose we use the identity: Copyright © 1998 IOP Publishing Ltd
242
Idem
utid
Methodx of Supersymmetry und Supergravity
(3.5.29) where A and B are arbitrary operators and E is an infinitesimal parameter. This identity can be easily proved with the help of the auxiliary function K(r, 6 ) = e-TAeT(A
-t C 5 )
which satisfies the equation
Defining operator LAby the rule
LAB
[A, B]
the relation (3.5.29) can be rewritten as eA
+ 8 5 = eA { I + ( L ~ ) - ~-( Ie-L41E B f .
(3.5.30)
Now, making use of equations (3.5.28,30)gives
(LV)-l(l - e-2L”)V= ie-2LVA- iA and acting on both sides by LyeLr
2 sinh(LV)6V=iLv (e-LvA- eLVA) = iL, cosh (&)(A - A)
- iLv sinh (Lv)(A+ A).
This leads to
6V=
1
--Ly
2
1 -
i 2
(A + A) + - L v coth (&)(A - A) 1
=-(A-A)--[V,A+A] 2 2
+O(V4).
+i[V,[V,A-A]]
(3.5.31)
In the Abelian case, this transformation law reduces to expression (3.4.10). We see that the supergauge transformations are highly nonlinear. However, since 6 V = $(A - A) + O ( V ) , these transformations can be used to impose the Wess-Zumino gauge. Those supergauge transformations which preserve the Wess-Zumino gauge are described by the parameters ~ ( x0,g) , = e i ~ ~ U ~ ~ 5~=~5’~ ( = x )
(3.5.32)
+
They act on I/ under the Wess-Zumino gauge condition by the rule
i 6 V = ‘(A - A ) - -[V,A 2 2 Copyright © 1998 IOP Publishing Ltd
+ A]
(3.5.33a)
Field Theory in Superspuce or, in components, = dot' -
yf
243
KJIV,J
6jVz'= -
(3.5.33b)
As for the matter multiplet 0, defined by equation (3.5.25), the transformation
(3.5.25) acts on the corresponding component fields
as follows 6Ai = i
(3.5.35)
Therefore, we recover ordinary Yang-Mills transformations. In accordance with equation (3.5.33b3, Vf2 is a gauge Yang-Mills field. Its superpartners 2; and D' transform, due to (3.5.33b), in the adjoint representation of the gauge group.
3.5.5. Super Yung-Mills action We proceed by finding the gauge invariant action for the gauge superfield. Let us introduce the following Lie-algebra-valued spinor superfields 1W, = - - D2(e-2VD,e2V) = W,'T' 8
(3.5.36) and study some their properties. First, in the Abelian limit W, reduces to the form (3.4.3). Secondly, W, is chiral and W kis antichiral, DkW, = 0
DzWk = 0.
(3.5.37)
Finally, W, and Pkchange covariantly under the supergauge transformations (3.5.26).Explicitly, we have
Since D2D,q = 0 for every chiral superfield q, we have the result Copyright © 1998 IOP Publishing Ltd
244
Idws (inti Metliotls of' Supersymmetry und Supergruviry W ; = e'AW,e-'A
- = eiiim*e-i& w:,
(3.5.38)
W, and W'+ will be called the 'Yang-Mills superfield strengths'. The relation (3.5.37) tells us that the (anti)chiral scalar superfields tr(W'&W)
tr(W'W,)
where tr denotes matrix trace, are invariant with respect to the supergauge transformations. Therefore, the superfunctional d62 tr ( WlW,)
+ C.C.
(3.5.39)
where g is a coupling constant, can be taken in the role of the gauge invariant action. Above we have supposed that the generators T' are normalized by tr ( T ' T J )= dIJ.
(3.5-40)
Let us investigate the action at the component level. Imposing the Wess-Zumino gauge, W , takes the form W, = - 'D2D,V+ .f-b2([V,D,V]) 4 4 To calculate the component content of W,, we point out that the Wess-Zumino gauge is characterized by the conditions (3.4.16). Then, one readily obtains
W,l = A, D"W,I = - 4 0
D(,W,,/ = 2iG,,
(3.5.41)
Here we have introduced the Yang-Mills field strength: G,b = a,V, - d,V, - i[V,, Vb] 1 1 1 G,, = 2 ( ( T ~ ~ ) , P G=, ~- - 8(,a . V P, + - [Veri, Vp']
'
(3.5.42)
and the Yang-Mills covariant derivatives
V,B' = 8,~' - i vOr( T')ljBj V,A" = d,i." - i[ V,, i."]. Based on the above expressions, one finds Copyright © 1998 IOP Publishing Ltd
(3.5.43)
Field Theory in Superspace
14 I d % tr ( WzW,) =
s
i :
245
1 d4x tr - - GZaG,p - - iE.@VzkI2 D2 2
+
Now, it is helpful to recall the identity
1
J
d4x tr (G'BGaa) = d4x tr (GkfiGifi)
modulo total derivatives. This means, in particular, that the relation
s
d6z tr (W'W,) = d6Z tr (W6W2)
(3.5.44)
holds in the Wess-Zumino gauge and, due to gauge invariance, in the general case. Finally, we can write the component action
- il,aaVaX+ 2D2
1
.
(3.5.45)
As is seen, we have obtained the supersymmetric extension of the Yang-Mills action.
3.5.6. Super Yung-Mills models Gauge invariant coupling of matter to a Yang-Mills superfield is described by an action superfunctional of the general form
s
{
S = daz$i(e2Y)ijW + Sd6zLfc(Qi)+ c.c.] (3.5.46)
's
+2g2
d6z tr (W'W,).
Here the chiral superpotential LfC(Oi)should be a group-invariant function,
a 2 ( T J y j @= 0. aa,
(3.5.47)
In particular, if the chiral superpotential is of the form 1 YC(@i) = -mij@i@
2
1 + -A..
3!
'Jk
@irpi€)k
(3.5.48)
the coupling constants mij and &,k should be invariant tensors of the group under consideration. It is instructive to analyse the component form of the action. Defining the component fields of mi by rule (3.5.34)and imposing the Wess-Zumino gauge condition (3.4.16)on I/, one arrives at the action functional Copyright © 1998 IOP Publishing Ltd
246
Itkccis
-
tiiiri
AJT')',;.'$' 1
- -(
gl
Methods of Super.sjqrnnietry und Supergrucity
+ F,F' +
4 c^A'dAJ
+
L G U ~ I G , , ~ ij.IQaVa7 ~ - 20'0')
where f , ( A ) is given by equation (3.2.24). After eliminating the auxiliary fields F'. F, and D', the scalar fields have the non-negative potential U ( A , A) = 1 1Y*i(A)I + g2 -C(AT'A)'.
(3.5.50)
2 1
I
3.5.7. R e d rrpresmtutions In conclusion, we would like to discuss one important question similar to that arising when constructing supersymmetric spinor electrodynamics. Namely, let YDi' be a multiplet of Dirac spinors transforming in some (complex) unitary representation of the gauge group, ~ \ Y ~=Ii <' I i r I ) P l y D l
(r')+ = y'
(3.5.51)
coupled to the Yang-Mills field in the standard fashion: S
'1
= --
2
d4xTD,(iy"Va
+ m)YD@'.
(3.5.52)
Suppose, one would like to find a super Yang-Mills model of the form (3.5.46), in which, after reduction to components, the spinor fields from ai and G j can be combined somehow into Dirac spinors such that the spinor part of the action (3.5.46) coincides with expression (3.5.52). The question is the must following: what is a group representation by which the multiplet {aL} transform? Let us decompose VDflinto a combination of two Majorana spinors:
(3.5.53)
Introducing a multiplet of Majorana spinors in the manner
(3.5.54) it transforms by the real representation:
(3.5.5 5 )
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
247
Since 9'are Hermitian, the matrices R e F ' are symmetric and the matrices lm,Fr are antisymmetric. As a result, T' are antisymmetric Hermitian matrices, (T')T = - T'.
(TI)' = T'
(3.5.56)
Now, it is clear how to resolve the problem raised above. First, one must take a set of chiral scalar superfields {ai)= (W,OP) transforming by the introduced real representation. Secondly, the multiplet (3.5.54)should be identified with the spinor component fields of ai and ai by the rule
Then, the gauge invariant matter action
s
S = d8:Ge2V0
"is
+ -2
d6:0 '0 + C.C.
(3.5.57)
leads to the desired dynamics of spinor fields. Therefore, in order to have a complex representation in components, one must use its real realization in superfields. When taking a complex representation for superfields, there is room for chiral fermions at the component level and, hence, for chiral anomalies in the quantum theory. In what follows, we will often restrict ourselves to the consideration of real group representations for a description of matter chiral superfield multiplets. As applied to the gauge superfield V = V'T', this requirement means V+=V
V'=-J!
(3.5.58)
3.6. Geometric approach to super Yang-Mills theories In this section we intend to give a clear geometric interpretation of supergauge transformations and the gauge superfield. This will be done after reminding ourselves of some relevant mathematical notions. 3.6.1. Complex and c-number shells of compact Lie groups
In formulating super Y ang-Mills theories, there arise several groups: the gauge group G, its complexification G' and their c-number shells 6 and ec. We are going to describe these objects.
Copyright © 1998 IOP Publishing Ltd
We begin with an excursion into the theory of Lie groups. Let G be a compact connected n-dimensional Lie group, and Y be the corresponding Lie algebra. The exponential mapping exp: 9 -+ G turns out to cover G (due to compactness and connectness), therefore real coordinates q'(1 = 1,. . . ,n ) in 59 with respect to a basis { E ' } , X
~ X=iq'E'
E
ER
(3.6.1)
can be used to parametrize G, g(q) = exp (#E').
(3.6.2)
It is worth noting that the map exp is on to but not one-to-one. Generators E' are assumed to satisfy the commutation relations = if'JKEK
[E',
(3.6.3)
with the structure constants f I J K being real and totally antisymmetric. Introduce the complex shell Y' of the algebra 9. The Yc is a complex n-dimensional Lie algebra arising from Y by taking the coordinates to be complex, X E ~ ' X = iw'E'
(3.6.4)
w'~@.
Supplying 9' with the operation of complex conjugation (involution) X + X * = ia'E'
a' = (a')*
(3.6.5)
makes it possible to identify the initial algebra 9 with the real subset of '3". The algebra Y' can be equivalently treated as a real 2n-dimensional Lie algebra, denoted by (Yc)R,with elements i(Re o'E'
+ Im 0's')
and generators E' and S'( = iE') under the commutation relations [E', E J ] = if'JKEK
[E',
sJ]= if'JKSK
[s',
sJ]
= -if'JKEK,
Then, the conjugation map *: Yc 9' acts as an automorphism on (Yc)R. This automorphism has a transparent meaning in the basis --f
(3.6.6) with respect to which every element of (Yc)Ris written i(o'Ei+,
+ a'Ef-)).
(3.6.7)
It is seen that (Yc)Rhas the direct sum structure, (9c)R = ?I @; Yt-.), +and ) on to S f - , and vice versa. the complex conjugation maps Y;+) There exists a unique (up to isomorphism) complex connected Lie group 9' satisfying two requirements: Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
249
(1) 9' is the Lie algebra of G'; (2) considering G' as a real 2n-dimensional Lie group, G is isomorphic to the subgroup in G' corresponding to the subalgebra 3 in 9'. G' is said to be 'the complex shell' of G (for example, if G = SO(n), then G' = SO(% C); if G = Sti(n), then G' = SL(n, C)). In a neighbourhood of the
unit of G', every element can be represented in the exponential form g(w) = exp (io'E')
(3.6.8)
The variables w' play the role of local complex coordinates on G', at least in a neighbourhood of the unit. When treating G' as a real Lie group, it is parametrized by variables w' and their conjugated 6' as follows
+ G'E:-JJ
g(w, 6)= exp [i(w'E:+, = exp (io'Ef,
))
exp (i6'E;- ,).
(3.6.9)
Therefore, G' possesses the direct product structure, G' = GP+) x GP-), where the subgroups Gf+, and G f - ) are mutually conjugated. The initial group can be identified with the diagonal in this product. Note that G' is non-compact, in contrast to G. Now, let us discuss c-number shells of the groups G and G'. Starting from the group manifold G, we embed it in the supermanifold G using the rule described in subsection 3.3.4. is naturally endowed with the structure of a real super Lie group which is induced by the Lie group structure on G (see also the discussion of supergroups in subsection 2.1.6). Elements of are parametrized as in equation (3.6.2),but in contrast to G, the local coordinates q' on are real c-numbers. Similarly, the group manifold G' is embedded in the complex supermanifold G', which becomes a complex super Lie group after inducing the group structure from G'. Elements of e' can be represented in the form (3.6.Q where w' are complex c-numbers. Every unitary finite-dimensional representation F of the compact group G with Hermitian generators T'( = dF(Ez)),
c
Y ( g ( q ) )= eiqJT'
(T')' = T'
(3.6.10)
can be continued to a holomorphic representation Y(g(o))= eiwIT'
or antiholomorphic representation g(g(o))e - iGJ(TJ)*
(3.6.11)
(3.6.12)
of the complex group G'. Choosing in equations (3.6.10-12) parameters $(U') to be real (complex) c-numbers, one obtains representations of G and respectively.
e',
Copyright © 1998 IOP Publishing Ltd
250
Idem und Methods of Supersymmetry and Supergravity
3.6.2. K-supergroup and A-supergroup The basic symmetry principle underlying Yang-Mills theories is invariance under local (or gauge) group transformations. Every gauge transformation is specified by a smooth mapping of Minkowski space into the group G, X : R4 + G. (3.6.13) In classical field theory, when fields are treated as Grassmann-algebra-valued functions on a space-time, it is more correct to consider mappings from Minkowski space into the c-number shell 6 of G,
X:R4+G
(3.6.14)
in the role of gauge transformations. The set of all such mappings forms an infinite-dimensionalgroup (with respect to the multiplication law (XI . X,)(x) = X , ( x ) X 2 ( x ) called the local G-group. It is the gauge group of classical Yang-Mills theory. Trying to keep a literal analogy with ordinary Yang-Mills theories, it may seem natural that in the supersymmetric case the role of gauge group should be played by the supergroup consisting of supersmooth mappings from the superspace [w4I4 in G, X : R4I4+ G.
(3.6.15)
This group will be called 'the superlocal' G-group (or the 'K-supergroup'). Every element of the K-group is determined by a multiplet of real superfields ~ ( xe,,0):
X ( X ,8,0) = exp (iK'(x, 8, 8)E')
R'(z) = Kr(z).
(3.6.16)
Unfortunately, the K-supergroup has not arisen in the previous section. The point is that in the supersymmetric case there exists another natural candidate for the role of gauge group. Recall that the complex truncated superspace C;l*, rather than the real superspace R4I4, is the fundamental object in supersymmetry; R4I4 is embedded into C;I2 in the super Poincarit invariant fashion. Therefore, it seems reasonble that C;12 should be taken in the role of the arena upon which rigid (?-invariance is promoted to a local invariance. Furthermore, localization should be done in a way consistent with the complex structure on C;I2. This means that we change to its complexification (?' and consider superholomorphic mappings of the form
e
Every such mapping is determined by a multiplet of holomorphic superfields ATY, e), Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspuce
(3.6.18)
i.(y, 0) = exp (iA'(y, @E')
which become chiral when restricted from Ctl'to R4I4(setting4 = x" i ( z ) = exp (iA'(z)E')
251
D,A'(z) = 0.
+ i688) (3.6.19)
and A2, their product (i. . i.Jy, , 0) = Given two superholomorphic mappings i, 0) is also superholomorphic. Hence, the set of all mappings (3.6.17) forms a supergroup, which will be called the 'A-supergroup'. The A-supergroup is the invariance group of the super Yang-Mills action (3.5.46). j.,(y, O)j&
3.6.3. Gauge superfield Now, we are in a position to demonstrate, following E. Ivanov, the geometrical origin of the gauge superfield. Consider the complex supermanifold + '12 = ~
Jd4
( 3.6.20)
4 1 2
which can be parametrized by the complex coordinates ya, 6" and 0' corresponding to C4I2and respectively. The A-supergroup naturally acts on Jl4 'I2 by means of left nonlinear shifts of the coordinates U':
e',
+
y'" = y"
0'' = 8"
g ( d ) = exp (iA'(y, B)E')g(w).
(3.6.21)
In addition, one can realize the K-supergroup as a transformation group of ~ 4 + @acting by right nonlinear shifts of the coordinates U':
ya=
= ,g"
g(w')=g(w)exp( -iK'(iy+zjj,0,8 1 1
(3.6.22)
It is clear that the A-transformations commute with the K-transformations. Next, we embed the real superspace R4I4 into A4 'I2 as follows: +
+ i9o"B = R'(x, e, 8)
y" = xa
w'
(3.6.23)
where R' is the complex scalar superfield which determines the embedding. In accordance with equations (3.6.21,22), the A- and K-supergroups act on R' according to the law exp (iR'(z)) = exp (iA(z)) exp (iR(z))exp ( - iK(z))
(3.6.24)
where we have introduced the shorthand notation: R = R'E' and so on. Let us discuss the transformation law (3.6.24).We decompose R' into real and imaginary parts. Obviously, real superfields ReRI and ImR' live in the subgroup 6 of GC and the coset space &/G, respectively. Since the Copyright © 1998 IOP Publishing Ltd
252
Idem and Methods of Supersymmetry und Supergruvity
K-supergroup acts on exp (in) by arbitrary local &transformations, one can gauge away Reo' by a proper choice of K-transformation. Therefore, the superfields Re R'(z) are purely gauge degrees of freedom. Imposing the gauge fixing condition
(3.6.25) ReR'(z) = 0 we work only with the A-supergroup which acts on the multiplet of real scalar superfields V'(z)
Im n'(z)
(3.6.26)
&/e.
spanning the coset space Hence, the K-invariance is auxiliary. Its meaning is that &directions of the embedding (3.6.23) are never observed dynamically. To find the transformation law of V'(z), it is worth noting that, to preserve the gauge (3.6.25), every A-transformation should be supplemented by a A-dependent K-transformation: exp ( - V ' ) = exp (iA) exp ( - V )exp ( - iK[ I/, A])
(3.6.27)
where K'[ V , A] are rather complicated functions. Instead of finding their explicit form, we take another course. In the gauge (3.6.25),we have exp (- 2 V ) = exp (in)exp (-in).
(3.6.28)
In the general case, this relation defines a set of real scalar superfields V'(z) (the reality of I/' follows from the fact that the conjugation map f:g(w) -, g ( 6 ) represents an antiholomorphic isomorphism of Gc on itself, f(g(w1)g(w2)) = g(G1)g(6&. Equivalently, in the expression of (3.6.28) the imaginary part of R is excluded by construction. Next, taking the conjugate of (3.6.24)gives exp (in,)= exp (iA) exp (in)exp (- iK).
(3.6.29)
Finally, from the relations (3.6.24,28,29)one obtains exp ( - 2V') = exp (iA) exp ( - 2 V )exp ( - iA).
(3.6.30)
As may be seen, we have recovered the transformation law of the gauge superfield (3.5.26). Therefore, the Yang-Mills superfield can be identified with the multiplet of real scalar superfields V'(z) residing in the coset space Gc/G and transforming by the law (3.6.30) under the A-supergroup (clearly, this interpretation differs drastically from the well-known geometrical interpretation of ordinary Yang-Mills fields). It is worth emphasizing, however, that in the geometrical approach developed the real multiplet V' represents the gauge-fixed version of the initial complex multiplet RI. This observation serves as the foundation for the following point of view of super Yang-Mills theory: ( 1 ) the full invariance group is the direct product of the A- and K-supergroups; Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
253
(2) the true dynamical object is the multiplet of complex scalar superfields O'(z) residing in Gc and transforming by the law (3.6.24) under the invariance group. Taking the above point of view, one can see a close analogy between formulations of two quite different theories: the super Yang-Mills theory and gravity. Objects corresponding to each other in the super Yang-Mills theory and in gravity, are given below: O'(z) c* e,'(x)
V'(Z) * gmn(x) K-supergroup c* local Lorentz group A-supergroup tt general coordinate transformation group e-2V
=
in
-iR
a
b
e e * g m n = em en v a b . In fact, the analogy can be promoted further. As is known, the vierbein e,'(x) plays the role of a converter which turns curved-space tensors (transforming under the general coordinate transformation group) into tangent-space ones (transforming under the local Lorentz group). The vierbein's superanalogue Q'(z) is seen to have a similar meaning. Consider a multiplet of chiral scalar superfields Qi(z) transforming according to the law (3.5.25) under the A-supergroup. We convert this multiplet into the nonchiral one
@ir) = (e-in)ijw' In accordance with equation (3.6.24), @{r)
=
Q = O'T'.
(3.6.31)
changes according to the law K = KIT'
(3.6.32)
under the K-supergroup and is unchanged with respect to the A-supergroup. Next, converting the antichiral multiplet in the manner
ai
one arrives, due to equations (3.5.25)and (3.6.29),at the transformation law (3.6.34) Surprisingly, it follows from equation (3.6.28) that the action (3.5.27) can be rewritten in the free form (3.6.35) as if the interaction were absent. Copyright © 1998 IOP Publishing Ltd
254
Idecis m i Methods of' Suprrs~~mmetry und Superyrncity
3.6.4. Guuye covuriunl derivutives Gauge superfields R' and Cl' can be used to construct gauge covariant derivatives-that is, first-order differential operators preserving the transformation laws of matter multiplets. In the rest of this section, we restrict ourselves to consideration of real G-representations in which the generators satisfy equation (3.5.56). Introduce some matter multiplet @'(z)and its conjugate @(z) (the G index is given in the upper position in accordance with the reality of the G-representation), transforming under the A-supergroup by the laws (3.6.36~)
0 + 0'= @T+GTt
= e1A-T @
(3.6.36b)
Taking a covariant derivative of 0 breaks the initial transformation law, + DAO' #
D,O
eiAD,O.
To preserve the transformation laws, it is worth changing D, by new sets and Sa-) of the form of operators 97)
with rg*)(z) being 'superfield connections'. One demands that the A-supergroup act on ai.) in such a way that the following relations cj$+)@ A -,C J ~ + ) W = e l A ~ A( + ) @ g$-)ifT
+
9 A(-@T
(3.6.38~)
= e l A ~ A( - l a T
(3.6.38b)
(+)e-lA
(3.6.39a)
hold. This means 6&+)
CJ(-) A
-, +
91') = e
95-1 = e l A g p e - l A
(3.6.39b)
The operators 95,) are said to be 'gauge covariant derivatives'. A particular solution of the above problem, based on the use of the gauge superfield V', is given as follows: g ( += ) e-2VDle2V g$+)= D+ (3.6.40~1)
(3.6.40b)
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
255
Since D,A = D,A = 0, making use of equation (3.6.30) leads to 9;+] - e - 2 V ’ ~ e2V‘ - e i Ae - 2 V e -iAD . eiA e2 Ve -iA I(
= eiA 9% (+) e-iA
= eiAe-2V~ae2Ve-iA
a$+) = Ddr= eiADdre-iA
= ei Ag- (d+ ) r e-iA
.
Note, if Q, is chiral, then it is gauge covariantly chiral with respect to @+). It is worth noting also that the two sets 9 y )and 9L-) are connected by (3.6.41)
g(+) A = ~ - ~ V C J JA( - ) ~ ~ V .
In the general case, the gauge covariant derivatives (3.6.37) satisfy the algebra
[ay)? @B+ ) } = 9(i)@+) - (A B -
~
~
~ CC
g+(
f)ensB@f)&)
B
A
(3.6.42)
iF(k) f A )B
F“’AB = p(*)(z)p* AB Here TABCis the supertorsion tensor defined in terms of the covariant derivatives D A : CDA,
=
TABCDC.
F‘& is said to be the ‘supercurvature tensor’. It is expressed in terms of the connection superfields by the rule: F (AB+ ) = DA p Bk ) - ( - 1 y DB r(+) A + i [ r (Af ), rCi) E },
(3.6.43)
It is evident that F!.&) transforms covariantly under the A-supergroup (in the adjoint G-representation): (3.6.44~)
~ y-,pi) ~ )= ,iAFyJe-iA ~a;) -,~2;) = eiAFL;)e - iA.
(3.6.44b)
In the case of the derivatives (3.6.40), the algebra (3.6.42) is {a;*) g(.*)} = -2i9(&) > a U 1
a”’}= 0
{g;*x”,gb”}= {GL+),
[ ~ i *~) , $ =3 2ighpwF) [a(+, 9($]= 2iEZPW(7 1
[&I
(3.6.45)
’
,g (p\j *]) = -&x&2y7)j+)) -&,&BL*)wp).
Here the supercurvature components WL?) and superfield strengths (3.5.36) by the formulae
wh“= w
Copyright © 1998 IOP Publishing Ltd
wr)=
Wi+)are connected with the
e-2vm.e2v 1
(3.6.46~)
256
tikw iind Mciitoiis of' Supersymmetry und Supergravity (3.6.46b)
Wi-1 = W h.
WL-) = e 2 v w , e - 2 v
Let us comment on the derivation of the relations (3.6.45). As a result of (3.6.404 we have
p+= ) -i(e-*"~,~~v)
q+) = o=> p t ) = ? ~ , r ( + ) , 2
This immediately leads to
z
x
[Gr', 9)#]= 2ichpWP. Similarly, one readily obtains [gL-)g(7i pp 1 = 2i&,/jW/j. 1
Then, making use of the relation (3.6.41), one obtains the commutators [C//h", 9)$]and [Gi-), Finally, a commutator of vector derivatives can be calculated by the rule
a)$].
[ g ( +a(+) ) i 3 = -[{9p, @*)}, ,,, ,@ 2
9gJl
i i ],9i*)} + (@+), 2 It is worth pointing out that this expression should be antisymmetric with respect to the double transposition r t + pand h-4. On the other hand, from expressions (3.6.45) we see that
[Qy, 4j7]].
= - {[g',",~$*lp,]
This requires the identity The reader can check this explicitly. Remurk. Since the superfield strengths W, and W hare chiral and antichiral, respectively, it follows from the relations (3.6.40,41,46) that the supercurvatures W'," and Wi*'are gauge covariantly chiral and antichiral, respectively, ZJ!,+)w!+) = 0
c$+JW(,*) 2 1
=0
(3.6.48)
In conclusion, we introduce gauge covariant derivatives 9,for a matter multiplet transforming under the K-supergroup by the law (3.6.32).They are given in the form 9, = e-lRgy)eiQ = e - ~ n q \ - ) e ~ n 2 x= e-'nD,e'n Copyright © 1998 IOP Publishing Ltd
a* = e-'Rbheln
(3.6.49)
Field Theory in Superspace
257
1
C/?jl
= - ; g 2'?z} ,
2 where we have used equations (3.6.28). In accordance with equation (3,6,24,29), the operators 9" are characterized by the transformation law
9,-+ 9; = eIKgAe-IK.
(3.6.50)
3.6.5. Mcitter equutions of motion Gauge covariant derivatives arise in dynamical equations for matter multiplets. For example, consider the dynamical system (3.5.57). It is characterized by the equation of motion
(3.6.51) From this equation one obtains
Recalling definition (3.6.40), we can rewrite the equations in terms of the gauge-covariant derivatives:
(3.6.52)
9k-)aT
= 0, these fourth-order equations can be Now, since Gr)Q,= 0 and easily reduced, after some work with the algebra (3.6.45), to the following second-order equations:
Evidently, the equations obtained are explicitly gauge covariant. Copyright © 1998 IOP Publishing Ltd
258
Idem
mid
Methods of Supersymmetry and Supergruvity
3.6.6. G m g r superfield dynamical equations In conclusion, we would like to obtain dynamical equations corresponding to the pure supergauge action (3.5.39). To this end, it is worth discussing how to do functional variation with respect to V in an optimum way. Let F [ V ] be a functional of the gauge superfield. Making an infinitesimal displacement in V changes the functional by 6 F [ V ] = F [ V + S V ] - F [ V ] = tr
If F [ V ] is gauge invariant, F [ V ' ] = F [ V ] , V' being as in equation (3.5.26), then the equation (3.6.55) proves to be gauge covariant, 6FCVI --o=>--6V
6F[V'] = 0. 6 V'
However, the variational derivatives 6 F [ V ] / SV transform into a nonlinear representation of the A-supergroup, since the variation 6V changes in a complicated nonlinear way: e2V' e2(V'
- e iAeZ V e -iA
+ SV') = eiAe2(V+
SV)e-iA
(3.6.56)
Instead of considering 6V, one can introduce covariantized variations A(*)V defined in the manner
which transforms covariantly: (3.6.58) in accordance with (3.6.56). The initial variation 6V is in a one-to-one correspondence with A ( + ) V or A(-)V, since it follows from (3.6.57) that
and similarly for A(-)V. Expressing 6V in (3.6.54) in terms of A ( + ) Vor A(-)V, one obtains Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspuce 8' *IF[ V I A(*'V(z) *
259 (3.6.60)
If F [ V ] is gauge invariant, then A ( * ) F [ V]/A(*]V transforms covariantly:
(3.6.6 1) Finally, equation (3.6.55) is satisfied if and only if the covariantized forms (3.6.62) hold. Now, let us vary the action (3.5.39). Taking into account equations (3.5.36?44), we obtain dSzG(e-'" D"e2')W,
Therefore, the dynamical equations are @ ( + ) W ,= 0.
(3.6.63)
Using the identity (3.6.47), they can be written in the form
gx[+)w(*) = g(,*)@k(k) -0
(3.6.64)
representing the non-Abelian generalization of the vector multiplet equation of motion (3.4.1 1).
3.7. Classically equivalent theories
It has been shown in Sections 2.6 and 2.7 that different superfields can be used to describe the same super Poincare representation. In the present section we would like to acquaint the reader with some supersymmetric field theories, each of which is characterized by classical dynamics identical to that arising in one of the previously considered models but realized in terms of other superfields. Copyright © 1998 IOP Publishing Ltd
260
I&ws trnd Methods of Supersymnzetry und Supergruuity
3.7.I . Mussice chirul spinor superfield model Consider the theory of a chiral spinor superfield y,,(z), conjugate & with action superfunctional n
n
D,x,
= 0, and its
I-
J
dszG2 - m2 d6zx2- m2 (3.7.1)
1 - -. G = -(D*xr + D ~ ")x = G. 2 Here the real scalar superfield G is linear,
D2G = D2G = 0
(3.7.2)
in accordance with the identities (2.5.30f).To find the dynamical equations of the model, one can use the variational rule (3.7.3) This gives 18
- D2D,G
+ m2xl = 0
1 8
+ m2fk = 0.
- D2DkG
(3.7.4)
Since DaD2D,G = DkD2DkG,one obtains the on-shell constraint
D"xa= DkF.
(3.7.5)
After this, the system (3.7.4) can be rewritten in the form
(U - m2)X" = 0 (3.7.6) where we have used the chirality of xa (D2D2x,= 1 6 0 ~ ~ ) . Recall that when imposing the first equation (3.7.6) only, the (anti)chiral superfields f, and xz describe two independent superspin-i states (see Section 2.6). Adding the second equation (3.7.6) expresses in terms of xa. As a result, the theory (3.7.1) describes the irreducible superspin-3 representation, similarly to the massive vector multiplet model (3.4.1). Therefore the theories (3.4.1) and (3.7.1) are dynamically equivalent. The equivalence of the two theories (3.4.1) and (3.7.1) can also be seen as follows. Let us introduce an auxiliary model with action
Fk
S [ V , x,f ] =
s
s
s
d8z(V2 - 2VG) - m2 d6zx2 - m2 d6Z2 (3.7.7)
where V ( z )is a real scalar superfield. We vary the action with respect to V, obtaining the equation Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
261 (3.7.8)
V-G=O. , gives Next, taking the variation with respect to 1
1-DzD,V+ m2xm= 0. 8
(3.7.9)
Due to equation (3.7.8), I/ can be eliminated from the action. This produces exactly the action (3.7.1),
scv,x,XI I ss,av= 0 = SEX, XI. Similarly, using equations (3.7.9),one can eliminate x, from the action (3.7.7).
This procedure (up to a factor) the action (3.4.1),
Hence, the action (3.7.7) plays the role of a converter from the massive vector multiplet model to the massive spinor superfield model and vice versa. On these grounds the models (3.4.1)and (3.4.2)are said to be dual to each other. 3.7.2. Massless chiral spinor superjeld model Let us set the mass parameter in (3.7.1) to zero. The action .
C
(3.7.10) turns out to be invariant under the gauge transformations
-
6 ~ =, iD2D,K
6xi = -iD2DhK
K = l?
(3.7.11)
where K is an arbitrary real scalar superfield. What are the superhelicity states that the model (3.7.10) describes? Note that the linear superfield G (3.7.1) is gauge invariant. The equations of motion derived from the action (3.7.10)
D2D,G = 0
DZDhG = 0
(3.7.12)
together with the identities (3.7.2) show that G is an on-shell massless superfield (see Section 2.7). G has no definite superhelicity. However, one can construct from G two secondary superfields: antichiral 1%= D,G and chiral ii = DhG. These superfields prove also to be on-shell massless, in accordance with the analysis of Section 2.7. Their superhelicities are K(;?;I)
=0
K(&)
=
- 1/2.
(3.7.13)
Recall that the same superhelicity content arose in the massless chiral scalar superfield model (3.2.5). Therefore, the theories (3.2.5) and (3.7.10) are dynamically equivalent. Copyright © 1998 IOP Publishing Ltd
262
1ricu.Y
triiti
Metliods of' Supersynzmetry and Superyruvity
In fact, the two theories (3.2.5) and (3.7.10) are dual to each other, because they can be obtained from the action S [ V , x,
a=
i
d8z{V z + V(D'xr
+ D,j ')}.
(3.7.14)
Indeed, eliminating V with the help of its equation of motion, one obtains the action (3.7.10). On the other hand, varying S [ V , x, 2 with respect to xu and ;rj! gives
DZD,V= D 2 D b V = 0
(3.7.15)
which can be resolved as follows
D,co=O
V=co+G
(3.7.1 6)
where @ is an arbitrary chiral scalar superfield. Substituting this solution into (3.7.14) and using the observation that D,Q)=O DZG=O+ .
P
(3.7.17)
P
one arrives at the action (3.2.5). In conclusion, let us discuss the component structure of the model (3.7.10). Following the prescription of Section 2.8, we define the component fields of %r by
U 4 = xxl
Br&x) = D,uXb,l
(3.7.18) From the symmetric spin-tensors BrP and &p one can construct an antisymmetric real tensor B o b by the standard rule B,, = (o,,),pBuP - (c,b)&fiP'.
Next, the component fields of the linear superfield G are expressed in terms of the fields (3.7.18) and their conjugates in the manner: 1 GI = - ( A A ) E V(X) 2
+
D,G/ = qE - ia,,x' 1
Il/,(x)
(3.7.19)
I = -iidb,BDlp + idrb&p = -(o"),&(B). 2 Here L,(B) is the antisymmetric tensor field strength defined in (1.7.43). It is worth noting that the component fields (3.7.19) are invariant under the gauge transformations (3.7.1l), due to the gauge invariance of G. Finally, the - [DZ,D,]G
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
263
component form of the action (3.7.10) can be calculated by the rule:
s
S = d4xL
which gives
s i:
S = d4x
1 2
- L"(B)L,(B) - -Fp3,q - i$a"d,$
].
(3.7.20)
This action represents the supersymmetric extension of the antisymmetric tensor field theory (1.7.43). In components, the equivalence of the theories (3.2.5) and (3.7.10) is trivial. The former theory describes two massless scalar fields and a massless Majorana spinor field. The latter theory describes three massless fields: one scalar, one antisymmetric tensor and a Majorana spinor. But free massless scalar and antisymmtric tensor fields are dynamically equivalent (see subsection 1.8.4). It is worth pointing out that in Minkowski space bosonic field theories can never be equivalent to fermionic ones, due to the spin-statistics theorem. The situation is different in superspace [w4l4. Here some bosonic superfield theory may appear to be equivalent to a theory of fermionic superfields and this fact does not contradict the spin-statistics theorem. 3.7.3. Superfield redefinitions In both theories considered above, their equivalence to theories realized in terms of different superfields, occurred due to the existence of proper duality transformations. Another way of obtaining dynamically equivalent models is superfield redefinitions. For example, one can always express a chiral scalar superfield through a real scalar superfield by the rule
(3.7.21) However, since I/ has more degrees of freedom than 0,this redefinition is accompanied by the appearance of gauge invariance V + I/' = V
+ -21 (D"t+hor+ D b P )
Dk$"
= 0.
(3.7.22)
Here $"(z) is an arbitrary chiral spinor superfield. Defining component fields of V as in expression (2.8.19), one can see that the fields A , IC/", $* and the Copyright © 1998 IOP Publishing Ltd
264
Ideus
Litd
Metltods of‘ Supersjw”trj and Supergruvitj
transverse part of I/, are purely gauge degrees of freedom. The other fields in expression (2.8.19) are related to the component fields of (3 as follows:
0 1 = F(x)
D$I = E.,(x) (3.7.23) 1 i - - D 2 0 , /= D(x) + - ~ ” V , ( X ) . 4 2 Remurk. The chiral parameters lCll in expressions (3.7.22) are defined modulo transformations K = R. I ) , -+ I): = $, + iD2D,K After making the redefinition (3.7.21), every chiral scalar superfield theory transforms into a real scalar superfield theory. For instance, the model (3.2.2) is changed to m SCVI = - d*zV - (D’, D2) - -(D’ + D’) 2 4
‘1 (Ik
Similarly to the initial model (3.2.2), its transformed version (3.7.24) turns out t o describe one superspin-0 state. Surprisingly, it is sufficient t o change the mass term in the manner m -(Dz + D’) + m2 4 and the resultant action (3.7.25) will describe two superspin-0 states. The reader can easily check this statement. 3.8. Non-minimal scalar multiplet One of the historic traditions popular among the practitioners of supersymmetry is to use term ‘scalar multiplet’ for the set of dynamical (super)fields in a supersymmetric theory realizing the superspin-0 representation on the mass shell. More precisely, this term is literally used for those supermultiplets which involve only physical scalar fields to describe the spin-0 states. The supersymmetric model, which has been discussed in subsection 3.7.2, also describes the massless superspin-0 multiplet, but with one of the physical scalars being traded for a gauge antisymmetric tensor off shell. That is why this model is often called the ‘tensor multiplet’. It is scalar multiplets which realize the matter sector in supersymmetric phenomenological models. Another tradition is that the scalar multiplets are identified with chiral scalar superfields and their conjugates. It is quite natural to ask whether such an identification is obligatory or, to a greater extent, a matter of habit and Copyright © 1998 IOP Publishing Ltd
265
Field Theory ir, Superspace
convenience? Certainly, the chiral scalar superfield provides the simplest and very convenient description of a scalar multiplet (‘minimal scalar multiplet’). But there exist alternative (or variant) scalar multiplet realizations like that given in subsection 3.7.3 (as compared to the minimal scalar multiplet, here one of the two auxiliary scalar fields is converted into the divergence of a vector field, see equation (3.7.23)). Therefore, such nonstandard multiplets provide alternative descriptions of supersymmetric matter and cannot be simply ignored. In the present section we consider several models of the ‘non-minimal scalar multiplet’ which was introduced for the first time by S. J. Gates and W. Siegel. The non-minimal scalar multiplet is described by a ‘complex linear scalar superfield’ which we did not meet in our previous study of supersymmetry.
3.8.I . Complex linear scalar superjield A complex linear scalar superfield
r(z)is subject to the single constraint D2r=o.
(3.8.1)
In contrast to the real linear scalar, no reality condition on r is required and thus D 2 r # 0. Constraint (3.8.1) defines a reducible representation of supersymmetry. For example, r can be expressed in the manner
r=Q+DaQ,
(3.8.2)
DbQ=DkR,=O
with Q and R, some chiral scalar and spinor superfields, respectively. Complex linear scalar r and antichiral scalar & are complementary to each other in the sense that an unconstrained complex scalar U can be decomposed as follows U=&+r D,&=D2r=0. (3.8.3) Similarly, real linear scalar G and chiral scalar @ are complementary to each other in the sense that an unconstrained real scalar V reads V=@+&+G
Dk@=D2G=0
G=G.
(3.8.4)
The general solution to equation (3.8.1) is
r=
I; = D ~ Q ,
(3.8.5)
being an unconstrained spinor superfield defined modulo with Q,(z) transformations of the form h Q , = D~6,crp,
(3.8.6)
where 6c,p,(z) is an unconstrained symmetric bi-spinor superfield. Therefore, any supersymmetric model described by constrained superfields r and l= can Copyright © 1998 IOP Publishing Ltd
266
Ideas and Methods of Supersymmetry and Supergravity
be treated as a theory of unconstrained potentials WO and with the gauge invariance (3.8.6). Remark. The complex linear superfield naturally originates if we consider a more general superfield redefinition than that given by relation (3.7.2 1). Namely, let us express the chiral scalar superfield through a complex scalar superfield by the rule 1@ = --D2U. 4 Since U possesses more degrees of freedom than @, this redefinition is accompanied by the appearance of gauge invariance
u -+ u'= u + r
D2r= o
with an arbitrary complex linear superfield r 3.8.2. Free non-minimal scalar multiplet
Let us consider a simple dynamical system described by the action superfunctional S[@, 6 ,U , VI =
s
d'z
{ 6@- V U }
(3.8.7)
where @ is a chiral scalar superfield, and U an unconstrained complex scalar are trivial, this model is obviously superfield. Since the dynamics of U and equivalent to the massless scalar multiplet model (3.2.5). But as long as U is unconstrained and complex, it is in our power to make the superfield redefinition
U-ui-6 which turns the action into
j[@,6,U , U ] = -
s
d'z { V U
+ U @+ U & } .
(3.8.8)
The superfields U and U can be eliminated with the help of their equations of motion. As a result, one arrives exactly at the model (3.2.5). On the other hand, varying with respect to @ gives
and, hence, U becomes a complex linear superfield. After that, the second and third terms in (3.8.8) drop out, since
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
267
The above analysis shows that the massless scalar multiplet model (3.2.5) is equivalent to special theory of a complex linear superfield r and its conjugate with the action superfunctional s [ r , ~ l = - / d ~ r .
(3.8.9)
It is instructive also to re-establish the equivalence by considering the dynamical equations S[T, leads to, which can be easily found by representing r and as in relation (3.8.5). The results are summarized in the following Table.
r]
Table 3.8.1. Minimal and non-minimal scalar multiplets.
Superfield type
Basic constraint
Equation of motion
antichiral scalar complex linear scalar
Du6=0 D2r = o
D26 = 0 D,T = O
For completeness, we also present a similar table establishing the duality between the minimal scalar and tensor multiplets, the latter considered in subsection 3.7.2. Table 3.8.2. Minimal scalar and tensor multiplets.
Superfield type
Basic constraint
Equation of motion
chiral & antichiral scalars real linear scalar
D*D,(@ + 4)= 0 D'G = 0
D2(@ 6)= 0 D2D,G = 0
+
Action (3.8.9) defines the free non-minimal scalar multiplet model. To determine its component form, it turns out to be reasonable to introduce the component fields of r in the following way:
B = rl 1 H = --D2rl 4
j d c= &,rl
Uu&= D)drDarl 1
W .- -DQD)cDurl.
Pa = Durl
a-4 Then, a simple calculation leads to the component Lagrangian
(3.8.10)
1
L = - aUBa,B - - x o ' a u j 2 1 1 - H H -O'Uu -(wP 2 2
+
Copyright © 1998 IOP Publishing Ltd
+
+ WP) .
(3.8.1 1)
268
Ideas and Methods of Supersymmetq and Supergravio
Comparing the relations (3.8.10) and (3.8.1 1) with similar ones for the minimal scalar multiplet model, see equations (2.8.4) and (3.2.13), we observe that the non-minimal scalar multiplet model involves extra auxiliary fields: the complex vector field U, and the two spinor fields w, and p,. Moreover, the auxiliary scalar field H enters the Lagrangian with opposite sign as compared to the auxiliary scalar F in (3.2.13). Remark. The action S[T, f'] is the simplest representative of action superfunctionals of the general form (3.8.12) with { a dimensionless parameter. It is a simple exercise to show that for all values of 1, except { = f l , these actions describe the scalar multiplet on shell. For 3' = i l the corresponding equations of motion imply nothing but r = ztp modulo an irrelevant constant.
3.8.3. Mass generation I The only way to introduce a mass for a single non-minimal scalar multiplet is to add to the massless action (3.8.9) new terms containing the bare potentials W, and \Ilk (3.8.5). So, let us consider the following action S[W,
a] = -
s
d8z
{ f'r + 2m(WaWa + \ I l k $ , ) }
which does not possess the gauge invariance (3.8.6). equations of motion 1 --D,r 4 1- --Dkr
4
determine Q, and consequences are
in terms of 1 --D2r 4
The corresponding
+ mW, = 0 + mW6 =0 r
and I=, respectively.
(3.8.14) Their obvious
+mf' = 0 r+mr
1-2-
--D 4
(3.8.13)
=0 .
(3.8.15)
We see that r is chiral on the mass shell and satisfies the dynamical equation (3.2.1) of the massive superspin-0 multiplet. In summary, the massive scalar multiplet can be described by an unconstrained spinor superfield. Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
269
3.8.4. Mass generation 11 Non-minimal scalar multiplets can acquire masses in tandem with chiral superfields. Following B. B. Deo and S . J. Gates, let us couple a single nonminimal scalar multiplet to a chiral scalar superfield my means of deforming the kinematic constraint (3.8. I ) to
I--D2C = 9(@) 4
(3.8.16)
with a holomorphic function 9. A general solution of this constraint is
for some local functional $[@I. An action superfunctional of the joint chiral-non-minimal dynamical system under consideration can be chosen in the form S=
s
d8z
{ 6Q - C C } +
(1
+
d 6 z g C ( @ ) c.c
J
(3.8.18)
with Y c ( @the ) chiral superpotential. To pass to components, we define the component fields of C in complete analogy with prescription (3.8.10), simply by replacing there r by C. With standard definition (2.8.4) of the component fields of @, a simple calculation then gives the component Lagrangian
where V ( A ) is defined by relation (3.2.12). Eliminating the auxiliary fields leads to the following scalar potential U(A,
A,B , B) = l 9 ( A ) I 2+ 1 Y ( A ) - B9’(A)12 .
(3.8.20)
Looking at the Lagrangian obtained, we see that the auxiliary fields of I: do not couple to the component fields of a. In particular, the auxiliary scalar H does not contribute to the potential. This is not accidental. The point is that the structures F F and H H enter the Lagrangian with opposite signs. If we had linear in H contributions to L,they would produce corrections to the potential with the wrong sign and, as a consequence, the potential would not be positive Copyright © 1998 IOP Publishing Ltd
270
Ideas and Methods of Supersymmetry and Supergravity
semi-definite. Another important property of the model in field is that one can generate a nontrivial interaction without having chiral superpotential at all! The deformed constraint (3.8.16) is responsible for such a coupling. In particular, if we choose Y e , ( @ =)0 and 9(@) = - m a , with m a positive parameter, and eliminate all the auxiliary fields, the final Lagrangian takes the form
L = - aUAa,,A - aiiBauB- m 2 ( A A 1-
--
2
\kD(iyua,
+BB)
+ m)*D
with all fields being now massive! Here of the physical spinor fields
*D
(3.8.21)
denotes the Dirac spinor constructed (3.8.22)
3.8.5. Supersymmetric electrodynamics In subsection 3.5.2, we have realized the matter sector of supersymmetric electrodynamics in terms of two chiral scalar superfields with opposite charges. Here we are going to present an alternative formulation for this theory . Supersymmetric matter will be described by a chiral scalar @ ( z ) along with a generalized complex linear superfield C ( z ) constrained by 1--D2C + m @ = O . 4
(3.8.23)
The gauge group is postulated to act on @ by the law Q'
&A = 0 ,
=
(3.8.24)
Then, constraint (3.8.23) enforces C to possess the same transformation law E' =
.
(3.8.25)
The unique renormalizable and gauge invariant action reads SSED=
s
d8z { &e2YV@ - Ce24VC
(3.8.26)
As compared to the non-gauge models (3.8.12) and (3.8.17), in the present case both the ( term and chiral superpotential are forbidden by gauge invariance. To pass to components in a manifestly gauge covariant form, now we have to be a bit more delicate than in the case of purely chiral matter. Let us start with introducing a generalized chiral representation (compare with subsection 3.1.6). In this representation one works with the matter superfields
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
27 1
whose gauge transformations involve only the chiral parameter A (3.8.28) and completely similar for the column of Ccc) and EC0; here q denotes the operator of charge. Therefore, adapted for handling such superfields are the gauge covariant derivatives ( 3 . 6 . 4 0 ~ in ) which V must be identified with Vq. Herewith, the superfields and $0 are covariantly chiral and antichiral, respectively , &+'@'C' =0 G&+)$C) =0 . (3.8.29) Similarly, the superfields respectively,
and
are covariantly 'linear' and 'antilinear',
The representation under consideration is chiral in the sense that the covarinat derivatives ( 3 . 6 . 4 0 ~correspond ) to the chiral transform of those defined by relation (3.6.49). We define the component fields of QCc)as gauge covariant space projections A = @(c)l
@,
1 F = --('(+))2@(c)l 4
= GJ:+)@(c)l
(3.8.31)
and analogously for B = C(c)I
H
= - -1
4
(a(+))* (C)I
p L Y = 9 , (+) (C)I
- (+)
jdc =ab,C'C'I Uak
-(+) (+) =9& ga C (C)I
0.-
!.~(+)a$r)~~)~(c
Q-4
,
(3.8.32)
In the Wess-Zumino gauge (3.4.16), definition (3.8.31) of the component fields of @ coincides with relation (2.8.4). The component fields of &(') and should also be defined via computing their gauge covariant space projections. Since the chiral representation is characterized by the conjugation rule
UCC, = e2"q
UCC)
(3.8.33)
in the Wess-Zumino gauge the component fields of &(') and coincide with the complex conjugates of those given by relations (3.8.31) and (3.8.32), respecti vel y . To find the component form of the action (3.8.26), it only remains to rewrite it in the form
Copyright © 1998 IOP Publishing Ltd
272
Ideas and Methods of Supersymmetry and Supergravio
and to note that YMAT is gauge invariant, hence
1
=16
/
d4x ( 9 ( + ) ) 2 ( G b ( + ) ) 2.. 3 ~ ~ ~ l(3.8.35)
When computing the latter projection, one should keep in mind the basic constraints (3.8.29) and (3.8.30) as well as make use of the covariant derivative algebra (3.6.45), in particular, its simple consequences
[so, 3'1= - 4 i 9 a u ~ +a swag [932,G21 = - 4i9°"[9D,,
[Ga,g 2 ]= 4 i 9 a a W - s w a g + I6(W"2& + waGa)q + 8 ( 9 " W u ) g .
(3.8.36)
Finally, one should make use of the projections (3.4.20) and the identity
( ~ ~ ~ ) u= (v', u) () ~I ' I v,, a,
- i V, q
(3.8.37)
which holds for any superfield U , with Vu the electromagnetic gauge potential, see relation (3.4.15). As a result, one obtains the component Lagrangian 1 L ~ E= D - -FUhF,h - ~ ~ D { i y " V m}\kr, u 4 2 - VUAVuA- m 2 A A - V'BV,B - m2BB
+
1
- -q(AA
2
- BB)2 - i i i M y ' a l , l i M 2
1 - -4 [ A ~ D (-I y 5 ) A ~ BEM(~ - y5)\kD
+
C.C. 2 + ( F + m B ) ( F + m B ) + 2 ( D i 5 q ( A1 A - B B ) ) '
-HH
+ ~2 f i ' l J u+ -21( p w + 6 6 ) .
I (3.8.38)
Here \ k denotes ~ the matter Dirac spinor (3.8.22), and AMthe Majorana spinor entering the massless vector multiplet. Comparing the two Lagrangians (3.5.19) and (3.8.38), we see that they differ only in their auxiliary field structure. Therefore, the formulation of supersymmetric electrodynamics given in the present section can be equally well used for the description of this theory. When m = 0 the basic constraint (3.8.23) tells us that C coincides with the complex linear superfield r which should be expressed through its potential by the rule (3.8.5). The corresponding gauge invariance (3.8.6) defines a gauge theory of an infinite stage of reducibility (see subsection 4.5.1). Covariant quantization of such a gauge theory turns out to be a nontrivial technical problem, and a complete solution of this problem was given by M. Grisaru et a1 I . For
'
M. Grisaru. A. Van Proyen and D.Zanon Nucl. Phys. 502B 345. 1997
Copyright © 1998 IOP Publishing Ltd
Field Theory in Superspace
273
# 0 one can avoid any quantization problems by adopting the point of view that the constraint (3.8.23)is nothing else but the definition of @. Then, we work with a single unconstrained complex scalar superfield C and its conjugate 2, in the role of matter superfields, and there is no extra gauge invariance associated with the matter sector.
in
3.8.6. Couplings to Yang Mills supe @eld Complex linear superfields can be coupled to a non-Abelian gauge superfield very much like to the chiral matter. A multiplet of complex scalar superfields r transforms with respect to the gauge group by the rule (compare with relation (3.5.25))
r' = e l A r
= Fe-lA
A = A'T'
A = AITI
&A!
=o
(3.8.39)
and this transformation is obviously compartible with constraint (3.8.1). The massless gauge invariant action reads
(3.8.40) When working with purely chiral matter, it is always a nontrivial question how to get Dirac fermions in components, see subsection 3.5.7. For such a realization of supersymmetric matter, the chiral superfields should transform in a real representation of the gauge group just in order to have Dirac fermions at the component level. Another situation takes place if chiral superfields are introduced in tandem with complex linear ones, the latter being constrained by equation (3.8.23), for example. Now, we have a natural prescription to construct Dirac fermions, which is given by relation (3.8.22). Remarkably, the left handed and right handed components of the Dirac fermions are automatically distributed between the different superfields, @ and C respectively. Moreover, @ transforms in the same complex representation of the gauge group to which the Dirac spinor fields belong.
3.8.7. Nonlinear sigma models The duality transform, which we implemented in subsection 2.8.2 to convert the minimal description of the scalar multiplet into the non-minimal one, can be extended to the case of nonlinear sigma models. Let us rewrite the supersymmetric sigma model action (3.3.10) in the following equivalent form
(3.8.41) Here U are unconstrained complex superfields, r complex linear superfields. This dynamical system is equivalent to the supersymmetric nonlinear sigma Copyright © 1998 IOP Publishing Ltd
274
Ideas arid Methods of Supersyntntetry and Supergravity
model, since the r-equations of motion require U to be chiral, and hence reduce the above action to that given by relation (3.3.10). On the other hand, if we vary the action with respect to U and 0, we obtain the equations (3.8.42) which have to be solved to express U and arrives at the dual action
0 through r and r. As a result, one (3.8.43)
where
f is the Legendre transform
of K
We see that the duality transformation is accompanied by the exchange of potentials K ( @ , 6)-+f ( r ,F) . (3.8.45)
In the non-minimal picture (3.8.43), the Kahler invariance (3.3.1 1) is hidden. At the component level, the nonlinear sigma model described by action (3.8.43) is radically different from that realized in terms of chiral superfields (3.3.10). We refer the interested reader to the original publication by B. B. Deo and S . J. Gates for a detailed discussion of the component structure of model (3.8.43). Very interesting sigma models arise if we consider, once again, chiral superfields in tandem with complex linear ones. Let us here briefly discuss so called 'chiral-non-minimal nonlinear sigma-model' which was proposed for the supersymmetric description of the low-energy QCD effective action *. The action superfunctional reads
and possesses the Kahler invariance (3.3.11). In contrast to the chiral sigma model (3.3.10), here the target manifold is the tangent bundle over the Kahler manifold where @' and @ take their values. The complex linear variables r' form a tangent vector at the point of the Kahler manifold with coordinates @' and & L , with the transformation law (3.8.47) with respect to the holomorphic reparametrizations (3.3.12). This transformation law is nicely compartible with the basic constraint (3.8.1). A nontrivial exercise for the reader is to find the component form of action (3.8.46).
*
S. J . Gates f h y s . Lett. 36JB 132, 1996; S. J . Gates, M. T.Grisaru, M.E. Knutt-Wehlau. M . RoEek and 0.A. Soloviev f h y s . L e f t 396B 167, 1997.
Copyright © 1998 IOP Publishing Ltd
4 Quantized Superfields If you can make one heap of all your winnings And risk it on one turn of pitch-and-toss, And lose, and start again at your beginnings. Rudyard Kipling:
!f
4.1. Picture-change operators In the previous chapter, the classical dynamics of the most popular N = 1 supersymmetric field theories was described. Now we turn our attention to investigating their quantum properties. Our primary goal will be to develop the superfield extension of the Feynman path integral perturbation theory. We anticipate that the reader is familiar with the functional methods of quantum field theory, although an introduction to these methods will be given below. As has been shown earlier, there are two equivalent formulations for supersymmetric theories: in terms of component fields and in terms of superfields. The only difference between these formulations is that dynamical variables exist on different spaces: in the former case-on Minkowski space, in the latter-on superspace R414 (or its chiral subspaces). Owing to the universality of the path integral quantization procedure, it can be formally applied in both cases. When using the component field formulation, the path integral rules turn out to be correct in the sense that they lead to the same physical results as canonical quantization. When working in the superfield approach, it is worth expecting that the superfield path integral rules will guarantee the supersymmetry to remain explicit at any stage of calculations. A natural question to ask is: What is the relation of the superfield path integral rules to physical scattering amplitudes? An answer to this question will be given in the following section. The chief goal of the present section is to introduce an auxiliary formalism, which 275 Copyright © 1998 IOP Publishing Ltd
276
Irkas unti Mi~tliodsof Supersymmetry and Supergravity
proves useful hereafter. We start by discussing (super) functional supermatrices and (super) functional derivatives. Then we describe picture-change operators which carry out transforms from superfields to component fields and vice versa. These operators will play an important role below. 4.1.1. Functionul supermatrices Many differential operators arising in field theories become linear if one considers their action on some linear functional spaces associated with spaces of histories. Under reasonable assumptions, it is possible to handle such operators as one does with ordinary supermatrices (see Sections 1.9 and 1.10). Here we are going to present the corresponding technique. There are several linear functional spaces related to the space of histories 0 of a dynamical system, which describe bosonic and fermionic tensor fields cpi on Minkowski space. Recall that @ is spanned by all possible sets (sp'(x)} of smooth tensor fields on Minkowski space such that: (1) the fields have fixed Grassmann parities &((pyx))= E i
Ei
=o,
(2) every field history includes together with each field conjugate, that is, (sp'(x))* = Cijsp'(X)
(4.1.1)
1 'pi its
complex (4.1.2)
Cij being a real matrix fixed for the system under consideration; (3) the fields may have an arbitrary behaviour in finite regions of space-time and obey boundary conditions determined by the dynamical subspace 0, (owing to these boundary conditions, in nonlinear theories Q, cannot be treated as a linear space and should be viewed as an infinite-dimensional supermanifold). Functional spaces discussed are the following. First is the 'extended space of histories' spanned by all possible sets {cpi(x)}of smooth fields on R4 under requirements (1) and (2) (and maybe restricted to be bounded only). Omitting also requirement (2) gives the complexification &c of 6.Second is the subspace A@ in 6 consisting of all histcries with compact support in space-time, that is, for every history { W ' ( X ) } E A ~every field 6cp' may take non-zero values on a compact set in R4. Sometimes it is also reasonable to consider its complexification A a C .The space A 0 can be called the space of field variations. Finally, for every point sp = {pi(x)}of 0, one can introduce the tangent space Uq to 0 at cp, its complexification (UJCand subspaces in these. Each of the spaces presented is naturally identified with the even or real even subset of some infinite-dimensional supervector space. For example, consider the supervector space &(A,) of all smooth mappings q: R4 ,pw)
where LP"q)is a (p,q)-dimensional supervector space, with p and q being Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
277
the number of bosonic and fermionic fields, respectively, which @ describes. Choosing a pure basis {Ei} in 5?(p*41, E ( & ) = E ~ , one finds that C-type elements in &(Ax) are of the form q(x)= Eicp’(X) where the smooth functions cp’(x) satisfy equation (4.1.1).Therefore, the set Next, imposing the complex conjugation rules {cp’(x))lies in 6,.
-
.
= ( - 1)E’E .CJi J
one observes that real C-type elements in &(Am) obey the equations (4.1.1) and (4.1.2).Therefore, &-points span the real event subset of &(Am). Recalling now the identification of CPiqand Rplq with the even and real even subsets of a (p, q)-dimensional supervector space (see Section 1.9), one can look on & (&,) as an infinite-dimensional real (complex) superspace. Let F be a linear differential operator acting on (a C-type linear operator on & ( A A &(Fcp’(X)) = Ei.
(4.1.3)
Associated with F is its ‘kernel’ (or ‘functional supermatrix’) FiJ(x,x’) and the ‘supertranspose kernel’ FSTC(x,x‘) defined by
s
F~’(x =) d4x’
for all cp E aC, One easily finds:
s
x‘)cp’(~’)= d4x’ cpj(x‘)FST;(x’,X)
&(PAX,x’))= &(FST)(X,x’))= E’
+E j
(4.1.4)
(4.1.5)
and FST?(x,x‘)=( - l)ElfEtEj Fji(x’,x).
(4.1.6)
It is instructive to compare the final relation with equation (1.9.58b) and equation (4.1.4) with equation (1.9.63).The kernel is evaluated by the rule P i x , x’) = F6iid4(X - x’)
(4.1.7)
where F is to act on the indices i and x only. The ‘functional supertrace’ of F is defined as
s
s
sTr F = d4x ( - 1 )EiFii(x,X ) = d4x ( - 1 )EIFSTii(x,x ) .
(4.1.8)
If F is a non-singular operator possessing logarithm, its ‘functional superdeterminant’ is defined as sDet F = exp(sTr(1nF)).
(4.1.9)
The last two definitions represent infinite-dimensional generalizations of Copyright © 1998 IOP Publishing Ltd
those given in Section 1.10. It should be remarked also that introducing the even-even, even-odd, odd-evcn and odd-odd parts of F'Ax,x'), one can rcpresent sDet F as in equation (1.10.63). Another remark is that not every linear operator possesses well-defined sTr and sDet. We restrict ourselves to consideration of operators for which the right-hand sides in equations (4.1.8) and (4.1.9) exist. Given two linear operators F , and F,, the following relations hold:
i^
( F F2)jj(x,x') = d4x" F:,(X, x")F:,(x", x')
(4.1.1Oa)
sTr(F,F,) = sTr(F,F,)
(4.1,lob)
sDet(F,F,) =sDet F,*sDet F,.
(4.1* 10c)
To prove the latter, we set F 1 = e A , F,=eB and use the Baker-Hausdorff' formula: sDet(eAeB)= sDet
=exp(sTr A ) exp(sTr B). To simplify subsequent formulae, we introduce a condensed notation based on unifying the symbols i and x, which specify qi(x), in the label In such notation, we have
c
Tensor-like objects
+
mean ( n m)-point functions . . Til ... in? . - Til .. in ,
. ....in + :i"+i...in+m-~, its +I
T: I,
...l n
-
.
i f l + ~ . . , j n + , v (' ~' 5 ~ ~' 0 ,
i,,+~.,.i~+,,, I1 ... 1" (Xl, ,
4.1.2. Superjiunctional supermatrices
.3
x,+
1,
x,, & + l ,
' '
5
' 9
e . ,
xn+J X"+,).
Consider a supersymmetric dynamical system. As is known, it can be described by the space of histories 0 or by the space of superhistories V,in relation to which @ is the space of component fields. In general, the superfield variables d(z) include unconstrained tensor superfields V y z ) , chiral tensor Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
279
superfields @yz) and their conjugate antichiral ones @ ( z ) , u'(z) = ( VYZ), o q z ) , a q z ) )
Dh@E=
(4.1.12)
= 0.
Throughout Sections 4.1-4.3, we use lower-case underlined letters from the beginning (middle) of the Latin alphabet to label unconstrained (chiral and antichiral) superfields; in the antichiral case, indices are in addition dotted. Superfield variables can be bosonic and fermionic, &(u'(Z))=&,
(4.1.13)
&'=0, 1
and are real in the sense that any superhistory U = {U'} with every superfield 'U its complex conjugate,
EV
includes together
O \ (4.1.14) 0 1
C', being a fixed real matrix. Similarly to @, the space V is an infinite-dimensional supermanifold, whose structure is specified by boundary conditions imposed on superhistories at space-time infinity. In contrast with the field variables, which are functions on Minkowski space, the Vqz), @qz) and Gqz) exist on different superspaces (parameterized by (x, 8, e), (x, 0) and (x, 8), respectively), and this is expressed in the explicit form of the unit operator on V: u'(z)=
where
s
d'J'z' 6'j(z, z')u~(z')
and
Copyright © 1998 IOP Publishing Ltd
s
d(J)z'uJ(z')6JI(z',Z)
(4.1.15)
280
Irieas
r i d
Melhorls of' Supersymntetry una' Superyruvity xp+ - =
fiOo'B.
The fact that the chiral delta-function (3.1.46) can be written as in (4.1.17) is a direct consequence of the relation (3.1.66) holding in the chiral represen tat ion. Associated with V are several infinite-dimensional superspaces: the 'extended space of superhistories' Q,the space of superfield variations AV, the tangent spaces U,.to V at U , for every U E V,and their complexifications AV,, (UJC.These are introduced by analogy with 6,A&, and so on. Note also that chiral and antichiral superfields describing points in Q,, AV, and (UJCare not related by complex conjugation. Let F be a linear differential operator acting on Q,.We define the 'superkernel' (or 'superfunctional supermatrix') of F as follows:
v,,
s
d'J'z' F',(z, z')d(z') = d8z' F'~(z,~ ' ) V ? Z ' )
+
z')W(z')
s -
d62' F',(z, z')@(z')
(4.1.I8a)
some components of the superkernel being chiral or antichiral: -
D&F",(z, z') = DhF',,(z, - z') = O D,F&,(z, z')= D;F',(z, z') = 0.
(4.1.18b)
The superkernel is evaluated by the rule (4.1.19)
F',(z, z')= F~',(z, z')
where F acts on the indices I and z only. Its supertranspose is defined by
s
Fv'(z) = d'J'z' vJ(z')FSTJ'(z',z )
FSTIJ(z, z')=(- I)E'+E'E'FJI(~', z).
(4.1.20)
Similarly to equation (4.1.5), one now finds z')) = E' E(F'J(z, 2')) = &(FSTIJ(z,
+E ~ .
(4.1.21)
The 'superfunctional supertrace' and 'superdeterminant' of F are defined in the manner:
s s
s s
sTr F = d")z ( - l)"'F',(z, z ) = d(')z ( - l)EIFSTIr(z, z) =
d ' 2 ( - 1)E'F'Az, z) + d6z ( - 1)"" FE,,, -(z, z )
Copyright © 1998 IOP Publishing Ltd
Quantized Superjields
+
s
d6;(
281 (4.1.22)
- 1)"lFkh -( z ,Z )
sDet F = exp(sTr(1n F)).
(4.1.23)
The properties (4.1. IOU-c) are trivially extended to the supercase. It may seem, looking at equations (4.1.18) and (4.1.20) that operating with superfunction objects is much harder than with functional ones. Some real complications exist: first, when introducing the superkernel, one must succeed in satisfying the (anti)chirality requirements (4.1.18b) (note, however, they fix the superkernel uniquely); second, it is required to integrate equations (4.1.1 8,20) over different superspaces. Nevertheless, it is worth keeping in mind that every superfield variable describes a large number of ordinary fields. So the shortcomings mentioned can be looked on as a reasonably small cost for the advantage of working with a multiplet of fields as a single object. In addition, the operations sTr and sDet possess remarkable properties (having no direct space-time analogues), which often greatly simplify superfield calculations. Some of such properties are formulated in two simple theorems. Theorem 1. Let F be a linear operator of the general structure
e, @(e- e)2(8- 8 ) 2 + ),
x; + ),
- ), x; -),
eKe - e 1 2
(4.1.24)
@(8-P)2.
Then, for every positive integer N , the operator F N has the same structure and sTr(FN)= 0.
If F possesses a logarithm, then sDet F = 1. The meaning of equation (4.1.24) is quite simple. It tells us that F, regarded as a differential operator, is of zeroth order in the partial derivatives in 8 and 8. As examples of operators of the type (4.1.24), we can suggest z-dependent supermatrices (4.1.25) -
Copyright © 1998 IOP Publishing Ltd
282
Ideus und Methods
of Supersymmetry and Superyruvity
or generalized d'Alembertians
11
+
6gb[G"b(z)a,B, L"(z)d,]
0=
SE, [G$
)(z)dadb
+ Qgh(z) -
+ Lf+)(z)aaI QF+)n(Z)
(4.1.26)
+Qp-)fi(Z)
dhfi [GSb)(z)a,B,+L;I-)(Z)a,]
where G,,,, L,,, and Q(+, - are chiral (antichiral) superfields. Owing to Theorem 1, we have sDet(%0)=1 (4.1.27) for an arbitrary c-number 1, with non-zero body. Theorem 2. Let K be a first-order differential operator of the general structure
11
JghKA(z)dA + K g-b ( z )
K = Pn[A"(~)d,+A"(z)e'"d,e-'~]
+A$@) -
(4.1.28)
6 afi [ ~ ~ ( z )+dR,(z) , e - ix?'eiX ] + R hfi (z) 2 = twva,
where all A are chiral and all R are antichiral. Then sDeteK=l. Proof. Since sDet(exp K) = exp(sTr K), we are to prove sTr K = 0. Introducing the superkernel of K, one observes that each of Kgb(z,z'), KE, (z, z') and Khi(z, 2') is proportional to (0-0') or (B-8') and vanishes at z=zr. To understand the importance of the above statement, suppose we have a linear representation T of some supergroup G (for example, the super Poincare group) on Q or on the generators being first-order differential operators of the form (4.1.28) (as is the case in practice, in particular, for the super Poincare group). Then every operator of the representation is unimodular, sDet(T(g))= 1 V gE G.
vc,
In quantum theory this implies the supergroup invariance of the path-integral measure! By analogy with equation (4.1.1l), we introduce the superfield condensed notation:
J
F f j v J = d(J)z' F',(z, z')vJ(z') In the case of tensor-like objects
Copyright © 1998 IOP Publishing Ltd
sTr F =( - 1)"'F'j.
Quantized SuperFehis
or
Til, i,>i,8+ I . .. T,, ->,,,
1.,.1n-w
- T I ,,,,, ,
-
( ~ 1 ,* * > Zn, z n + 1 ,
-
- 3
203
zn+m)
it will always be assumed that for Zr=??(tjr) the expressions on the right are chiral (antichiral) with respect to the argument z,. 4.1.3. (Super)functional derivatives
Up to now, (super)functional derivatives, as they were introduced in Chapter 3, have been left ones. Below it will be convenient to operate with both left and right (super)functionalderivatives. Here we give the necessary definitions. Let Y: @ + A , be a functional on a.Its left functional derivative
and its right functional derivative c
are defined by
s'
B
d4x 6 q i ( x )-'€'[q] = d4x Y [ q ]-6q'(x) (4.1.30a) 6q'(x) 6V'(X) or, in condensed notation (4.1.30b)
Sq'(x) being an infinitesimal variation. Functionally differentiating q gives the unit operator on 0: (4.1.31)
To simplify expressions involving repeated functional derivatives, it is useful to introduce the following notation:
Symmetry properties of such tensor-like objects can be readily found by considering the nth variation of Y[q] dn 6"Y[q]=,Y[q+t6q]
IL
dt
and then representing it in several different forms. Thus one finds 6n+mY[q] = 6 q ' l . . * 6q'"-"';,,+,,,,.,~ ] , Y [ q I
,6q;,. =Y,;,,*
., (),,,,
&$ti;
il,y*:l , > - , , ? . . .I , , I [q""".
,/,,,
T
7 , [q]Gq". , . 6q'"'"'.
Copyright © 1998 IOP Publishing Ltd
. , 6q'"+"'
284
Ideas and Methods of' Supersymmetry and Superyravity
(4.1.33a) (4.1.33b) (4.1.33~) (4.1.33d)
Let Y be a bosonic functional. Equation (4.1.33b) tells us that, for every ( P E @ , the supermatrix ; , Y j [ q ] is supersymmetric in the sense of definition (1.9.78).Associated with such a supermatrix is the c-number-valued bi-linear form on A@: 6qz)=(6qz2 6 q 1 ) = 6 d i,Y,j[q18$z~
(&I:
In the case of a real bosonic functional Y :Q, + R,,the above bi-linear form turns out to be real c-number-valued, hence the operator ;,Y,j[q] is self-adjoint on A@. Now, let Y :V + A m be a superfunctional on V. Its left superfunctional derivative 8/So'(z)Y[u] is defined as
=jdaz6VTz)-
+ where
-
s
8
+
Y[v] 6V T z )
3
s
8
d6z 6 W z )-'YCVI 6W(Z)
d6Z6$e(z) -=-- Y [U] 6@(Z)
8
D&-Y[u]=O SO(Z)
(4.1.34)
8
Y[u] =o. Dx6W(Z)
The right superfunctional derivative Y[u]8/6ur(z) is read off from 6Y[2;]in which 6 d ( z ) is to be placed on the right. In condensed notation, we have (4.1.35) Superfunctionally differentiating
U
gives the unit operator on V: (4.1.36)
Copyright © 1998 IOP Publishing Ltd
285
Quantized Superfields
In perfect analogy with equation (4.1.32),we introduce the notation
Relatitions (4.1.33)are trivially extended to the case under consideration. 4.1.4. Picture-change operators
In developing the general formalism for supersymmetric dynamical systems, it proves reasonable to have an analytic realization of the fact that 0 is the component field space for V.This is the purpose of the present subsection. Associated with every superhistory (v'(z)} E V is the unique history {cp'(x))~@spanned by the component fields of Va, Qm and @% This correspondence is one-to-one and can be uniquely extended up to a correspondence between Q and 6.So we have two mappings
(4.1.38) carrying out transforms from superfields to their component fields and vice versa. In accordance with the results of Section 2.8, the operators P and Q are linear. Introducing their kernels, we can write
s s
$(x) = d"'z' P',(x,z')v'(z')
D&P',(x, - z')=D;P'k(x, - z')=O and
v'(z')= d4x' Q'XZ, x')c~'(x')
(4.1.39~)
(4.1.39b)
D&Q"i(z,x') = D,PBi(z, x') = 0
for every v E Q and the corresponding cp €6. Using condensed notation, the above relations take the form:
(4.1.40) The operators P and Q will be called the 'picture-change' operators. Kernels of the picture-change operators are easily found in practice. Consider, for example, a chiral scalar superfield @(z). Its component fields are given by equation (2.8.4).Then one acts in the manner: P
J
I
Copyright © 1998 IOP Publishing Ltd
n
I L 0 =J
A ( x )= @ 1 = d6z'6 + (z,z')@(z')
d6z' S+(zl, z')@(z') (4.1.41)
286
Ideas and Methods of Supersymmetry und Supergrucity
=!4 [d6z'{D''D~6'(z1 1
F(x)=--DZ@,/ = 4 =
-z')}O(z') d'z' 6'(2/ -z')D'~@(z')
-14 ~d8zf{D.i6'(z~
16
d6zf{D'zD'268(z/ -z'))@(z')
where ZAl +a,
ea,
8),1 =(Xu, 0, 0).
It is an almost trivial task to extend this exercise to all admissible superfield types. The simplest properties of the picture-change operators are: i E(P'~) = E(Q ;) = ei (Pil(X,
+
(4.1.42~)
z'))* =( - l)E'+&f&~CijPjJ(X, Z')CJ,
(Qii(x, z'))* =(
- l)ElfEzE'C\QJ,(x, z')CJ,
(4.1.42b) (4.1.42~)
They follow from equations (4.1.3, 14).Together with Pliand Q,: we introduce their supertranspose supermatrices by the rule (4.1.43) Then we have (4.1.44) The fact that the picture-change operators are linear has many important consequences. First, note that the mappings (4.1.38) are uniquely extended up to the linear mappings P:
Q,+&,
Q: @ = + Q c
inverse to each other. Let F be a linear operator on operators on &c:
Q,.It induces the linear
F = PFQ
(4.1.454
~ Ji : = pi ~i ' J QJ j .
(4.1.45b)
Obviously, we have
Copyright © 1998 IOP Publishing Ltd
Quantized SuperJieIds
287
The use of equations (4.1.21)and (4.1.42~)leads to
sTr F = ( - l)e'Fi;=(- 1)"'PifF;Q: t 3 : t =(- l)E'FJQ ;I",= ( - 1)"'Ff therefore
sTr F = sTr F.
(4.1.46)
Next, if F possesses a logarithm, the same is true for F and their superdeterminants coincide
sDet F = sDet F .
(4.1.47)
As a result, it is inessential which picture, superfield or component field, is used for computing supertraces and superdeterminants. Furthermore, consider a superfunctional "[U] on V.Associated with Y[v] is the functional Y[q] on @
W q l ='Uv(q)l= WQfpI.
(4.1.48)
Representing an arbitrary variation of Y in several forms,
or
8
8
6Y=Y76vf=Y-:6q 6V 6fp'
;
and using equations (4.1.44),one obtains the relation between the functional and superfunctional derivatives:
(4.1.49a)
(4.1.49b) In particular, the following identities hold:
y.dvl=y,;C~lP'p
Y,;CqI = Y,iCvlQ~
(4.1SOa) (4.1.506)
Y being arbitrary.
Provided one of the operators P or Q is at our disposal, its inverse can be obtained by the following scheme. Introduce a 'PoincarC invariant Copyright © 1998 IOP Publishing Ltd
288
1dc.a.r and Metkods of' Supersymmctry and Supergruvity
supermetric' on A@ of the form: P
(4.1.5 1)
where (1)
tii,
is a constant invertible matrix subject to the requirements:
vi,is supersymmetric and does not possess the even-odd pieces,
and odd-even (4.1.52)
(2) q i j satisfies the reality condition ('lij)* = ( - l)"'EqkiCkiClj
(4.1.53)
which implies that the supermetric takes real c-number values; (3) qij is a Lorentz invariant tensor, that is, it is a descendant of the invariant tensors (oU)%* and (8")". The inverse of q i j will be denoted by vi', qikqkj=6). Next, introduce a 'super Poincare invariant supermetric' on AV of the form
Here (4.1.55)
is a constant invertible matrix subject to requirements completely similar to those imposed on q i j (with obvious modifications). The inverse of qIJ will be denoted by p l x J , q,K@J = Since A@ is the component field space for AV, it seems reasonable to restrict the supermetric on A@ (4.1.51) to be induced by the supermetric on AV (4.1.54), ds2 = gql
r:
QST i g,jQ$6&. -
(4.1.56)
As a result, the supermetric on A@ proves to be super Poincare invariant. It is easily seen that the induced supermetric satisfies all the criteria formulated above. Comparing equations (4.1.5 1) and (4.1S6) leads to
Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
289
(4.1S 7 b ) Therefore, P is expressed via the supertranspose of Q. Supermetrics qij, q" and q r J , q'J may be used for lowering and raising indices of fields and superfields, respectively. Let us give an application of equation (4.1.57).Consider a non-generate bi-linear form on AV (dol, do2)=d1;,i Hrjdu,j
where H i j is a c-type non-singular differential operator. It induces the bi-linear form on A@
(4.1.58) Now, defining H Ij -= q fIzHRJ
H 7j = q ;lHgj
(4.1.59)
and using equation (4.1.57b),one finds .
\
*
H = pii~T-e J J. J
(4.1.60)
sDet(q"H$) = sDet(q"HR7).
(4.1.61)
therefore This relation extends the identity (4.1.47)to the case of supermatrices with indices in the lower position.
4.2. Equivalence of component field and superfield perturbation theories This section is devoted to demonstrating the equivalence of two perturbation theories for computing correlation functions, based on the use of the component field and superfield formulations, respectively, in a large class of non-gauge supersymmetric dynamical systems.
4.2.1. (Super)jield Green's functions Consider some relativistic field theory with a classical action S [ q ] . In this section we make three basic assumptions about the structure of S [ q ] . First, S [ q ] is a local functional of the form (3.1.11). Second, it is non-gauge, which means the dynamical subspace (Dopossesses a history 'po
S.iCv01= O Copyright © 1998 IOP Publishing Ltd
(4.2.1)
290
Ideas und Methods of' Supersymmetrv and Superyruvity
such that the Hessian at this point I,S,jCqol
(4.2.2)
is non-singular, that is, the equation ;,S.jCqolb' = 0
(4.2.3)
has no solutions with compact support in space-time,
(4.2.4)
V 6q~AO\0. { ;,S,j[qO] 6 q j ) # O Third, S [ q ] is analytic in a neighbourhood of qo in O:
(4.2.5)
Introducing the linearized action
(4.2.6) and the self-coupling
(4.2.7) the action can be written in the form SCqI = SCVOI+ SoCAq; pol + S,NTCA(P; ~ 0
1 .
(4.2.8)
Let us remark that the theory with action (4.2.6) leads to the dynamical equations (4.2.3).The union of all sets of bounded fields (Acp'(x)}, on which the linearized action takes finite values, defines the tangent space U,, to 0 at cpo. Because the Hessian (4.2.2)is non-singular it possesses Green's functions defined by ;,S,j[qo]GjL= -8f
G'jj,S,~[q,]= -6'L.
(4.2.9)
If G'j is a Green's function, its supertranspose
..
(@T)'j=(-
1)EIEIG;
(4.2.10)
turns out to be a Green's function too. To pick out a unique solution of equation (4.2.9),one must impose proper boundary conditions. The Green's functions which play a crucial role in classical theory are the retarded and advanced ones, G,,, and Gadv.We anticipate that the reader is familiar with the boundary conditions appropriate for these Green's functions. Here it is worth recalling that G,,, is the supertranspose of Gad" and vice versa. The Green's function most important in quantum theory is known as the Feynman Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
291
propagator G,. Its defining property is that G, propagates only positive (negative) frequency modes in the remote future (past). Next, G, is known to coincide with its supertranspose (4.2.11 ) Both these properties make it possible to operate with the Feynman propagator as with an ordinary supermatrix. Let us recall that if H M Nis a supersymmetric supermatrix(l.9.78),as is the case for j,S,j[cp],then its unique inverse satisfies the supertransposition property( 1.9.81), which is as equation (4.2.11). Next, if H,, changes infinitesimally, H + H + 6 H , then its inverse acquires the displacement
6 H - = - H - ' ( 6 H ) H - '. Similarly, when the operator Hij= ;,S,,fcpo] suffers an infinitesimal disturbance, i , S , j [ ~ o+l ~ s , j C ' ~+oE ~ v I
b€A@ where E -,0, its inverse H - ' = - G, changes by the above law. Suppose the dynamical system under consideration is supersymmetric. Then it can be described in terms of the superfield variables u'(z), of which cpi(x)are the component fields, by re-expressing cp in S [ p ] via U in accordance with the instruction (4.1.394: S [ v ] = S[cp(v)] = S [ P v ] .
(4.2.12)
Equation (4.1.50~) tells us that every stationary point of S [ q ] corresponds to a unique stationary point of S[u] and vice versa,
S,$~ol
S,iC~ol= O
00 i
-- Q [~ C PIO .
(4.2.13)
In accordance with equation (4.1Sob), we have (4.2.14)
{ i~S,j[vo]GvJ} 20
V 6 v E AV\O.
As a result, the Hessian (4.2.2)is non-singular if and only if the super Hessian (4.2.14)evaluated at uo = Qcpo is non-singular. Finally, the analyticity of S [ q ] implies the analyticity of S[u], which is of the form
(4.2.15) where the superfunc.tiona1 1 So[A\v; u O ] =- A J i , S q j [ ~ 0 ] A u J 2 Copyright © 1998 IOP Publishing Ltd
(4.2.16)
292
Irie(i.crs ( r i d Methods of' Supersymmetry and Superyrmity
represents the superfield version of the linearized action (4.2.6), and the superfunctional (4.2.17) is the superfield form of equation (4.2.7). Let G'i be a Green's function of ;.S,j[qo]. Introduce the superfield t wo-point object
Gi.?=Qf;GijQST':" J '
(4.2.18)
Because of relations (4.2.9)and (4.2.14)this satisfies the equations ~.s.J[V~IGJ^'
= - 6;
i
G" j , ~ ,[U03 g = -6 R .
(4.2.19)
As is seen, G" is a Green's function of jqS,j[~,J. When Gii^is subjected to some boundary conditions, like the Feynman, retarded or advanced ones, the same boundary conditions are fulfilled by its superfield version G f j .The superfield version GFrJof the Feynman propagator G$ is called the 'Feynman superpropagator' and satisfies, due to equation (4.2.1I), the relation GFJ'=( - yJQ, (4.2.20) ^^
Remark. Let GU be a Green's function of 7,SJ['p0] and GI3 be its superfield version. Introducing the operators H>=qBt,S,j[qo] G i c;Lullj
Hi j = ul 1z ,S,JC~Ol G f, = G i R q g j
(4.2.21)
- l zj= -6 ij. H'KG
(4.2.22)
one obtains L H'$L G ;.- --#! J
Since H > and H; are connected by law (4.1.60), the last relations lead to G g = Q i;G Ij P J' j . (4.2.23) 4.2.2. Generating functional Under the basic assumptions made in the previous subsection, it is well known that the in-out vacuum amplitude and the mean values of time-ordered functionals of quantized fields, like n-point Green's functions, are given by the Feynman functional integral: (outlin)=N*
Copyright © 1998 IOP Publishing Ltd
s
~q~e('l~)~C~]
(4.2.24)
Quantized Superfields here h is the Planck constant, volume element is given by
293
.4k is a normalization constant, and the
The symbol q in the first line of equation (4.2.25)is to be understood as a qunatum operator, in the second as an integration variable. The integrations in both expressions are over field histories satisfying boundary conditions appropriate to the in- and out-states chosen. In the wKa-approximation used in practice, the integrations are restricted to some complexified neighbourhood which , is taken in the role of the classical of a dynamical history q o ~ O O ground state, the neighbourhood being specified by the Feynman boundary conditions: pure positive (negative) frequency in the remote future (past). The Feynman propagator is the unique Green’s function of ;,S,Jqo]corresponding to these boundary conditions. When operating with functional integrals (and later with superfunctional ones), we shall formally assume that all the properties of integration theory over Rp124 listed in subsection 1.1 1.5 generalize to the case under consideration, simply replacing functions by (super)functionals, superdeterminants by (super)functional superdeterminants and so on. As is well known, instead of computing chronological averages (4.2.25), for each Y [ q ] ,it proves sufficient to evaluate the generating functional for Green’s functions (4.2.26) where { J ; } is an arbitrary external source under the requirement & ( J i ( X ) ) = &i.
(4.2.27)
Because of the variational law
one finds (4.2.28) for every functional Y of quantized fields. The standard way of perturbatively computing Z [ J ] is to represent S [ q ] in the form (4.2.8) and to make use of the identity
Copyright © 1998 IOP Publishing Ltd
where ( S [ A q ;cp,]
+J ; A q ’ )
}
(4.2.30)
is the generating functional of the linearized dynamical system (4.2.6). The final Gaussian integration can be easily performed resulting in
(4.2.31a) or, equivalently
{
Z o [ J ]= sDet”’(AG,‘j) exp - - J;JjGF j ’ }
2h
(4.2.31b)
with H > and GFij being defined as in expressions (4.2.21). Here we have used the normalization condition
(4.2.32)
4.2.3. Generating superfunctional By analogy with the standard Feynman rules formulated above, one is in a position to develop a superfield quantum theory. First, one can introduce quantized superfields u:,,,~ related to the quantized fields qbuant by the law i “quam
ir:
= Q ;(Pquant*
Next, given a superfunctional ’€’ of quantized superfields, its quantum average can be defined by the superfunctional integral
JPvWuI exP{(i/h)SCulj (4.2.33)
((‘€‘ycvl))
exp((i/h)SCul)
where 9 v = 9v”5
and superfunctional integrals, similarly to functional ones, should be understood as sequences of Gaussian integrations. In particular, the ‘n-point superfield Green’s function’ is defined to be ((v’d2..
. urn>>
n > 2.
(4.2.34)
Introduce an external ‘supersource’ { Ji) representing a set of unconstrained, Copyright © 1998 IOP Publishing Ltd
Quuntized Superjelds
295
(4.2.35) and consider the ‘generating superfunctional for Green’s functions’
s It
Z[J] = Nv 9u exp
-(S[u]
+ Jp‘)
(4.2.36)
A‘al being a normalization constant. Then every average (4.2.33) can be
represented in the form
with 8 / 8 J being defined as *
(4.2.38) Therefore, the generating superfunctional contains all necessary information about the superfield quantum theory. To compute Z[J], we proceed in complete analogy with the calculation of Z[J]. Namely, we represent the classical action superfunctional in the form (4.2.15) and make use of the identity
where
s It
Zo[J] = Nv 9 v exp
- (S[Av; uo]
}
+JfAvL)
(4.2.40)
is the generating superfunctional of the linearized theory (4.2.16).Choosing the normalization condition
(4.2.41) the above Gaussian integral takes the form
Zo[J]
= sDet-
Copyright © 1998 IOP Publishing Ltd
(i’) {it -H
exp - - J~JJC,”
}
(4.2.42~)
296
I t i u ~m i d Methods of' Supmymnretry und Supergracity
or, equivalently (4.2.42b) with H : and GFij being defined as in expressions (4.2.21). Recalling the property (4.1.27) of the superfunctional superdeterminant, the final relation can be rewritten as
Zo[J] =sDet'I2(GFfj)exp
(4.2.42~)
4.2.4. Coincidence of Z[J] and Z[J] We now show that Z[J] coincides with Z[J], the latter being considered as a functional of (properly defined) component sources of Ji. To start with, let us define the component sources J; of the supersources J i by the rule:
J;= J&
=r J
~ J=; P ' ~
(4.2.43)
Such a choice of component sources provides us with the relation J p f = J;c~!
(4.2.44)
Considering now supersources and sources with upper indices defined as J' = JjqJf
Jt-
Jjqj'
(4.2.45)
we note that they turn out, due to equation (4.1.57),to be connected to each other by the law .
*
J'= pifjf
J~=Q$J'
(4.2.46)
that is, in the same fashion as of and (pi are connected. Recalling the definition of PsTand QsT (4.1.43),one finds J i = ( - l)Eft&IQST;iJr Jf=(-l)&+& PsT ! iJl. i'
(4.2.47)
Now, the use of the relations (4.2.23) and (4.2.43,47)gives JfJjGFJi= J;JjGFi'.
(4.2.48)
Furthermore, it is not difficult to prove that (4.2.49) By definition, we have
Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
291
and
On the other hand, because of equation (4.2.43)we have (4.2.50)
which confirms equation (4.2.49). As the next step, it is worth recalling that the relation (4.1.45) implies (4.1-46).The Feynman propagator is connected to its superfield version by the law (4.2.18),which leads to relation (4.2.23),and hence sDet(hG,'~)= sDet(hG,'j). (4.2.51) Finally, equations (4.2.12)and (4.2.44)tell us that S[u0]
+J p : =
+
(4.2.52)
S[cpO] J ; ( P ~ .
Now, the relations (4.2.48, 49, 51, 52) mean Z [ J ] =Z [ J ( J ) ] JAJ) = J$"p
(4.2.53)
Our final result is seen to be of great importance. Its obvious consequence is the fact that any chronological average (Y[cp]),Y being some functional, can be read off from the corresponding superfield analogue ( ( Y [ u ] ) ) , where Y [ q ] and Y[u] are connected as in equation (4.1.48). From equations (4.2.28,37,50,53) one deduces (WcpI) =((Q[@(u)l)) = ( ( W P V l ) ) . (4.2.54)
. .
In particular, the n-point Green's functions (cp'lcpi2. . . vi"),n = 2, 3, expressed via the superfield Green's functions (4.2.34)as follows:
. . ., are
and so on. In conclusion, it is worth saying some words about the general structure of Z [ J ] and Z [ J ] . Assuming that Z [ J ] can be expanded in a functional Taylor series, one immediately finds (Z[O])-'Z[J]=l+-J;(cp')+ 1
h
. .
Jg ...J~~J;,(cp"cp" ...cp") (4.2.56)
where the quantized field average (4.2.57)
Copyright © 1998 IOP Publishing Ltd
298
Ideas and Methods of Supersymmetry and Supergravity
is said to be the 'mean field'. Analysing the relations (4.2.29)and (4.2.31),one readily notes that = cpl,
(cp')
+ O(h)
(4.2.58)
so the mean field differs from the classical field cpo by quantum corrections. Further, the two-point correlator (cpcp) turns out to be of the form (4.2.59) In perfect analogy with Z [ J ] ,the generating superfunctional has the form i (Z[O])- Z[ J] = 1 + - Jr( (U'))+ A
Jr, . . . Jf,J', ((u f d 2 .. . v r n ) ) .
(4.2.60) The quantized superfield average ( ( U ' ) ) = -In Z[J]
16Jf
(4.2.61) ILO
will be called the 'mean superfield'. The mean field and the mean superfield are connected as follows (cpi)=P;f((uf)).
(4.2.62)
In other words, the union {(cp')) presents the full set of component fields for {((U'))}.Similarly to equation (4.2.59), the two-point superfield correlator is of the form h ( ( u ' ~ ~ ) ) = ( ( u ' ) > ( ( u ~ ) ) + ~ G ~ ' ~ + O ( h ~ ) . (4.2.63) 1
In summary, we have shown that ordinary perturbation theory and the superfield version defined above are equivalent for non-gauge supersymmetric systems. From now on, all computation work can be done using superfields, without any reference to components. Return to component fields should be carried out only at the final stage, with the help of the projection rule (4.2.54).
4.3. EBective action (super)functional
In this section we introduce a central object of modern quantum field theory-the effective action, in both component field and superfield approaches. A brief review of the renormalization theory will be given. We also present a wide class of (pathological) supersymmetric models finite at each order of perturbation theory, with trivial S-matrix. Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
299
4.3.1. Egective action
We begin by recalling that associated with the generating functional for Green's functions Z [ J ] is the generating functional for connected Green's functions W [ J ]defined by Z [ J I= e(i/h)WC-'l.
(4.3.1)
The expansion of W [ J ]in a power series in J reads
.. .
..
GCili2...in being the n-point connected Green's function. Gcij is usually called the full propagator and is related to the two-point correlator (4.2.59) in the manner:
(4.3.3) As is seen, the full propagator coincides with the Feynman propagator in the limit h - 0 ; on these grounds, GF is often called the tree-level propagator. By functionally differentiating WCJ],one obtains the functionals
(4.3-4) known as the mean field in the presence of sources. This coincides with the mean field when the sources are switched off
@ IJ=O
= (cp).
(4.3.5)
Owing to the non-singularity of GFij, the full propagator considered as a power series in h turns out to be perturbatively non-singular. Then relation (4.3.4)can be uniquely resolved with respect to J resulting in a functional of @, J ; = J x @ ] . Now, making the Legendre transform of W [ J ] ,one arrives at the following functional I-[@] = W [ J C ~-~;[@343' II
(4.3.6)
which is called the effective action. It possesses the obvious property t
(4.3.7) Setting here J =0, hence @ = (cp), gives
Copyright © 1998 IOP Publishing Ltd
300
Ideas and Merhods
of
Supersymmetry and Supergravity
therefore the mean field is a stationary point of r [ @ ] . Next, by functionally differentiating equation (4.3.7)and using the identity
which immediately follows from (4.3.4),one obtains ;,r.j[(cp)]G)'= -6; tc.
(4.3.9)
Comparing the relations (4.3.8)and (4.3.9)with (4.2.1)and (4.2.9),respectively, and also recalling that (cp) has been interpreted, due to relation (4.2.58),as the quantum analogue of cpo, it seems reasonable to interpret r [ $ ] as a quantum analogue of the classical action S[cp]. Clearly, knowledge of r[@] makes it possible to restore W [ 4 , with the help of the inverse Legendre transform, and therefore to calculate arbitrary chronological averages, using the prescription (4.2.28).On the other hand, it is well known that the S-matrix is uniquely read off from r[@]. As a result, all necessary information about quantum theory is encoded in the effective action. That is why the effective action is said to be the central object of quantum field theory. r [ @ ] is often called the generating functional for vertex functions, the n-point vertex function being defined as
r::...:=l--:+ :[+]I @=
I,,
,lll>...l,,
n22.
(4.3.10)
These functions appear in a functional Taylor series for r [ @ ] :
A@ = @ - (cp). Every vertex function is known to be represented by one-particle irreducible Feynman diagrams without external lines (recall that a connected Feynman diagram is said to be one-particle irreducible if it cannot be transformed into two disconnected pieces by cutting one internal line). Making use of the relations (4.2.26), (4.3.1) and (4.3.6,7) leads to the functional integral representation for r[@J:
ewor[~~ =
gcp e(ilhWSCfpl - r.X@l(@-@i))
(4.3.12)
which proves to give a self-consistent way for perturbatively computing r[@], One starts by performing the change of integration variables pi= @+hli2x' (whose Jacobian is assumed to be unity, which is the case for supersymmetric theories) and expanding S [ q ~ ] = S [ @ + h ' / ~ xin] a power series in h"*. This Copyright © 1998 IOP Publishing Ltd
Quantized SuperJields
301
gives
Ar[@]= r[@]- S [ @ ] . Next, adopting the ansatz
r[@]= s[+]+
m
n= 1
tinrcn)[@l
(4.3.14)
the above integral relation makes it possible to determine r("+l), provided represents the nth quantum correction all rei), i = 1, . . ., n, are known. to the effective action. It can also be shown that Feynman diagrams contributing to r(n) contain exactly n internal loops, so the expansion (4.3.14) is known as the 'loop expansion'. As a simple example, let us calculate the one-loop contribution to I-[@]. For this purpose we keep only A-independent terms in both sides of (4.3.13):
(compare this result with (4.2.31~)). Then we write 1
r(')[@] =- ln(sDet(q'L&,S,;[@]) 2
sTr ln(ViL~,s,;[@])
therefore the effective action up to second order in A is
I-[@] = S[+]
i
+-2 h sTr ln(q" L , S , ; [ + ] ) + o(v).
(4.3.16)
It should be remarked that relations (4.3.13) and (4.3.14) allow us to represent quantum corrections to r[+]as one-particle irreducible Feynman graphs without external lines arising in a perturbation theory in which the role of propagator is played by the Green's function of ;,S,j[@] under the Feynman boundary conditions, the functions S,;,...I, [@Iplay the role of vertices, @ being considered as an arbitrary background field. In the case of a supersymmetric dynamical system, the effective action turns out to be the component form of a superfunctional defined in perfect analogy with r[@].Let us introduce the generating superfunctional for connected Green's functions
W[J]
A
=T In 1
Copyright © 1998 IOP Publishing Ltd
Z[J]
(4.3.17)
302
Ideus cind Methods of Supersymmetry und Supergrucit})
define the mean superfield in the presence of supersources f
.U’=-
6Ji
(4.3.18)
W[J]
and perform the Legendre transform of W[J]
r[q=W[J[q]
-Jdfi]d
(4.3.19)
where Js should be expressed via v’s with the help of equation (4.3.18).The superfunctional r[q will be called the ‘effective action superfunctional’. Using the relations (4.2.50)and (4.2.53),one immediately has
=r[a(g)l
v’f(+)=~f;$.
(4.3.20)
The trivial consequence of this relation is that the component loop expansion (4.3.14) implies the superfield loop expansion
(4.3.21) n= 1
and vice versa. One can easily obtain the following superfunctional integral representation for r[G]: ,(iih)r[al=
,k* ”
vi^
2ve(i!h~S[u] -rmW-
69).
(4.3.22)
At the one-loop level, r[q is of the form
r[v’]= S[q +-1 h sTr ln(qfR~,S,j[C])+ O(h2). 2
(4.3.23)
Relation (4.3.20) tells us that all necessary information about the supersymmetric quantum theory is encoded in the effective action superfunctional.
4.3.2. Super PoincarP invariance of W[J] and r[C] Let us discuss the transformation properties of the superfunctionals W[J] and r[Gl with respect to the super Poincare group. At a purely formal level, the question is resolved as follows. In a supersymmetric dynamical system, the classical action is invariant,
S[g.v] =S[v]
(4.3.24)
under arbitrary super Poincare transformations -
vf=ei(-bUP.+[Ii2]K“hJ,h+~Q-&Q) U f = ( T ( g ) ) f j v J
Copyright © 1998 IOP Publishing Ltd
(4.3.25)
Quantized Superfields
303
the super Poincare generators being given in (2.4.36).Any mapping (4.3.25), considered as a replacement of variables in a superfunctional integral, is characterized by the Jacobian
sDet-'( (g.u')
L) = sDet-'(T(g)). 6VJ
Owing to the explicit structure of the super Poincare generators, the operator in the exponential in expression (4.3.25) is of the type (4.1.28). Then, in accordance with Theorem 2 from subsection 4.1.2, we have
(
LJ)
sDet (g.u')-
VgESll.
=1
(4.3.26)
As a result, all the super Poincare transformations are volume-preserving: 9(g*u)=9u
v g€Sn.
(4.3.27)
Now, setting the supersource J to transform contragradiently to (g.J)dg.u)'=Jfvf
U,
(4.3.28)
V g€Sn
we obtain, using equations (4.3.24, 27, 28), n
I.
(S[g-
-U]
+Jdg- ' .u)~)
therefore W[g*J]=W[J]
V g€Sn.
(4.3.39)
This implies, as a result of relations (4.3.18,19), super Poincare invariance of the effective action r[g.iq=r[iq vgem. (4.3.40) The final relation is the quantum analogue of equation (4.3.24)and expresses the fact that the dynamical system under consideration is supersymmetric at the quantum level. The previous discussion is not completely strict and requires some comment. First, the superfunctional W[J], as it has been constructed above, depends parametrically on the dynamical history uo E V chosen in the role ofthe classical ground state, that is, W[J] = W[J; vol. Second, when obtaining W[J; uo], we have integrated over some neighbourhood of uo, which is specified by the Feynman boundary conditions. So, the strict definition of Copyright © 1998 IOP Publishing Ltd
304
Idem und Methods of Supersymmetry and Supergravity
W[J; vO] is given by
+
( S [ u o Au] - S[uo] - Jf At.')
I
. (4.3.41)
Let g be an element of the super Poincare group. It moves vo into the dynamical history g uo E W O and the domain Xuo onto X18.uo.On the basis of arguments similar to those used to obtain relation (4.3.39), one now easily obtains W[g. J ; 9 ' uO] = W[J;
~ 0 3
V g E Sn
(4.3.42)
which represents the corrected version of (4.3.39). Suppose the dynamical subspace W O possesses a 'super PoincarC invariant dynamical history' vo, as is the case in practice, so that g'u,=u,
v g€Sn.
(4.3.43)
Setting here g = exp{i(t."Q, + chQdr)},Q, and Qibeing the supersymmetry generators, we conclude that Q,v&)=
Q,o',(z)
= 0 => d,o',(z) = 0 =>
0'0 =constant
(4.3.44)
where the anticommutation relation {QZ,
Qk}
= - 2i(o")d,
has been used. Next, setting g = exp([i/2]K"'J,,) in equation (4.3.43), one finds that only scalar components of {U;} may have non-vanishing values. Under equation (4.3.32), the domain E,,,turns out to be super Poincare invariant, and the relation (4.3.42) takes the form W[g*J;vo]=W[J;t.o]
V gESl7.
(4.3.45)
Then, the mean superfield proves to be constant ((2:'))
=constant.
(4.3.46)
In what follows, as a rule, perturbation theory will be based on the use of a super Poincare invariant d namical history in the role of uo. By redefining the superfield variables 2: ( z )+ v'(z)and the classical action S[u] +S[u] - S[oo], we arrive at the typical situation
vi
Y
U() = 0
S,+-O]= 0.
S[O] = 0
In addition, we shall adopt ((vi))
=o
which is really the case under reasonable physical assumptions. Copyright © 1998 IOP Publishing Ltd
(4.3.47) (4.3.48)
Quantized Superfields
305
4.3.3. Short tJ.ucursion into renormalization theory The effective action is usually calculated perturbatively within the framework of the loop expansion. It is quite typical that vertex functions are given by Feynman integrals divergent at large momenta. In this case the theory considered is said to contain ultraviolet divergences. Renormalization is a special procedure for reconstructing the theory, leading to finite vertex functions (that is, to a finite quantum theory). Normally, the renormalization procedure consists of two parts. The first is regularization. Regularization is a prescription according to which all initially divergent Feynman integrals determining vertex functions, are replaced by other finite integrals depending on some (regularization) parameter A. These integrals are said to be regularized. It is required that the regularized integrals formally reduce to the initial ones when the regularization parameter goes to some definite value A'. We say the limit A -+ A' corresponds to the regularization being taken away. There are a lot of popular regularization schemes in quantum field theory, Pauli-Villars regularization, dimensional regularization, higher derivatives regularization and so on. Each of them has advantages and disadvantages, the choice of scheme is a matter of convenience. The second part of the renormalization procedure consists in constructing counterterms. In more detail, suppose we have a theory of fields ip' (in this subsection index i corresponds to f in our previous notation) described by an action S [ @ ; RI, where d denotes all parameters of the theory (i.e. masses and couplings). The fields are assumed, for simplicity, to be bosonic. Introduce another theory of fields q idescribed by an action S [ q ; R , A ] , with R being the corresponding parameters, of the form S [ q ; R, A] = S [ q ; R ] +AS[q; R , A].
(4.3.49)
Here S [ q ; R ] is the original action expressed via new variables cpi and R , and functional AS[p; R , A ] is of the form A S [ q ; R, A] =
2 h'AS,[q; R , A]. -
(4.3.50)
r= 1
The ASrs are chosen in such a way that the effective action of the theory 50)is finite at each order of the loop expansion with the classical action (4.3.49, after taking away the regularization. The functional AS is called the action of counterterms (or simply counterterms). A fundamental result of renormalization theory is that in a local field theory counterterms ASr exist and are local functionals of the fields qi. It quite often turns out that the functional A S [ q ; R , A ] has a structure duplicating that of the action S [ q ; R ] in the following sense. Let
Copyright © 1998 IOP Publishing Ltd
306
Iiiiws and Mcthods of Supersymmetry und Supergravity
then bo
S [ q ; R ] = ~ S $ < q i i q i ? + R ~ ~ ~ ~ , , , i , , ( p i I q i ~ . . (4.3.52) .qi~l
2
n=3
with R!Y/2,,,i,, being parameters. AS[q; R, A ] is said to have a structure duplicating that of S [ q ; R ] if it is of the form
(4.3.53) It is important to notice that the coefficients of S 2 )and R(")are the same in equation (4.3.52) and equation (4.3.53). In this case the functional (4.3.49) takes the form S [ q ; R, A]=:
Sl:]JS{;/;
+(AZ 1r1i2) i l l 2 qiiqi2
n=3
Here di/i!::;k = d$,@;. . . St) is a symmetrized product of delta symbols. Let us denote
(4.3.55) Then equation (4.3.54)can be rewritten in the manner bo
-
S [ q ; R , A ] =~Sjf~~(Z,)i:/~cpii(pi?+ ER"! j l j ~ . . "('n)ilil...i" .j
jij2...jnqii,.+,i:.
., (pin
n=3
(4.3.56) Suppose that ( Z 1 x $ has the following structure ( Z 1 )jlj2=(Z;/2yl 1112 ( il(2 11' 2 Y'2 i 2 ) .
(4.3* 57)
This implies that the action S [ q ; R , A] can be obtained from the initial one S [ @ ;R ] with the help of the transformation
(4.3.58) where
Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
307
The relations (4.3.58) are known as the renormalization transformation, the quantities Z , (4.3.57) and Z , (4.3.59) are said to be the renormalization constants. The following terminology is often used in practice: the initial objects S [ $ ; I?], $ and I? are called the bare action, fields and parameters, respectively; the final ones S[cp; R, A], cp and R are called the renormalized action, fields and parameters, respectively. As is seen from the above discussion, the renormalized action is obtained from the bare one with the help of the renormalization transformation. Every theory possessing this property is said to be multiplicatively renormalizable. It has been pointed out that regularization is an important element of renormalization. A chief requirement of any regularization scheme, besides those stated above, is to preserve, as much as possible, symmetries of the theory at hand. Consider, for example, the most popular dimensional regularization. In this scheme a four-dimensional theory is continued to a d-dimensional space-time; the role of regularization parameter A is played by the space-time dimension d, A coincides with d = 4. Dimensionally regularized Feynman integrals are finite at d f 4 and may possess poles of the form (d-4)-k, where k = 1, 2, , , ,. Counterterms must cancel these poles resulting with finite Green's function in the limit d + 4. Clearly, dimensional regularization breaks down those symmetries of the initial classical theory which can be formulated only in four dimensions. For example, d = 4 massless theories are known to be scale invariant, but this invariance does not survive under continuation to d # 4 . As a result, one can expect the appearance of scale anomalies after renormalization. Similarly, applying dimensional regularization to supersymmetric field theories creates an analogous problem. The point is that supersymmetry can be formulated only in some exceptional dimensions. Regularization schemes suitable for supersymmetric field theories are those which preserve supersymmetry at every stage of the renormalization procedure. Such regularizations are said to be supersymmetric. Since only the superfield formulation explicitly displays supersymmetry, it seems evident that supersymmetric regularizations should be defined in terms of superfields.
4.3.4. Finite pathalogical supersymmetric theories One of the most remarkable properties of supersymmetry is that it leads to a great suppression of ultraviolet divergencies. Moreover, some super Yang-Mills theories are known to be completely divergenceless. The basis of this phenomenon is the fact that some divergent contributions, coming from bosonic and fermionic fields in a supersymmetric theory, cancel each other. A simple example of such cancellations is given by equation (4.1.27), which is equivalent to sTr(O)=(- l)"d'r=O. Copyright © 1998 IOP Publishing Ltd
(4.3.60a)
308
Ideas iind Methods of' Supersymmetry und Superyruoity
This can be rewritten, using equation (4.1.46), as ( - ]p6i;=(-1y16ii
1
s
d4xS4(0)=0.
(4.3.60b)
In spite of the divergent factor d4x S4(0), the full expression vanishes, since in every supersymmetric field theory the numbers of bosonic and fermionic fields coincide, hence ( - 1)c'6ii=0.
(4.3.61)
It is worth noting that many divergent structures arise with the factor ( - l)',J6ii,As will be demonstrated here, there exists a wide class of completely finite supersymmetric models which are, however, physically pathological. Of course, to construct realistic supersymmetric theories free from ultraviolet divergencies is much harder, but possible. Consider the model of a real scalar superfield V ( z )= V ( z )with the classical action (4.3.62) 9 being a real function of ordinary real variables. For simplicity, we assume B(0)= B'(0)= 0.
In our case, a super Poincare invariant supermetric of the form (4.1.54) is uniquely fixed modulo a constant set to unity:
s
ds2= d82 6 V ( Z S) V(Z).
(4.3.63)
By superfunctionally differentiating S [ VI twice, one obtains 62S[ V
]
= - ( 0 - B"(V ) ) P ( Z- 2')
6 V(Z) 6 V(2')
In accordance with equation (4.3.22), the effective action is e ( i / h ) r C ~= l
Jv
where F'[U](V-U)=
s
c j j e(i/h)(SCvl-r'[Ul(f"~
s
U))
d8z-
U being a scalar superfield. In accordance with equations (4.3.23) and Copyright © 1998 IOP Publishing Ltd
Qiiantized Superfields
309
(4.3.63,64),the one-loop correction to I-[ U ] reads
Since the operator (4.3.64)is of the type (4.1.26),we conclude as a consequence of Theorem 1 from subsection 4.1.2. But this is not the end of the story. With the help of the supergraph technique described in the following sections, one can readily find that all higher-loop corrections to T[U] also vanish, I-(~’[U]=O
n = 2 , 3,
. . ..
Therefore, the theory under consideration is finite and trivial as a quantum system: its effective action coincides with the classical one
r[ul= S[U]
(4.3.65)
quantum corrections are absent. From the physical point of view, the theory with the action (4.3.62) is pathological, since the kinetic terms of all component fermionic fields include second-order differential operators. O n the other hand, the corresponding dynamical equation 0V = Y ( V )
is non-trivial, due to the arbitrariness of P(V). In this respect it seems unexpected that the effective action is trivial (hence, so is the S-matrix). The example described can be generalized, for instance, to the case of models of (anti)chiral superfields 6 ( z ) and Wz),DdrO=O,of the form
More generally, every supersymmetric theory with an action S [ Vg, @E, 6&], which does not possess V - 0 , V-6 or 6-0 mixings or does not contain terms with spinor covariant derivatives acting on superfields, proves to be finite and trivial, since in such cases the super Lagrangians will have a structure similar to that discussed above.
4.4. The Wess-Zumino model: perturbative analysis
4.4.1. Preliminary discussion In this section the superfield perturbation theory described above is illustrated using the example of the standard Wess-Zumino model (3.2.11 ) . As will be Copyright © 1998 IOP Publishing Ltd
shown later, a general Wess-Zumino model (3.2.8)fails to be multiplicatively renormalizable unless the chiral superpotential takes the form 1 6p,(@)= - m@' 2
+ -31 i.Q3
(4.4.1)
where I H and i are constants. This is why our consideration is mainly concentrated on the standard case. We begin by discussing some general aspects essential to every supersymmetric model (3.2.8).The space of histories V of such a system is parametrized by pairs of superfields: a chiral scalar O(z) and its conjugate antichiral scalar @(z). So, the super Poincare invariant supermetric of the form (4.I .54) is fixed, modulo a complex constant. as follows
s
ds2= d6z S@(z)S@(Z) + C.C.
(4.4.2)
Its component form defind by (4.1.56)reads
where A, )%I and F are the component fields of 0 (see equations (2.8.4)and (4.1.41)). By superfunctionally differentiating S [ @ , @] (equation (3.2.8)),with the help of the variational rules (3.1.46),one obtains 6S 1---D2G+g;(@) 6@(z) 4
-=
6S 6@(z)
-=
--
4
DZ@+B;(@). (4.4.3)
Taking the second superfunctional derivatives gives Y:(@)~+(z, z')
6G(z)S@(z')
1-
--D26-(z,z') 4
6G(Z)S@(Z') (4.4.4)
The final expression represents the superkernel of the linear operator
(4.4.5)
Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
31 1
which maps the tangent space T(,,m) to V at (@,a) into its dual space. For a fixed Y, the H(v) can be considered as a linear operator acting on 9, due to the explicit form of the supermetric (4.4.2). The operator H“y)will play an important role in subsequent considerations. Here it is worth remarking that H(*) arises in the quadratic model of (anti)chiral scalars 2 and x coupled to external (anti)chiral scalars ‘P and 4‘ ’ with the action
] D,x
(4.4.6)
= D,Y = 0.
When Y=Y:(@,),(Do being a stationary point of expression (3.2.8), this superfunctional represents the linearized version (defined, in general, by equation (4.2.16)) of the Wess-Zumino action. A Green’s function (4.4.7)
of the operator H(yu)is defined to satisfy the equation where O(Z,
z‘) =
1.
0 z’) S-(z, O z’)
(4.4.9)
The signs (i) indicate that G(y) is chiral (+) or antichiral (-) with respect to the corresponding argument; in particular, DkG + +(z, z’) = DkG + +(z, z’)= 0. Below GCy)will always mean the unique Green’s function of H(y) under the Feynman boundary conditions. 4.4.2. Feynman superpropagator In the case of the standard Wess-Zumino model, there are two super Poincare invariant dynamical histories determined, due to equations (3.2.9)and ( 4 . 3 4 , by the equation -I;”;((Do)
= moo
+ -21A@;
=0
(4.4.10) Copyright © 1998 IOP Publishing Ltd
Both histories can equally well be taken in the role of classical ground state, since they are the only points minimizing the scalar potential (3.2.18). Because of the identity
the use of the latter solution in equation (4.4.10) is equivalent to the use of the former corresponding to the chiral superpotential (4.4.1) with m + - m (physically, only the combination lmI2 proves relevant). Therefore, we choose = 0, and the decomposition (4.2.15) for the Wess-Zumino model takes the form
S [ @ , 61=So[@, So[@, 6 1= Jd8z6@
61+ S,NT[@, 61
+
{:
m {d6rB2 + c.c.
'?^
SIN,[@,61= - i d6z (D3 3!
]
(4.4.1 1)
+ C.C.
As is seen, the linearized action is determined by the operator H(").
To find Green's functions of H"), it is worth recalling the identity
Di@=O
* DZD2@=160@.
(4.4.12)
Now, one easily verifies H(m)H(- *)-(U
+I2).
(4.4.13)
As a result, the formal solution of equation (4.4.8) with Y = m reads
1
G ( m ) = -H(-*)
(4.4.14)
- Iml"
The Feynman superpropagator is specified by the well-known &-prescription: G=G(m)= -H(-fi)
1
-/mlz+iE
E+
+o.
(4.4.15)
From now on, we set m to be real. Then G can be explicitly written as
This relation shows that the elements of G are expressed via the 'chiral scalar Copyright © 1998 IOP Publishing Ltd
3 13
Quantized Superfields superpropagator’
(4.4.17a) and the ‘antichiral scalar superpropagator’ 1 6-(z, -mmZ+is
G,(z, z’)= -
2’).
(4.4.17b)
Both G , and G , admit a representation similar to the well-known proper-time representation for the scalar propagator invented by J. Schwinger: -
1 -m’+ie
:j
a4(x - x‘) = i
ds U ( x , x’; s)
(4.4.18)
where U ( x ,x’: s) is the unique solution of the equation (4.4.19~) under the initial condition
+
U ( x ,x’; s + 0) = 64(x - x’).
(4.4.19b)
U ( x , x’; s) reads
{
U ( x , x’; s)= - i exp i [(””)’ ~(411s)’ 4s
m2s]].
(4.4.20)
Using the explicit form for 6 +(z,z’) given by equation (4.1.17),one now has P3u
G,(z,z’) = i J
ds U&, z’; s)
0
(4.4.21)
1
U,(z, 2’; s) = - -(6- 0’)’ exp (4ns)’ Obviously, U&) is the unique solution of equation (4.4.19~) under the initial condition U&,
2’; s +
+ 0) = 6
+
(z, 2’)
and is chiral in both superspace arguments.
4.4.3. Generating superfunctional Introduce an external chiral-antichiral supersource (J(z),J(z)) Copyright © 1998 IOP Publishing Ltd
DhJ = DJ =O.
(4.4.22)
3 14
Ideas und Methods of Supersymmetry and Supergruoity
Using the notation S , = {d6z J(z)@(z) s d 6 z J(z)@z) = J@+E
+
(4.4.23)
the generating superfunctional of the Wess-Zumino model is defined as
s
z[J, 51= ,J/ 9 @ g@ei(SO+fS'hT+SJ),
(4.4.24)
After integration, this reduces to the general form (4.2.39,42), the specialization to the case under consideration being as follows: Vo=(@o,@o)=O Hj=H'"' I G,"=GG. Note that sDet (H("))=sDet-'(G) is a constant independent of J and 5. In fact, it can be shown that
sDet(H(") = 1.
(4.4.25)
This leads to
(4.4.26)
where
(3 s
(J,J)G - = d6z d6z' J(z)G, +(z, z')J(z')
s s
+2 +
d6z d62' J(z)G+ -(z, z')J(z')
d6zd62'J(z)G- -(z, z')J(z').
(4.4.27)
Using (4.4.16), the final expression can be rewritten in the manner [d6z J
m --'+is
J + 2 [d6z J
Since J is chiral, we have
1
4 l d 6 zJ
1 D2J=-"Z+ie
As a result, we arrive at the expression
Z,[J, 51= exp{i(
Copyright © 1998 IOP Publishing Ltd
1D2J+ [d6i 5
1 U-m2+i&4
-j d ' z
s
d82J
5
1 -m2+iE
' J -m2+iE
m 5. -m2+iE
5.
Quantized Superfields
3 15
‘ J -m2+ic +m 2 sd6iJ
J)}.
--’+is
(4.4.28)
Let us introduce into consideration chronological averages for the linearized theory with the action So (4.4.11) using the general rule:
s
( ( A [ @ ,61))~
MEDDA[@, GI eis0[*,’1
A being a superfunctional. Then components of the Feynman superpropagator are given as the correlators:
G + - (z,z’) = G - + (z’,z ) = i( ( @ ( ~ ) @ ( z ’ ) ) ) ~
(4.4.30)
G + +(z,z’)= i( (@(z)@(z’)))o G - -(z, z’) = i( ( @ ( z ) @ ( ~ ’ ) ) ) ~ . It is clear that Z[J, 33 is expressible as a superfunctional Taylor series in J and 5 with the coefficients being complicated linearized correlators of the form (4.4.29). In practice, instead of using equations (4.4.27,28)involving integrations over the (anti)chiral superspaces, it proves more convenient to express the components of G according to the law
(-a) /
G + +(z, z’) =
(
r\2
D2D’2K+ +(z,z’)
J-
G+-(z,z’)= -- D2D”K+_(z,z‘)
(4.4.31)
where 1 -m2
K + -(z, z’) = K - +(z, z’) K ( z , z’)= -~ = - 64(8 -
e’)
D2 K + +(z,z’)=m-K(z,z’) 40 Copyright © 1998 IOP Publishing Ltd
~
6 * ( Z - z’)
1 64(x -XI) -m2
(4.4.32~)
D2
K--(z,z’)=m-K(z,z’). 40
(4.4.32b)
3 16
Ideas und Methods of Supersymmetry and Supergravity
Here we have omitted the (is)s in the denominators, which are assumed here and below. Note also that in deriving equations (4.4.32) we have used equations (2.5.30g)and the identities D,d8(z - z') = - D:S8(z - z') (4.4.33) D$(Z - z') = - D;68(z - z'). Now, the relation (4.4.27)can be rewritten as
J(z)K+ +(z,z')J(z')
+ 2J(z)K
+
- (z, z')J(z')
+ 3(z)K- - (z, z')J(z')
and the free generating superfunctional takes the form
{
s(
1
m
DZ
J + 3 DZ J)}. 8 ~ ( 0 - m ~ ) (4.4.35) Both expressions involve only integrals over the full superspace.
Z,[J,J]=exp
-i
d'z 3-
m-m2J+8Jo(n-m2)
Remark. The labels (k)carried by the Ks do not indicate chirality or antichirality in the corresponding argument, as was the case of Gs, but only the relationship between Ks and Gs (4.4.31). The representations (4.4.27) and (4.4.34) lead to different supergraph techniques, which we now describe. 4.4.4. Standard Feynman rules To obtain Feynman rules corresponding to the original representation for Z,[J, 33 given by equations (4.4.26,27),one must give the explicit structure of interaction (4.4.1l), which implies
and the variational identity
Then, the Feynman rules in coordinate space are: 1. Propagators: QQ
- iG + +(z,z')
QZ
-iG+-(z,z')
QQ
-iG--(z,z').
Copyright © 1998 IOP Publishing Ltd
(4.4.37)
Quantized SuperJields
3 17
In the massless case, the @@- and 66propagators vanish, as is seen from equation (4.4.16).
2. Vertices: There are chiral (03) and antichiral (a3) vertices denoted by 0 and 0,respectively. Each chiral vertex is integrated over d6z, each
antichiral over d%,
s
0 =iA d6z
s
@=iX d62.
(4.4.38)
3. The usual symmetry factors for graphs should be taken into account. Our chief goal consists in finding the effective action r[@,6] of the Wess-Zumino model defined by 1 r[@, 61= -:In
Z[J, 31 - J @ - E
(4.4.39)
1
where
(:)=(
6/6J-)-In 1 Z[J,J]=('++ '+-)(!)+loop 6/6J i G-+ G-- J
corrections.
To read off r[@,6] from Z[J,J], one considers only the one-particle irreducible diagrams, amputate all the external lines and insert in their places @ and 6 , according to the rule
J
.
/
G,
( G * A
/ +
i.
- iG.s
G,
=
-iG.8
\/
-iG.*
I
'p
- iG+8
where A, B = f,and similarly for 6-insertions. In addition, each diagram must be multiplied by (-i) (see equation (4.4.39)). The Feynman rules described here were originally developed by A. Salam and J. Strathdee. As a direct consequence of these rules, let us prove the following assertion.
Theorem I . Every L-loop ( L a 1) supergraph containing only chiral (antichiral) vertices vanishes. Proof. Consider a connected L-loop supergraph of chiral type. It involves only G , +-propagators, chiral in both arguments. On these grounds, it is convenient to use the chiral representation (see subsection 3 . 1 4 , in which chiral superfields depend on x and 0 only, to analyse the 'diagram. Using Copyright © 1998 IOP Publishing Ltd
3 18
I d w s unci Mcfhfs
of Supersymmetry
and Supergruuify
equation (3.1.66), in the chiral representation we have m m G++(z,z‘)=6 + ( z ,2’) = 6 2 ( e - e,) ~
64(x - x’).
-m
-m2
As is seen, the variables x and 0 are separated. By assumption, the diagram contains a closed cycle formed by k points
When k > 1, its contribution to the diagram is proportional to + +(zl, z2)G++(z2,2 3 ) .
*
.
G + + ( Z k - 1, zk)G+ -62(e1 -e2)62(e2
zl)
-e3). . . -o,)~(o,-e~) =d2(e1 -e2p2(e1-e3).. . s2(el-8k)d2(8k-&)=O
where we have used the identities 6 2 ( e - e’)j-(e’) = 6 2 ( e - e’)j-(e)
62(ek-
(s2(e))2
= 0.
(4.4.40)
The case k = 1 corresponds to the tadpole
which is zero, owing to the identity 6 2 ( e-
e)= 0.
(4.4.41)
Remark. Just the same line of arguments can be used to prove equation (4.3.65).
Corollary. Every supergraph containing a closed purely chiral (antichiral) cycle vanishes. For example, the following two-loop supergraph
produces zero contribution to the effective action. Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
3 19
In accordance with Theorem 1, non-trivial supergraphs are those involving both chiral and antichiral vertices. From the viewpoint of practical calculations and general analysis, to handle such diagrams requires some improvement of the Feynman rules. The point is that neither chiral nor antichiral representations are helpful now, and one must treat S+(z,z') and S-(z, z') in the standard fashion 1-
S+(Z,z')= --D2J8(z-z') 4
1 S-(Z, z')= --D2S8(z-z') 4
with all the superspace coordinates being on the same footing and the most reasonable separation being of the form S8(z)= S4(x)S4(8). Nevertheless, similarly to the proof of Theorem 1, it is reasonable to expect to get some simplifications caused by the properties of the Grassmann delta function S4(S) like those given in equations (4.4.40)and (4.4.41).To make use of such properties, one must convert the integration measures d6z = d4x d28 and d62=d4x d28, associated with vertices, into daz =d4x d48 in accordance with the laws
(4.4.42)
since the delta function S4(8) corresponds to the measure d4S. 4.4.5. Improved Feynman rules Following M. Grisaru, M. RoEek and W. Siegel, we now consider the reformulation of the above Feynman rules based on the use of the representation for superpropagators (4.4.42), the identity
(D2U1)(D2U2)...( D2U,)=D2{U1(D2U2). ..(DW,))
and its conjugate, and the conversion rules (4.4.42). Together these imply that for each chiral vertex with three internal lines attached we have
where
Copyright © 1998 IOP Publishing Ltd
Ideus cind Meihods of' Supersymmetry und Superyruvity
320
and @ A ) means that W )acts on the second argument of K+A; for each chiral vertex with one external and two internal lines attached we have
= Sd'z
@(z)K+ A ( ~zI)( , -
a
3'")
{ (- D2)K
+ ,(z,
z2)( -
a
B(B))}
and so on. Equivalently, one could start with the representation for Z,[J, 31 (4.4.35)and use the following integral version of the variational rule (4.4.36)
= Sd'z S8(z- zl){
-a
D2d8(z-z2)
ii :--
D2d8(z-z3)
I
for calculating Z[J, J]. As a result, we get the improved Feynman rules for the effective action in coordinate space. 1. Propagators:
a i - line:
2
Z'
--
I
a-m2
~'[Z-Z'I
(4.4.43)
(5'1
(4.4.44)
2 . Vertices. chiral
(03)
= tX/d'z
antichirol
0 =lx/dOz
.
Associated with every chiral (antichiral) vertex with n internal lines attached are n - 1 factors -iD2 ( -aD2) acting on n- 1 internal lines. All external lines (CP- or &-ones) arise without any factors. 3. The standard combinatoric factors for graphs. Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
321
Remark. The following property of @@- and &%-lines is often useful:
(421
(-;E*)
4
mIiE2)
=+
One of the main advantages of the improved Feynman rules is the fact that for a graph with V vertices one can easily integrate V - 1 vertices in 0,O. Each internal line represents itself as the superpropagator
or im
o(o - m 2 )
P ( -~z / )= 64(e - el)
im 64(x - XI) o(n-m2)
with some number of D and D factors acting on z o r z‘. All these factors can be transfered onto other lines (internal or external) with the help of integration by parts
- After this, a4(0- el) factors out, and the integral over, for example, d48’ is immediately done. The validity of integration by parts turns out to be obvious by rewriting the Feynman rules in momentum space. Let us make the transformation to momentum space for the mean superfield U = (@, 0)
the superpropagators X = ( - iK + -, - iK + +, - iK - -)
and the delta function
Copyright © 1998 IOP Publishing Ltd
322
Ideas und Merhods of Supersymmetry und Supergravity
Here and below in this section we take the convention that 8 means all the odd variables 8" and Qjl. In momentum space the covariant derivatives D, = (2,,D,, Di) take the form
(4.4.47) and their algebra has the form
(4.4.48) The Feynman rules in momentum space read as follows: 1. Propagators:
(4.4.49)
2. Each vertex is integrated over d48. 3. Associated with every independent loop is the momentum integral j d4P" 4. Associated with the external momenta is the overall factor
The other factors are the same as in coordinate space. The operators D,(p) and D6(p) can be handled as ordinary covariant derivatives. It is worth remembering, however, that the coordinate space expressions 4
Z' Copyright © 1998 IOP Publishing Ltd
0,:
Quantized Superjields
323
correspond in momentum space to
respectively. The momentum space analogue of the relation
D,d8(z - z') = - D;S8(z- z') reads
D , ( ~ ) s ~-( 0o'1
= - D;( - p
) ~ ( -e
e')
(4.4.50)
which implies
(4.4.51 a) hence
(4.4.516) and so on. Using the basic property of the Berezin integral,
f being arbitrary, we have the rules for integration by parts:
(4.4.52)
The coordinate space version of the former coincides with equation (4.4.46). It is easy to see from the previous consideration that the momentum dependence of factors D(p) and D(p) arising on internal or external lines can be omitted without any misunderstanding. Copyright © 1998 IOP Publishing Ltd
324
Itkeas tint1 Mcilrodr of‘ Super.yymmetry und Superyruvity
4.4.6. Example of supergraph calculations
Let us calculate the one-loop correction to the two-point vertex function. The relevant supergraphs are:
(4.4.53b)
(4.4.534
In accordance with Theorem 1, the second and third graphs vanish. The contribution from the first graph is i
2
1).[d’z
d’z’
a(;)[ ~
1 d8(z- z’)][ D2D” -m2 16
1
-(D2D’2)64(8 - 8‘)
16
2
~
= 1.
e=@
d4x d4x’d48G(x, e)@(x‘,B)[G(x, x‘)]’ G(x, x’)= -
1 ~
-m2
1
1 68(z - z‘) @(z’). -m2
(4.4.54~)
64(x - x’).
Repeating the calculations in momentum space, where the graph ( 4 . 4 . 5 3 ~ ) reads
Copyright © 1998 IOP Publishing Ltd
325
Quantized Superjields 1
=i - Z i
2
!y j(2n)4 jd4R d48’a(- p , 8)@(p, e‘) 1
1
D2(k)D”(- k)d4(8 - el)
Applying the above arguments reduces this expression to
I
(4.4.54b)
where A(p2,m2)=i
-
1
1
+
( p - Q2 m2 k2 + m 2
4::({
(4.4.55)
is an ordinary loop integral divergent at large momenta. Therefore, we must introduce a regularization scheme in order to make our diagram sensible. 4 . 4 . 7 . Supersymmetric analytic regularization We shall mainly use the so-called analytic regularization based on the replacement of each factor 1 p2 +m2-is
=i
1;
ds e - is(p*+ ma)
(4.4.56)
in equation (4.4.49)by its regularized version
an analytic function of a complex variable w (regularization parameter), o -+ 0 at the end of calculations; p is a real parameter with dimensions of mass (normalization point) introduced to fix the dimensions of propagators. There is no need to regularize l/p2-factors in equation (4.4.49),since, in accordance with Theorem 1, neither 00-nor G - l i n e s form closed loops. It is obvious that every non-zero closed loop has to involve at least two @$-lines. By simple - power counting one can see that closed loops containing 00-or 00-lines produce finite contributions. Note also that the l/p2-factors in superpropagators are often cancelled by rule (4.4.49,as is the case for the two-loop supergraph (4.4.58)
contributing to the two-point vertex function. Copyright © 1998 IOP Publishing Ltd
326
Ideas urd Methods of' Supersymmetry unci Supergrmity
The Fourier transform of (4.4.57) reads G(.u. x' 1 U ) = ip2"'
[I
ds (is)'" U(x, x'; s)
(4.4.59)
with U(x,s';s) being the kernel (4.4.20), and represents the regularized form of the propagator (4.4.18).Then, owing to the presence of S4(6-6') in expression (4.4.32u), the regularized form of K(z, 2') can be written as
(4.4.60) where wO(z,~ ' ) = ( X - ~ ' ~ + ~ ~ C T ~ ( B - F ) - ~ ( ~ - ~ ' ) C T ~ B . It has been shown in Section 2.4 that the two-point function wa(z,z') is invariant under supersymmetry transformations. Therefore, the regularized superpropagator is explicitly super Poincare invariant. As a result, the regularization scheme described preserves supersymmetry at every stage of perturbation theory. Remark. Both expressions (4.4.59) and (4.4.60) are well behaved on the light cone when Re 0 > 2. In conclusion, let us point out that divergences arise within the framework of analytic regularization as poles in U . For example, the regularized form of (4.4.55)reads
and its direct calculation leads to 1
1
A(p2,m21w)=--+finite (44240
terms.
(4.4.61)
To cancel this divergence, we must introduce into the bare action the following divergent structure ----[d'z@D, /./. 1 (4422 0 7"
4.4.8. Non-renormalization theorem
Looking at the one-loop contribution to the two-point vertex function
(4.4.54), one can see that the integrand is local in the odd variables 6 (a function of only one 9" and 8,) and non-local in the odd variables x or p (a function of two xs or, equivalently, a non-polynomial function of p ) . The Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
327
locality in B turns out to be a general property of arbitrary supergraphs, in accordance with the theorem proved by M. Grisaru, M. Rotek and W. Siegel. Tlzeorenr 2 (Non-renormalization theorem). Each term in the effective action can be expressed as an integral over a single d4B. Proqf: Consider a one-particle irreducible L-loop supergraph. Recall that each vertex is integrated over d4B, each line represents d4(8-8’) with some number of factors D and D acting on it (we suppress p-dependence). Choose in the graph a fixed loop involving, say, n vertices. Associated with this loop is a cycle d4(B1 - 8,)d4(02 - 8,). . . d4(8, - with derivatives Ds and Ds acting on delta functions. Integrating by parts, one can transfer all the covariant derivatives acting originally on d4(8, - 0,) onto d4(8, or d4(8,, - 0,) or external lines to the loop. Now, performing the integral in O2 removes 64(8,- e*)and replaces 8, by el everywhere. Let us continue this procedure n- 1 times. Then, the cycle is reduced to a single delta function, and one obtains an expression for the graph of the form
e,)
e,)
j d 4 e Aj d 4 6 ,
.w,,e A ) { ~ . . . ~ 8 4 ( e n - - 1 ) } ~ B , , = 0 1
(4.4.62)
where the label A enumerates the vertices external to the loop. The last term here is easily evaluated. As a result of the anticommutation relations for the covariant derivatives, any product of D and D factors can be reduced to an expression involving at the most four such factors. Since
d4~~,-B1)=~~,-~1)2(8n-81)2 one needs exactly two Ds and two D s for expression (4.4.62)to be non-zero. In this case, we have 1 -DDZD2d 4 ( B f l - 8 , ) l ~ , = ~ , = l . 16 As a result, the entire loop has been contracted to a point in &space. Continuing the above procedure loop-by-loop reduces the entire graph to a point in &space, and the total contribution takes the form
I!Jd4pi I d 4 0 F(pi, e) where p i are independent external momenta. Corollary 1. The general structure of the effective action of the Wess-Zumino model is as follows
d4x, . . . d4x,
jd.0 <(xl,
. . ., x,)F1(xl, 8).. . F,(x,,
e). (4.4.63)
Copyright © 1998 IOP Publishing Ltd
328
Idotis
cind
Metliods of Supersymmetry und Supergravity
Here Js are translationally invariant functions on Minkowski space, Fs are G and their covariant derivatives, local functions of 0,
Fi=Fi(@,
a, DAO,DAG,,. ..).
(4.4.64)
Remark. Each term in structure (4.4.63)is represented by the integral over d48, but not over d28 or d2G. Certainly, this does not mean that there are no purely chiral or antichiral contributions to the effective action. For example, the following term is possible:
where we have indicated explicitly 8- and &dependencies and used the fact that in momentum space ~ ( p6,, 8)= esUu8p~@(p, e).
The contribution from the graph (4.4.58) is exactly of the given type. Corollary 2. All vacuum supergraphs vanish.
Proof. In accordance with Theorem 2, every vacuum supergraph can be represented as
d [d4B where d is a loop momentum integral. The entire expression vanishes, since
s d40 =0.
This last result implies that the generating superfunctional (4.4.21) is naturally normalized by
Z[J=O, 3=0]= 1.
(4.4.65)
Other consequences of the non-renormalization theorem will be discussed below.
4.5. Note about gauge theories The equivalence of Feynman rules in the component and superfield approaches has been established in the Section 4.2 for non-gauge supersymmetric theories. In principle, the proof can be extended to the case of gauge supersymmetric theories, but here it will depend on the class to Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
329
which a given dynamical system belongs. As is known, there exist several different types of gauge theories (reducible or irreducible, possessing closed or open gauge algebra and so on), and for each of them there are specific features to be borne in mind when formulating Feynman rules. We find it reasonable to take the following line: to expose the main types of gauge theories, but to recall the Feynman rules only for irreducible theories with closed algebras. This is precisely the family to which the super Yang-Mills models considered in Section 3.5 belong. The Feynman rules for general gauge theories with linearly dependent generators and open algebras have been obtained by I. Batalin and G. Vilkovisky, and we refer the interested reader to the original publication'. 4.5.1. Gauge theories Let v i = cp'(x) be the field variables, and S [ q ] be the classical action of a dynamical system. Suppose that on the space of histories 0 there exists an
infinite set of pure right supervector fields L
E(R,)= E,
'
&(Rig) =EL
(4.5.1)
+E ,
non-vanishing along the dynamical subspace m0, R&l,=,,ZO
v VoE@o
(4.5.2)
and satisfying the requirement SRa = S , ; [ q ] R ' , [ q ]EO.
(4.5.3)
Here B =(a, x'), the index a running over some finite number of values. Under the above conditions, the theory is said to be gauge. Associated with R , are the infinitesimal gauge transformations
(4.5.4) leaving the action S [ q ] invariant, where the parameters ("(x) are arbitrary pindependent functions over space;time (but reasonably constrained at infinity). The R g or, equivalently, R',[q] are called generators of gauge transformations. Equation (4.5.2) implies that the gauge freedom does not reduce to a trivial invariance of the form
'?I Qj'
..
gy= -(- 1pnij.
(4.5.5) Obviously, every such transformation does not change S [ q ] and coincides with the identity map restricted to aO, &,i=s,.[J
'LA. Batalin and G.A. Vilkovisky, Phys. Reo. D 24 2567, 1983 Copyright © 1998 IOP Publishing Ltd
330
I d e m arid Methods of Supersymmetry and Superyruvity
Owing to equations (4.5.2) and (4.5.3), the Hessian ;,S,j[cpO], at each point
cpo of Qo, is degenerate (that is, it possesses zero-eigenvalue eigenvectors of
compact support in space-time) and has no Green functions, since
~ , s , j [ c ~ o I R ' ~=O* [~oI
(4.5.6)
We assume the set { R B }to be complete in the sense that each infinitesimal displacement 6cpi(.w) leaving S[cp] unchanged can be represented as a cp-dependent gauge transformation modulo the dynamical equations:
S,j[cp]Scp'=O
=.
6cp' = R 'k [cp] ( e [cp] + S ,j [ cp] Q j [cp]
(4.5.7)
with the Rs being constrained as in equation (4.5.5). This means that all the degeneracy of ; . S , j [ ~ ois] due to the gauge invariance (4.5.4). Introduce the super Lie bracket of generators
[R,, R j } = RcR/i-(- 1)"""'RfiRfi
(4.5.8)
leading to a pure supervector field on Qo. For arbitrary infinitesimal parameters (; and (:, E((:,*)= E%, the transformation
6cp;= cp;[Rk,R p } ( f ( i leaves S [ q ] invariant, as a consequence of (4.5.3). Therefore, equation (4.5.7) tells us that
*
Z
[R,, Rp) =R,C',p+-S,jE"jlp
?
(4.5-9)
6cp'
where the Cs and Es satisfy the relations
(4.5.10) The coordinate form of equation (4.5.9)reads
R'k,$cp] R$[ cp] - ( - 1)B'BPR'),;[ cp] Rj,[ p ] = R\[cp] CiB[cp]+ S ,j [ p]E'&p['p]
(4.5.11) The equations (4.5.9) and (4.5.10) define the gauge algebra of the theory. The functionals C\g[p] are called the structure coefficients of the algebra. The gauge algebra is said to be closed when Ej;B[cp] = 0
(4.5.12)
otherwise, it is called open.
Remark. A famous example of a gauge theory with an open algebra is Einstein supergravity (see Chapters 5, 6). This theory is described by fields Copyright © 1998 IOP Publishing Ltd
Quantized Super5elds
331
'pi= (e,", Y m T T",', B, B, A"}, where eomis a vierbein, Y"') is a spin-? field called the gravitino, (B, B, Am) are supergravity auxiliary fields. The supergravity action functional (6.1.12) proves to be invariant under the general coordinate and local Lorentz transformations and the local supersymmetry transformations (5.1.4548).Before eliminating the auxiliary fields, with the help of their equations of motion (6.1.14), all the transformations form a closed algebra (with some field-dependent structure coefficients). However, after eliminating the auxiliary fields one discovers the presence of the terms SSjSY", and SS/6Tm' in a commutator of two supersymmetry transformations. The gauge theory is spid to be irreducible if for every dynamical history 'poE a0the generators Ri,[q0] are linearly independent, that is, the equation
Rj'['po]p=O
has no non-trivial solution with compact support in space-time; otherwise, the theory is called reducible. In the case of a reducible gauge theory, there exists an infinite set of* zero-eigenvalue eigenvectors Z2d['p0] for each functional supermatrix ~ ' , [ ' p ~ l ,
(4.5.13) Where &=(a, x"), the index a running over a finite number of values. The off-mass-shell version of equation (4.5.13) is
+
R{['plz2d['pl s,j['p]Bjd['p]= 0 v
B",['p] = -( - l)"'EIB'jd[(p]
(4.5.14)
and tells us that choosing in equation (4.5.4)the parameters
5"25[47]P
&(+E,
3: being arbitrary, results in a trivial transformation of the form (4.5.5).When the equation
Z$['p0]ld =o
has no solution with compact support (other than %=0), the theory is said to have the first stage of reducibility. Otherwise, there are non-trivial zero-eigenvalue eigenvectors for Z$['po], which also may be linearly dependent, and so on. In other words, there can be gauge theories having any finite stage of reducibility and even infinitely reducible ones (the most famous example is the Green-Schwarz superstring*). Of course, one can always choose in the set ( R 2 } a linearly independent subset, hence making the theory irreducible. In actual theories, however, *M.B. Green and J.H. Schwarz, Phys. Lett. 136B 367, 1984. Copyright © 1998 IOP Publishing Ltd
332
Ideas and Methods of Supersymmetry and Supergrauity
the original generators R’,[cp] are local and covariant operators of the fields; after singling out a linearly independent subset, the locality or explicit covariance may be lost. In our book we have met reducible gauge theories (both in components and superfields). Recall, for example, the antisymmetric tensor field model (1.8.34): which is invariant under the gauge transformations (1.8.35). Here we have A
‘pi= Bah(X)
1
R’k = -
2
(a,&
- ab&)s4(X - X’)
where :=(ab, x) and &=(e,x’). The generators are linearly dependent, with zero-eigenvalue modes
z%= dc64(x’ - x”) where d=(x”). The reducibility is manifested in the fact that the gauge parameters in transformations (1.8.35) are defined modulo the transformations jvb
Ab f
dbP
p being arbitrary. Obviously, the theory has the first stage of reducibility.
Furthermore, consider the chiral spinor superfield model (3.7.10) invariant under the gauge transformations (3.7.1 1). The theory has the first stage of reducibility, since the gauge parameter in transformations (3.7.11) is defined modulo the displacements K+K+A+A
DbA=O
A being an arbitrary chiral scalar superfield. One more example is given by the superspin-0 model (3.7.24) invariant under the gauge transformations (3.7.22). The theory has the second stage of reducibility, since the gauge parameters in transformations (3.7.22) are defined modulo transformations of the form (3.7.11). Note also that the superfield supergravity (see Chapter 6 ) is a reducible gauge theory. The linearized supergravity action (6.2.5) is invariant under the gauge transformations (6.2.6),where L, is defined modulo the redefinitions
L,+L,+v,
D&=O
q, being an arbitrary chiral spinor superfield.
4.5.2. Feynman rules for irreducible gauge theories with closed algebras We now reproduce the Feynman rules for irreducible gauge theories with closed algebras. Together with the requirement of irreducibility and equation (4.5.12), we shall also assume the validity of the relation ( - l p ;,R‘,[q] +( - l)W$,[cp] =O
whose role will soon become clear. Copyright © 1998 IOP Publishing Ltd
(4.5.15)
Quantized Superfields
333
Under the above requirements, the in-out vacuum amplitude is known to be given as
s
(outlin) = N 9 q eiscqlS[x'[q]]
sDet(F\[q])
(4.5.16)
where ~'[q], E ( x ' ) = E,, are bosonic and fermionic functionals such that the functional supermatrix
F2jtCpl=x2 ,i[dRj9[4)1
(4.5.17)
is non-singular at the point qoE (Dochosen in the role of the classical ground state and hence in a neighbourhood of CD. The ~'[q] are called gauge fixing functions. Note that their existence is due to irreducibility of the theory. The equation (4.5.16) represents the well-known Faddeev-Popov ansatz. Remark. The delta function in equation (4.5.16) is to be understood as a functional analogue of the ordinary Fourier representation for 6 p + 2 q ( z ) (1.1 1.34) The S[f[q]] is required in the right-hand side of (4.5.16) to break the degeneracy, owing to the gauge invariance, of the naive integral f 9 q exp(iS[q]). On the other hand, its introduction results in arbitrariness having no relation to the dynamical content of the theory. Were the amplitude (out I in) well defined, it should not depend on ~ [ q ] .Hence, the relation (4.5.16) may pretend to a physical correctness if its right-hand side, which will be denoted by (outlin),, satisfies the identity (out I in), = (out 1 in),
(4.5.18)
where Az'[q] is an arbitrary infinitesimal change of ~'[q].Let us recall the proof of identity (4.5.18) given by B. De Witt. In the functional integral (outIin),+A,=JIT
s
9 q eiscqlS[x+Ax] sDet(F+AF)
one makes the replacement of integration variables qi = @i - R',[@-g"@]
ti[@]=( F - l[@l)$ AxB[@] representing a field-dependent gauge. transformation, hence SCqI = X@l.
One also easily obtains
Copyright © 1998 IOP Publishing Ltd
334
Ideas und Methods of Supersymmetry and Superyruvity
and sDet(F[cp]
+ AF[cp])= sDet(F[$l){ 1 + d[$]}
.a+'[$]= (- 1Fx(F-'[$])'p{xq;[$]R',[$]
-r~',d$]Ri;[$]E"$]}.
Next, the change in the integration measure must be calculated in accordance with rule ( 1 . 1 1.32) which gives
8[$] = (- l)";,R',[+]('[$]
i
+ ( - lP(F-'[$])'p Axq;[$]R',[$]
-(-
l)rrr ~~~;[$]R',[$]Ei[$]).
Now, making use of equations (4511, 12) and (4.5.17)leads to (outlin),+A,=Jlr
s
9$e"[+] S[f[$]]
x { I -((-
I)"',RaC(?I+(-
sDet(F'&$]) l)'fiC$,C$l)t"$lj (4.5.19)
and this coincides with (out lin), under the fulfilment of equation (4.5.15). Let Y [ q ] be a physical observable, that is, a gauge invariant functional of cp, Y,;[cp]R',[cp] =O. On the basis of the arguments given above, its quantum average
s
(outlT(Q[p])lin) = N 9cpY[cp] eis[q] S[x'[cp]]
~Det(F'~[cp]) (4.5.20)
turns out to be independent of the gauge fixing, (out/ T ( ~ [ q l ) l i n ) ~ +=A(out1 ~ T(Y[ql)/in),.
(4.5.21)
Obviously, this fact is necessary for physical correctness. At the same time, it serves as a starting point for representing each average (4.5.20)in a form completely similar to that which appeared in the non-gauge case (see equation (4.2.25)),namely
s
(out I ~ ( ~ [ c p ] I)in) = const 9cptotal Y [cp] e i s ~ * ~ ~ l [ ~ p t ~ ~ ~ i l where the integration is performed over some extended space of histories = (cp, . . .), and Statal is a functional on this space. To parametrized by qtotal obtain such a representation, the following two tricks may be applied. First, Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
335
using equation (4.5.21),one can make in expression (4.5.20)the replacement (4.5.22)
where the f s are arbitrary external bosonic and fermionic fields on the space-time, and then integrate the modified version of expression (4.5.20) over these fields with the weight exp( ~f"Bip[qO]fa) sDet"2(@Bg,j[qoJ)
(4.5.23~)
B being a supersymmetric functional supermatrix &(Bap)= E,
B ea -(-
+Ep
l)e,+EB+E.EBB
Be
-
(4.5,23b)
The integration over the f s is assumed to be normalized as
for some fixed supermatrix q (compare with equation (4.2.32)).The second trick is due to the identities r
s
~Det(F'~[q])=M"' 9 c ' 9 c exp(iciFkB[q]c~) &(Ci)
=
=E,
+ 1.
(4.5.24b)
As may be seen, we can represent the superdeterminant in expression (4.5.20) by the Gaussian integral over fields ck and 'C of opposite statistics to the gauge parameters. As a result, the average (4.5.20) takes the form
s
(out I T(Y[q])lin) = ~Det"~(~~~Bgp[q,,])>BqBc'9cqY[q] eistotalcp,c'*cl (4.5.25) where
Copyright © 1998 IOP Publishing Ltd
336
t d m s unci Methods of Supersymmetry and Superyruvity
The fields c6 and C' are called the Faddeev-Popov ghosts. It should be clear from the above consideration that the expression (4.5.25) does not depend on the explicit form of f[q] and Big[qo]-external objects introduced for technical convenience. Each quantum average (4.5.24) can be read off from the generating functional of Green's functions n
const = sDet "z([email protected][q0])J7 where J ; is an external source, &= ( ci. .I As;) opposed to the physical correlators (outlin) and (out1 T(Y[q])lin), Y[q] being a physical observable, the generating functional explicitly depends on the choice of ~'[cp] and Bip[cpo]. The same is true for the effective action l-(x,B)[$]constructed from Z,,,,[J] according to the rule of Section 4.3. But the S-matrix derived from Z,,,,[J] is known to be gauge independent. From the viewpoint of practical perturbative calculations, it is useful to replace Z(,,,)[J] by its extension Z(,,,[J, Jcg,J,] corresponding to the case when sources for the ghosts are added in the exponential in expression (4.5.27). Next, one represents Statal as a sum of its free and coupled parts: StotalCV,
1
c', cl = SCVOI+ SOCAP,c', C; ~
s o =2 AcpiI,s.iC~ol+I ,
0 1 + SINT[@, c', C;
~ 0 1
X " ~ o l ~ ~ p ~ ~ o l X a , ~ c pco~l X~ ~~ ,c.pl [~~+o l R g ~ % l c ~ (4.5.28)
where Acp = q - qo and SINTis at least cubic in Aq, c' and c (assuming the normalization x'[qO]= 0). Further manipulations are exactly the same as in the non-gauge case (see Section 4.2). When establishing gauge independence, we have essentially used the validity of equation (4.5.15). Hence, the ansatz (4.5.16) proves not to be truly correct for gauge theories in which equation (4.5.15) does not take place. In this case, however, the modification needed turns out to consist merely in replacing the naive measure 9 q by an invariant one, 9 q p [ q ] , where p [ q ] is a functional transforming in such a way as to compensate the non-invariant terms in expression (4.5.19). Discussion of these theories is beyond the scope of our book, It should be remarked that both terms in expression (4.5.19, unless they vanish, are ill-defined in local field theories. The point here is , f=(i, x) and d =(a, x'), involve cp (and maybe that the generators R i B [ q ]where its derivatives to a finite order) only at the point x, hence the quantity ( - l)E'i,R',[q] contains delta functions with coincident arguments, and similarly for ( - l)"V?&[q]. Fortunately, in many theories of interest both Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
331
these objects vanish ( - I )':';.R',[v]= 0 ( - 1)e/{Cfi,&]
= 0.
(4.5.29~) (4.5.29b)
For example, in the case of a pure Yang-Mills theory, with vi= V,'(x), gauge transformations are given by the first expression in relation (3.5.33b), hence
R",[v] = d ' J 2 , d 4 ( ~ - ~ ' ) + f ' K J V , K ( ~ ) d 4 ( ~ - x ' ) @',[q] = 89-'KJd4(X
-X")84(X-X')
where 2 = ( J , x') and l = ( K b , x"). Here equations (4.5.29) are satisfied, since the structure constants of the gauge group are totally antisymmetric. Remark. Equation (4.5.29~) implies the invariance of the functional measure 4 being cp-independent.
9 q under the gauge transformations (4.5.4)with
4.5.3. Supersymmetric gauge theories Consider a dynamical system, which is simultaneously supersymmetric and gauge. We shall assume that the full gauge freedom can be described by transformations of the corresponding superfield variables uf = u ' ( z ) like
hi=R',[u][~ E ( [ ~= )
S,dv]R',[u] E O P E(R $) + cP
(4.5.30)
Here [ b = ( P ( z ' ) are tensor superfield parameters being, in general, unconstrained, chiral and antichiral; in the dependence upon superfield types of is; the summation over ,iiin relation (4.5.30)includes the integration over d'z', d6z' or d6Z' (these conventions are exactly the same as for the superfield variables). The R14 [ U ]will be called supergenerators of gauge transformations. By supposition, the correspondence (? + 8 v P maps tensor superfields into tensor fields, hence the operators R[v] are super Poincare covariant; in particular, they (anti)commute with the supersymmetry generators. In general, the supergenerators constitute an algebra of the form
R'~[u] Rji.[u]-(-
1Y'RPT,,~[u]Ryb [vI=Rf~[uIC~i,[~] 7 +S,~[U]ESip,[~]
(4.5.31) the Cs and Es being superfunctionals under equations like (4.5.10). Gauge transformations (4.5.30)can be rewritten in terms of components. For this purpose one must introduce not only the component fields q ' ( x ) of U'@) in accordance with law (4.1.40),but also the component fields t"(x) of i P ( z ) defined with the help of proper picture-change operators Pib and Q": (4.5.32) Copyright © 1998 IOP Publishing Ltd
338
Ideris und Methods of' Supersymmetry and Superyruvity
Then, the component form of the gauge transformations (4.5.30) is given by equation (4.5.4), where
R'Jq] = P'fRffi[o]Q",.
(4.5.33)
Note: we use letters from the beginning of Greek alphabet for component gauge parameters, and from the middle to denote superfield gauge parameters. It follows from equation (4.5.33) that the superfield gauge algebra (4.5.31) induces the component version (4.5.11 ) with
As a result, the component gauge algebra is closed if and only if its superfield counterpart is closed. Furthermore, if the theory is irreducible in components, the same is true in terms of superfields. Finally, from the formal identities ( - 1)!;R'2[q]=( - l)"f,Rffi[o]Q$
( - l)"flC'fi&[q] =( - l)E'Cttp[c]Q$
we see that the relations (4.5.29) imply (- l)"T,R f p [ ~ = ] 0 (- l)"c"tp[u] = 0.
(4.5.35~) (4.5.35 b)
and vice versa. Therefore it proves unimportant which picture (superfield or component), is used for analysing gauge algebra. Now, let us specialize our consideration to the case of irreducible gauge theories with closed algebras and under equations (4.5.35). Due to irreducibility, there exists a set of superfield gauge conditions K ~ [ u ](having the same superfield types as the gauge parameters c p , E ( ( ) ) = cp, such that the supermatrix F f i 9 [ ~=]i?Ju]R f$[U] (4.5.36) is non-singular at some stationary point u,EV, and hence, in its neighbourhood in V.The K ~ [ U ]will be called gauge fixing superfunctions. The components of K ~ [ u ]defined by f [ q ]= P&pb[u]
(4.5* 3 7)
constitute an admissible set of gauge fixing functionals. Under the choice (4.5.37), the supermatrices (4.5.17) and (4.5.36) are connected as Fi$3] = PifiFF,[~]Q'p.
(4.5.38 )
Hence, non-singularity of F[uo] implies non-singularity of F [ q , ] at the point Copyright © 1998 IOP Publishing Ltd
339
Quantized SuperFelds
q;=P‘,~h. As an immediate consequence of equation (4.5.38)we obtain sDet(F‘&q]) = sDet(Ffii,[v]).
(4.5.39)
In the class of supersymmetric theories under consideration, the quantization scheme described in the previous subsection can be applied to superfields and components, and in both approaches it leads to equivalent physical results in the following sense. The generating functional (4.5.27) turns out to be the component form of its superfield counterpart
z(%.B)[J] = const
s
9U2c’gjc e i ( S ~ o d ~ . c ’+, cJh? l
const = sDet l’z(qfi’Bi,;[~o])
(4.5.40)
1
+2
Stotal[t’,C‘, C] = S[U] - Kfi[t’]Bpi,KD[U]+cb~%v]R$t’]c”
provided that (1) the gauge fixing functions ~‘[crp] are chosen by the rule (4.5.37); (2) the functional supermatrices B48[’p0] and q h represent ~ the component versions of the superfunctional ones, that is
B--,[u0] fl’ = PST;’B~,$crpo]PB,,. psT 2 P vfii*= p q‘BP t‘ We suggest that the reader fills in the necessary details. Before we conclude, let us make one important remark. The equations (4.5.35) are identically satisfied for a large family of local supersymmetric theories. In the local case, the supergenerators Rrb[v], where f=(Z,z ) and jl= (,U, z’), are functions of U and its covariant derivatives to a finite order at the point z. Suppose that the R’$u] d o not depend on covariant derivatives of U , then I= (J,z”) j,Rifi[u] C 5 ( J ) ( ~ ” , Z )
-
the delta function being defined in equation (4.1.17). Since P ) ( z , z ) = 0, we obtain equation (4.5.354. As an example, consider the general super Yang-Mills model of subsection 3.5.6.Here we have the superfield variables UT=
(V’(Z), @(z), Oi(Z))
and the gauge parameters
ifi=(AJ(z’),AJ(z’)) ([+,
c-)
From equations (3.5.25)and (3.5.31) we find the supergenerators 1
- - dXJ6+ ( z , z’)
Ri+ [U] =
2
i( P)’,@j(z)S
Copyright © 1998 IOP Publishing Ltd
+
+ f’J( V(Z))6 ( 2 , z‘)
(z, z’)
+
340
Idem und Methods of Supersymmetry and Supergrut.ity
- iQjzN T J ) J-,(~ z,z').
Both equations (4.5.35) are identically satisfied.
4.6. Feynman rules for super Yang-Mills theories We proceed by obtaining superfield Feynman rules for supersymmetric Yang-Mills theories, based on the general considerations of Section 4.5. 4.6.1. Quantization of the pure super Yany-Mills model
We begin with superfield quantization of the pure super Yang-Mills theory (3.5.39).For later convenience, it is reasonable to rescale the coupling constant by g - $ g and the gauge superfield by V+gV thus setting the action S,,, =
4j 49
d6z tr( W"W,)
(4.6.1)
where (4.6.2) The original normalization was useful when analysing the component content of the theory and the underlying geometry. The one above turns out to be convenient for perturbative superfield calculations. In particular, after the rescaling V-gV, the action admits a well-defined limit g+O, since SsyM=
16
+
Sd'z tr( VDaD2D,V) O(9).
The action (4.6.1) is invariant with respect to the gauge transformations (4.6.3) obtained from the original transformations (3.5.31) by rescaling V-. gV and A + 2igA. As we have seen in Section 3.5, the gauge transformations form a group, therefore the gauge algebra of the theory under consideration is closed. The corresponding supergenerators turn out to be linearly independent. To prove this statement, it is sufficient to consider the Abelian case, where Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
341
equation (4.6.3)reads 6 V =A +
A.
Setting here 6 V = 0 implies that
D,A=O
A=-A=const
hence the supergenerators have no zero-eigenvalue eigenfunction with compact support in space-time, other than A = A = 0. Further, we have also seen that equations (4.5.35)are satisfied for the theory in question. So, it can be quantized by the scheme described in the previous section. In the role of gauge fixing superfunctions we choose
.[
+
V ] = - D2V(z) f(z)
4 1
E[ V ] = -- D2V(Z)+T(z) 4 f =f' T'
(4.6.4)
D,P =0.
Here f' are external chiral scalar superfields, inert under the gauge transformations (4.6.3). K and il- constitute an admissible set of gauge fixing conditions, at least at the stationary point V=O, where they transform under transformations (4.6.3)as
The differential operator arising here is non-singular on the space of chiral-antichiral superfields, since its zero-eigenvalue eigenfunctions satisfy the Klein-Gordon equation and cannot be localized in Minkowski space. The next step consists of deriving the ghost action that the gauge fixing (4.6.4) leads to. This is done most simply with the help of the standard observation that the ghost term in expression (4.5.40)can be represented in the manner
s,,
=C p c f i [ V ]
where 6 . is the variation of K under the transformation (4.5.30)with ( b being replaced by cb. In our case, both gauge parameters and gauge conditions are (anti)chiral, and they give rise to (anti)chiral ghost superfields:
Copyright © 1998 IOP Publishing Ltd
The above observation leads to
+
= [d8z tr((c‘ C’)6V) J
and with the use of transformations (4.6.3)we finally obtain
i
+
SGH= d8z tr((c’ C’)L,,[c -C
+ coth L,,(c + C)]).
(4.6.5)
Now, in accordance with the results of subsection4.5.3, the in-out vacuum amplitude reads (out jin) =
s
~v~~’~c’~~~ceissyM+scH
[
x S+ f -
,],[
D2
-
f
T - D2V ]
(4.6.6)
where
and does not depend on the external (anti)chiral superfields T and f. So, one can average over them with some appropriate weight. The standard choice is enpi: {d8z tr(Tf)]
(4.6.7)
y being a real constant. As a result, one arrives at
(4.6.8) where d8z tr((B2V)(D2V)).
Copyright © 1998 IOP Publishing Ltd
(4.6.9)
Quantized Superfields
343
Remark. Integrating the expression (4.6.6) over f and 7 with the weight (4.6.7),we obtain exactly (outlin), due to the identity s H 9 f e x p { ; {d’z tr(Tf)} = 1 which follows from equations (4.1.27),(4.4.25)and the relation
s
219~ e’s[X&v,Tl=sDet- 1/2(H(v))
(4.6.10)
with S [ x , 2; Y , T] and H(v) being defined by equations (4.4.6) and (4.4.5)’ respectively. Therefore, the integral (4.6.8) is independent of y, and this constant can be fixed from considerations of convention. The relation (4.6.8) motivates us to define the generating superfunctional for Green’s functions in the manner
s
z [ y ]= 9 where
ei(stotai+rd8z tr(/V))
(4.6.11)
3 = yf(z)Tr,3‘ are real scalar superfields.
4.6.2. Propagators and vertices We are now going to develop a perturbation theory to calculate Z [ f ] (4.6.11). For this purpose, we choose the stationary point V=O of SsVMin the role of classical ground state and represent the action Statal as
Smi = So + SIN,
where S,[V, c’, c, E’, E ] is the quadratic part of Storal in the gauge and ghost superfields, while SlNT[ V, c‘, c, E’, E ] describes an interaction of the superfields. Then one can write
x ZO[A
VI,
V’
- -
VI,
V11~’=v=9’=q=o
(4.6.12)
where Z , is the free generating superfunctional defined by
s
Z,[f, q‘, V , ij’, fj] = 9V9E’9c‘%9c exp i(S,[V, c’, c, E‘, E ]
s
d6z(qIcI+ qc) + tr d6z(ij’E’+ V E ) } .
(4.6.13) Here the supersources q‘ = q”(z)T’ and = q’ (z)T’ are anticommuting chiral Copyright © 1998 IOP Publishing Ltd
344
Iiicus and Meilloils of' Supersjnznzetrj, and Superyracity
scalar superfields. The variational rules for supersources are:
Let us determine So and SINT.We start by discussing the gauge action (4.6.1), which can be rewritten in the form S,,=
--
l 64g2
tr d8z e-2gVDxe2gVD2(e-2g"D,e2gV). (4.6.14) I
Consider the identity (e- 2sVD"ezSv)= (exp(- 2gL,)D,). 1
which we use to expand SsVM in a power series in V. Up to fourth order, the action reads S,,,,Sdsz
t r i G1 VDxj32DxV+-g(D2DaV)[V,(D1V)] 1 8
1
1
-- g2[V,(D"V)]D2[V, (DxV])-c g2(D2DaV)[V,[ V, (D"V)]] 16
I
.
(4.6.15)
Next, to the same order, the ghost action (4.6.5) reads i?c -Ec'
I
+ g(c' + V)[ V, (c -e)] +-31 g2(c' + e')[V, [V, (c +e)]] . (4.6.16)
Notice that the terms c'c and i?E have been dropped due to their chirality and antichirality, respectively. The last step is to unify the quadratic part S , of action (4.6.15) with the gauge breaking equation (4.6.9): 16 Copyright © 1998 IOP Publishing Ltd
345
Quantized Superjelds
Here we have used the identity (2.5.306). The relations (4.6.15-17) lead to the following expression for St,,
Statal = SO+ SIN,
“=f!
ik( +-))
{
d’z tr - V u V+-
1
V(D’, D’}V
(4.6.18~)
]+ [
d’z tr(5’c - Cc’) (4.6.186)
S l , , = ~ d s z tr{Ag(D2D”V)[V, (Dui‘)] +g(c’+C’)[V,(c-C)] 1 16
- -g2[ V, (D”V)]D’[
1
V,(D, V)] - -g’(D’D* V ) [V, [V, (D”V)]] 12 1 +-g’(c’+S‘)[V, [V, (c+C)]] 3
+ 0 ( g 3 ) . (4.6.18~)
As may be seen, the action So involves fourth-order derivatives, for y # - 1, and takes the most simple form when y = - 1. After establishing the explicit structure of So, the free generating superfunctional (4.6.13) is easily evaluated as:
ZOEA 4’3 rl, ii‘, iil = exp i
{
- - [d’z
d’z’ f’(z)GLJ(z,z’)yJ(z‘) -Jd8z tr( ij’;
v-q
Here GbJ=G’JGV,and the Green’s function G,, of a real scalar superfield satisfies the equation G,(z,z’)= -6’(~, z’)
(4.6.20)
and the Feynmann boundary conditions. Note that the ghost sector of Z, is calculated in complete analogy with the derivation of expression (4.4.28), but keeping in mind the anticommuting nature of ghosts. To find G,, let us introduce the orthogonal projectors
on spaces of linear, chiral and antichiral superfields, respectively. Their chief properties are given by the last line of expression (2.6.6)and equation (2.6.7). Now, equation (4.6.20) can be rewritten as
Copyright © 1998 IOP Publishing Ltd
346
i d e m und Methods qf Supersymmetry and Supergruuity
and we will find its solution in the form 1
GV =-
U
b q o ) + LW? + ) +?-,I.
Making use of the projectors' properties gives 1 G" = 0
c-
q o ) + ?(?
+)
+ q41.
(4.6.21)
In the case y = - 1, the Green's function takes the form G,(z, z')= --
1 +iE
P(z, 2 ' )
E -b
+0
(4.6.22)
and the generating functional (4.6.19)reads Z0[& q', q, ij', i j ] = exp
U
U
In what follows, we make the choice y = - 1. Now we turn our attention to the vertices. Consider the coupling (4.6.18~). It describes the self-interaction of real scalar superfields V' and their interaction with (anti)chiral ghost superfields E ' I , E', c'' and c'. The explicit form of S,,, dictates, in standard fashion, the structure of the vertices. The only problem which may appear concerns vertices involving chiral and antichiral superfields. But that question has been discussed in detail in Section 4.4. 4.6.3. Feynman rules for general super Yang-Mills models The previous consideration can be readily extended to the case of a general super Y ang-Mills model describing interaction between gauge superfields V' and matter chiral superfields Qk (and their conjugates @'=(@I)*) transforming in some representation R of the gauge group. We assume this representation to be irreducible and start with the classical gauge invariant action S =S s m + S M A T
where "
n
(4.6.24)
/. (4.6.25)
m and i. being mass and coupling constants, respectively (l'1f2'3 = (i.ll,21J*) Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
347
The representation R should be real in the massive case, m f O , and can be complex otherwise. Recall also that the coupling constants must form an invariant tensor of the gauge group in order that the action (4.6.25)be gauge invariant. Notice that SMAT can be viewed as a multi-component Wess-Zumino model coupled to the gauge superfield. The gauge theory under consideration is quantized in complete analogy with the pure super Yang-Mills one, and the corresponding generating superfunctional reads
u = (K @,
a, C’, C, E’, e)
(4.6.26)
D&Jk = D,J’ =0
where
+ SGB+ SMAT + SGH
Stotal = SWM
(4.6.27)
with SGB and S,, being defined by equations (4.6.9)and (4.6.5),respectively. The next step is to represent Stota,as a sum of its free and interacting parts. One thus finds
Statal = SO+ SIN,
(4.6.28)
the free action being of the form
s0
S(@)+$gh)
- o + o
0
(4.6.29)
where I
”
(4.6.30~) (4.6.306) (4.6.30~)
(4.6.31)
(4.6.32~) (4.6.32b)
Copyright © 1998 IOP Publishing Ltd
340
I(itu7 and Methods of Supersymmetry and Supergravity
1
--g2(D2DZV)[V, [V, (D,V)]] 12
S I
Sf;;ph’= d8z tr g(c’+e’)[I/, (c - C)]
+ -31 g2(c‘+ E ‘ ) [
+
I
V, [ V,(c t)]]
.
(4.6.32d)
Here we have written out the expressions for Si# and Sikigh)up to fourth order in the superfields. Based on the above relations, one now readily deduces superfield Feynman rules for calculating the generating superfunctional (4.6.26)and the effective action r [ V , a, 61 obtained as the Legendre transform of (l/i) In Z[g J , 51. The free generating superfunctional for the theory with action (4.6.27-32) is given as the product of the pure super Yang-Mills one (4.6.19)and that for matter
(the latter has, in fact, been derived in Section 4.4). The structure of the vertices follows directly from the explicit form for interaction (4.6.21,32).As in the case of the Wess-Zumino model, it is convenient to treat the ghost and matter (anti)chiral superfields within the framework of the improved supergraph technique described in subsection 4.4.5. Then all the vertices coming from expressions (4.6.32) will be integrated over d8z in coordinate space or over d40 in momentum space. We leave the reader to deduce the rules for attaching the factors (-$D2) and ( -aD2) to the ends of the internal (antilchiral ghost and matter lines. Notice that, up to sign, the ghost propagators coincide with the massless @@-propagator(4.4.43). As for the VI/-propagator, it reads in coordinate space v’v’-
I rne
-
= - b I Jo1 ~ 8I z - z ’ I
z
or, in momentum space,
P
= b”i2 b4(e- e’)
and has the opposite sign to the @&propagator. Copyright © 1998 IOP Publishing Ltd
(4.6.34~)
i
(4.6.34b)
Quantized SuperJeids
349
As a simple example, let us calculate the one-loop gauge superfieid contribution to the two-point @&-vertex function. This contribution comes from the following part of SINT Sjg+"=2g
s
d*z6t(TI)'k@kVI+. ..
where dots mean terms relevant for two-loop and higher corrections; (T')'k are the generators of the representation R in which Qk transform. The corresponding supergraph is P-k
and leads to
1
D2(k)D2(- k)d4(6- 8') (T') k' 2 a' k 2 ( p )0).
(4.6.35aj
Note that, in any representation, the operator dim G
c,= 1 (T')Z 1=1
commutes with every generator T J(since the structure constants in expression (3.5.24)are totally antisymmetric). Therefore, it is proportional to the unit operator, i.e. C, E U, in any irreducible representation. Since the matter representation R has been assumed to be irreducible, we have ( Tr)'kl( Tr)kIk = C,(R)d'k
the constant C,(R) is known as the second Casimir of the representation. Using this fact and performing the integration over 8',the expression (4.6.35~) takes the form -4g2C,(R)
s*
s
d4P A'(p2, m2) d48 6 k ( -p, 8)Qk(p,8)
where
(4.6.356)
(4.6.36) Copyright © 1998 IOP Publishing Ltd
350
Ideas und Methods of' Supersymmetry and Supergravitjs
I t is worth comparing the quantum corrections (4.4.54) and (4.6.35). In the massless case, the momentum functions (4.4.55) and (4.6.36) coincide, and the expressions (4.4.54h) and (4.6.35h) take similar structures but differ in sign. As a result, for some special values of the chiral coupling constants ik+ and the gauge constant g, the one-loop corrections to the two-point cancel each other. @@-vertex functions, which come from Si:; and Si:;), Under such a choice, there are no one-loop @@-divergences. 4.6.4. Non-renorviuiization theorem
The non-renormalization theorem described in subsection 4.4.8 turns out to preserve all its power in the case of the general super Yang-Mills theories just considered. Indeed, the only essential properties, which have been used in the proof of the theorem, were the facts that (1) each vertex is integrated over d48; (2) each propagator presents d4(8- e') multiplied by some &independent momentum function together with a number of D-factors acting on the delta-function. Both properties are implied by the Feynman rules given in the previous subsection. As a result, each supergraph forming the effective action of the super Yang-Mills theory can be represented by a single integral over d40. In other words, the effective action has the following structure
r[v,@,@] =E n
S
S
d4x1... d4x, d4e %(x1, . . ., xn)Fl(xl,e).. . F,(x,,
e) (4.6.37)
where the Ys are translationally invariant functions on Minkowski space, and the F s are local functions of V, @, @ and their covariant derivatives Fi=Fi( V, @, G, DAV,DA@,DAG),. . .).
(4.6.38)
4.7. Renormalization
As is well known, renormalization is one of the essential elements of quantum field theory. Formally speaking, renormalization consists in finding suitable counterterms and adding them to the initial action in order to substract the divergences loop-by-loop. A chief aim of this section is to investigate the general structure of the counterterms in super Yang-Mills theories.
4.7.1. Superficiul degree of divergence Let us consider some super Yang-Mills theory described by chiral Ok, antichiral and gauge V = V'T' superfields interacting among themselves, the interaction determined from the classical action (4.6.24, 1, 25). As has been shown, its effective action is given entirely in terms of the supergraphs. Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
351
The corresponding Feynman rules were formulated in subsection 4.6.3 (see also subsection 4.4.5). We pick out here some specific features of supergraphs. First, each vertex contains an integral over d48. Second, each vertex without external (anti)chiral lines attached includes four spinor covariant derivatives-Dfactors; if an (anti)chiral external line is attached to the vertex, then the number of D-factors is reduced from four to two. To secure finiteness of the supergraphs at all the intermediate stages of calculation, we have to introduce some regularization procedure. The problem of regularization in supersymmetric theories is a separate one. For the moment, we shall assume the existence of a regularization preserving supersymmetry. Then, at the end of this section we shall return to the discussion of supersymmetric regularizations. The superficial degree of divergence of a Feynman diagram is defined to be the degree of homogeneity of the diagram in momenta. To calculate the superficial degree of divergence w(G), G being an arbitrary supergraph, one has to take into account the momentum dimensions of the objects used to write the contribution of the given supergraph. These objects are propagators, vertices and momentum integrals. Let us consider an arbitrary L-loop supergraph G contributing to the effective action, with V vertices, P propagators and E external lines. Let C of the P propagators be of @@- or =-type and E , from these E external lines be (anti)chiral ones. According to the superfield Feynman rules (see subsections 4.4.5 and 4.6.3) the contribution of any such supergraph has the form
J d4p, . . . d4p, d48, . .d48,
[ .. .]
(4.7.1)
where the square brackets include the above number of propagators and include some definite number of D-factors associated with vertices. It is obvious that only the integration over momenta may lead to divergences. The momentum integrals (4.7.1)contribute the quantity 4L to w(G). Taking into account the explicit dependence of the propagators on momenta, we have the contribution - 2(P + C). Recall that the @@- and GG-propagators have the extra l/p2 factor in comparison with the and VV-propagators (and the ghost ones). However, the quantity
@a-
4L - 2(P + C)
(4.7.2)
does not represent the final momentum dimension of the supergraph contribution (4.7.l), since the D-factors also depend on momentum. Let us find the total number of D-factors depending on the internal momenta. The superfield Feynman rules tell us that each vertex without external (antichiral) lines includes four D-factors; only two D-factors are Copyright © 1998 IOP Publishing Ltd
352
tiecis and Methods of Supersymmetry and Supergravity
associated with a vertex possessing an external line of @@- or GG-type. Therefore, the number of D-factors associated with the I/ vertices in expression (4.7.1) is equal to 4V-2EC. We also know that each propagator of a@-or m - t y p e contains two D-factors. Hence, the total number of D-factors depending on the internal momenta in expression (4.7.1) is given by
4V- 2 E , + 2C
(4.7.3)
Now, the reader should recall the non-renormalization theorem (see subsections 4.4.8 and 4.6.4). According to this theorem, equation (4.7.1) can be transformed into an expression involving a single integral over d46. That is, ( V - 1) &integrals can be taken explicitly, since each internal line contains a Grassmann delta-function J4(6,- Oj). After taking ( V - I ) &integrals, ( V - 1) of the P delta-functions are cancelled, and we have ( P - V + 1) remaining 6-functions. But owing to the well-known topological relation I/+ L - P = 1
(4.7.4) the number of such 6-functions is equal to L. As a result, after transforming equation (4.7.1) to the form with a single 6-integral, we have an expression including L Grassmann delta-functions. Since S4(6 - 6) = 0, this expression does not equal zero if all the S-functions are completely reduced by means of relations like
D2D2S4(8- e’)= 16.
We see that 4L D-factors are required to cancel the remaining &functions. In summary, after the transformation of equation (4.7.1) to the form dictated by the non-renormalization theorem we have the following number of D-factors depending on the internal momenta
4V - 2 E ,
+ 2C -4L
(4.7.5)
What can we d o with the remaining D-factors? Note that in the process of reducing equation (4.7.1) to the form with a single &integral one should perform integrations by parts, as in equation (4.4.52). Therefore, some D-factors can be transferred to the external lines. Those D-factors, which are not transferred to the external lines, must be converted into momenta via the relation { D, D} p. Suppose that all the remaining D-factors are converted into momenta by the law (D, 0)- p (no D-factors are transferred to the external lines). Then, they produce the following maximal number of internal momenta
-
1
-(
2
4 v - 2 ~+~2c -4L)= 2(V- L)- E , + c.
(4.7.6)
Then, the maximal superficial degree of divergence is given by the sum of Copyright © 1998 IOP Publishing Ltd
Quuntized Superfields
equations (4.7.2)and (4.7.6).Therefore, we have w(G)= 4L- 2P - 2C + 2 V - 2L- E , + C =2(L+ V - P ) - E , - C = 2 - E C - C
353
(4.7.7)
where equation (4.7.4)has been used. Assuming that N , D-factors are transferred to the external lines, the superficial degree of divergence is given as 1 w ( G )= 2 -- N , - E , - C .
2
(4.7.8)
This is the final expression for the superficial degree of divergence. 4.7.2. Structure of counterterms
To ascertain the explicit structure of the counterterms, it is necessary to take into account the superficial degree of divergence, the non-renormalization theorem and the fact that the counterterms are local functionals in x-space. Also, we will assume that the regularization procedure chosen preserves supersymmetry. Therefore, the counterterm AS should be a local superfunctional of @, and I/ invariant under the supersymmetry transformations of these superfields. The locality of AS, supersymmetry and the nonrenormalization theorem imply that the counterterm must have the form AS=
s
dsz A Y ( @ , G, V, . . .)
(4.7.9)
where A Y ( @ ,G, V, . . .) is a function of the basic superfields and their covariant derivatives to a finite order. Let us consider some elementary consequences of equation (4.7.9). First, there are no counterterms to vacuum energies since
s
AS= dsZ AYJq,,=a=v=o=O.
Therefore, there is no need to take into account the supergraphs for vertex functions without external superfield lines. In fact, all such supergraphs vanish, as a consequence of the non-renormalization theorem. Second, the counterterms cannot contain terms having the structure of a chiral superpotential, such as r since their counterparts in the full superspace
have nonlocal form. Copyright © 1998 IOP Publishing Ltd
354
I d w s [ind Methocls of' Supersymmetry und Supergravity
Before we proceed to the study of the structure of counterterms, let us discuss the problem of gauge invariance of the counterterms. This problem is common to all gauge theories. The fact of the matter is that the quantization procedure demands that we introduce gauge fixing conditions which break the classical gauge invariance. On these grounds, the effective action turns out, in general, to be non-invariant under the initial gauge transformations. Certainly, the S-matrix is always gauge invariant. Thus it should not be a surprise that counterterms in the super Yang-Mills models may, in general, be non-invariant under supergauge transformations. The standard way to classify counterterms in conventional gauge theories is based on the Slavnov-Taylor identities, which allow one to decrease the number of admissible counterterms. Analogous identities can also be formulated for superfield gauge theories. However, in many cases of interest it is possible to realize the quantization procedure in such a way that the corresponding Slavnov-Taylor identities imply invariance of the effective action under the classical gauge transformations. This formulation of quantum gauge theories is known as the background field method (it can be applied under the existence of gauge invariant regularization). Concrete realization of the above formulation is based on specific features of the given theory, and it is not clear from the outset that the super Yang-Mills theories are tractable within the framework of background field method. Fortunately, the background-field formulation of general super Yang-Mills models does exist, and it has been suggested by M. Grisaru, M. RoEek and W. Siegel'. For our aims, it is essential to know only that there exists a formulation of the theory, in the framework of which the effective action is invariant under the initial supergauge transformations. Now, let us find all the admissible counterterms. Divergent supergraphs are characterized by the condition w(G) 2 0. In accordance with equation (4.7.8),there can be three possible basic types of divergent supergraphs: 1. E , = 2 , C=O, N,=O, w(G)=O
(4.7.10)
2. E,=0, C=2, N,=O, w(G)=O
3. E,=0, C=O, ND<4,O<w(G)<2. Notice, the above conditions give no restrictions on the number of external V-lines. Consider the first variant in expression (4.7.10).Divergent supergraphs of this type have exactly two (anti)chiral external lines and may carry an arbitrary number of external V-lines; their divergence is logarithmic. Hence, suitable counterterms should have the form AZ
s
d8z @F( V ) @
'M. Grisaru, M. RoEek and W. Siegel, Nucl. Phys. B 159 429, 1979. Copyright © 1998 IOP Publishing Ltd
(4.7.11)
Quantized Superfields
355
all the indices being suppressed. Here F ( V ) is some function of the gauge superfield V, AZl is a series in couplings and the regularization parameter. We suppose that the calculation procedure is fulfilled within the framework of the background field method. Then, all the counterterms should be supergauge invariants. This implies that the only possible form for F(V) is exp(2gV). Therefore, in the first variant of equations (4.7.10)the counterterm reads
s
AZl d'z 6 e2g"0.
(4.7.12)
The supergraphs belonging to the second variant in equations (4.7.10) include only external I/-lines. The corresponding counterterm should be proportional to m2 ld8z/(V)
(4.7.13)
f(V) being some function of V. But if we carry out the calculations in the
framework of the background field method, then such a counterterm is forbidden by supergauge invariance. More precisely, the only admissible choice is f (V )= const and hence the expression (4.7.13) vanishes. As an example, the reader can explicitly check that the total divergent contribution produced by all possible one-loop diagrams of the form
proves to be proportional to
s
m2 j d ' z tr(e2g"(e2gV)T) = m2 d8z tr(e2g"e-2gV) = 0. Finally, consider the third variant in equations (4.7.10).The most general form for the corresponding counterterm reads
s
d'z G( V, D, V, . . .)
Here G is a function of the gauge superfield and its covariant derivatives to fourth order; in addition, G depends parametrically on the dimensionless and the regularization parameter. Note that coupling constants y and jVklklki the dimension of G is equal to two, while I/ is dimensionless. Obviously, one Copyright © 1998 IOP Publishing Ltd
356
Idciis tmci Meiitotls of Supersymmetry und Supergravity
can write
5 c d'z G =
d'z V.dV+O(V3)
(4.7.14)
where .d is a V-independent differential operator of the fourth order in D, and D4. Assuming validity of the background-field method, the counterterm should be invariant under the gauge transformations (4.6.3).Then, its quadratic part in V must be invariant under the linearized transformations
D*A=O.
SV=A+A
Therefore, the most general form of the operator d is
1 .d=- AZ2D1D2D, 16 AZ2 being a series in the couplings and the regularization parameter. Now, the full counterterm can be recovered from the requirement of supergauge invariance to obtain AZ2
'1
89
d'z tr(e-2g"D' eZgYW,)= AZ
-$
S d 6 z tr(W'W,). (4.7.15)
In summary, we have shown that in the theory under consideration there are only two possible types of counterterm. They are given by the expressions (4.7.12)and (4.7.15). Let the classical action be written as S=
c
's
d8zGeeZgV@+- d6z tr(W'W,)
4gZ
(4.7.16) with the group indices suppressed. The above analysis allows us to find the vRand parameters mR, j*R,gR in the form renormalized action S,,
aR,
where Z , = 1 + AZl, Z 2 = 1 + AZ2 are the renormalization constants, WY; is the superfield strength expressed through the superfield VR and the parameter gR. Comparing the expressions (4.6.16) and (4.7.17),we see that
0= ( Z p D R
v = (Z2)'!2VR.
(4.7.18)
Hence, Z , is the wave function renormalization for the matter superfields Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
357
These relations are direct consequences of the non-renormalization theorem as well as of supergauge invariance. Let us formally introduce renormalization constants for mass and couplings as follows
m=ZmmR
j.=Zj.jvR
g=Z,g,.
(4.7.20)
Then, the relations (4.7.18) and (4.7.19) tell us that ~ , ( ~ , ) 1 / 2 =I
Z,Z, = 1
zj(z,)3’2= 1.
(4.7.21)
We summarize the results. In super Yang-Mills theories, there are only two independent renormalization constants, Z and Z,. Renormalization is determined by the wave function divergences. 4.7.3. Questions of regularization Previous considerations in this section were based on the assumption that there exists at least one regularization scheme preserving supersymmetry and supergauge invariance as well as being adapted for use in supergraph calculations. We review here several often-employed regularizations. In each case, fulfilment of the above requirements is not guaranteed by construction and has to be checked explicitly.
a. Supersymmetric Pauli-Villars regularization. Let us introduce a matrix which is formed from the propagators (4.4.49) of the Wess-Zumino model and defined as
As is known, the chief idea of the Pauli-Villars regularization consists in modifying, in each loop, the initial propagators in such a way as to obtain propagators possessing less singular behaviour on the light cone. This is achieved by adding to the initial propagators, in each loop, propagators of some auxiliary massive fields with suitable coefficients. In our case, it seems reasonable to regularize the superpropagator (4.7.22) Copyright © 1998 IOP Publishing Ltd
358
liiecis c i i d Methods of’ Supersymmetrj- and Supergravity
in the manner
=I
CiMi
1
x 64(8- 8’)
(4.7.23)
where Mi are auxiliary masses, and C iare dimensionless coefficients. To cancel the leading singularities of X on the light cone, it is sufficient to choose C iunder the conditions
1C i M i = -m.
cci=-l
(4.7.24)
i
i
All the auxiliary masses should be taken to infinity at the end of the supergraph calculations.
Remark. There is no need to regularize the factors l/p2 in equation (4.7.22) since, as we have seen, their origin was purely kinematical: to convert the vertex integrals over d28 or d2Binto ones over d48. In fact, one could even work with the unregularized propagators K , + and K - -, because in the Wess-Zumino model the only divergent supergraphs with internal @@- or @@-linesare vacuum ones, due to equation (4.7.Qand they vanish. But this would lead to the breakdown of supersymmetry. motivates us The explicit form of the regularized superpropagator (4.7.23) to associate with each mass M i some auxiliary chiral superfield vi and its conjugate. This idea allows us to realize the Pauli-Villars regularization directly in the Lagrangian approach by adding to the classical action an action of auxiliary superfields as well as some coupling terms. A correct choice for the total action is of the form
+
(Sd”; [5 m2 +- 1 1 2
1
i Ci MivZ
+i 3! (@ +F v i ) 3 ]+c.c.]
(4.7.25) Cs being solutions of equation (4.7.24). The action corresponds to a dynamical
system of several (anti)chiral superfields. The corresponding perturbation consideration can be fulfilled in the standard way in terms of supergraphs. Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
359
Notice that supergraphs with external 0-and @lines only should be taken into account. The auxiliary superfields work in the loops. Obviously, in this regularization scheme the supersymmetry remains unbroken at all stages of the quantum calculations. Unfortunately, Pauli-Villars regularization is not well adapted for gauge theories, in particular, for the super Yang-Mills theories. The problem is that the scheme demands that we introduce auxiliary massive (super)fields for all physical (super)fields, in particular, for the massless gauge ones. This can be accompanied by the breakdown of (super)gauge invariance. b. Regularization by dimensional reduction. Any regularization procedure should answer the question: in what sense are the divergent Feynman integrals to be understood? The standard requirements of the regularization scheme consist in that it must lead to convergent Feynman integrals and preserve as many properties of the initial classical theory as possible. In supersymmetric theories it is natural to demand that the regularization scheme preserves supersymmetry. As we have noticed, the problem of existence of such a regularization should be discussed case by case. For instance, in the component approach one could use so-called dimensional regularization which is most popular in modern quantum field theory. Dimensional regularization demands the theory to be formulated in space-time of an arbitrary dimension from the very beginning. However, supersymmetry can exist only in space-times of special dimension. Therefore, it is conceivable that naive application of dimensional regularization will lead to a breakdown of supersymmetry. Regularization by dimensional reduction invented by W. Siege1 is a generalization of the standard dimensional regularization adapted to supergraph calculations. Let us consider a divergent Lloop supergraph. The regularization consists in two steps. First, we fulfil all necessary manipulations with covariant derivatives (D-algebra) in four-dimensional space-time and integrate over 8, as in the proof of the non-renormalization theorem. As a result, we obtain a single integral over d48 and L momentum integrals. The integrand depends on external superfields and their covariant derivatives and therefore the full expression is manifestly supersymmetric. Second, to the remaining momentum integrals we apply standard dimensional regularization. All the momenta should be considered as %dimensional ones. The divergences appear only in the form of poles 1/(9-4)k, k = 1, 2, , . ,. Regularization by dimensional reduction appears to preserve both supersymmetry and supergauge invariance. This scheme is well adapted to massless theories, since it does not demand the introduction of auxiliary masses. To illustrate the use of regularization by dimensional reduction, let us consider the supergraph (4.4.534. After fulfilment of the %algebra and Copyright © 1998 IOP Publishing Ltd
360
I t k u ~cind Metliocis
of Supersymmetry and Superyruvity
8-integration, we arrive at equation (4.4.54b)in which the function A ( p * , m 2 ) reads A( p 2 , m 2 )= i (k2
1
+ m2)((p- k)2 + m2 j
This integral is divergent. According to regularization by dimensional reduction, we have to replace the above expression by 1
A,(p2, m2)=ip4-9
(4.7.26)
p being some parameters of mass dimension. It has been introduced in order
to make A&’, leads to
m2) dimensionless. Direct evaluation of the integral (4.7.26)
Av(p2, m*)= --p 4 - 9 r ( 2 - r ) (449’2
jol
dz [ m 2 + p 2 a ( l - c ( ) ] - ~ + ~ ’ ~ .
In the limit Y +4, one obtains A,(
p 2 , m2) =
2 +finite terms. (47r)2(9- 4)
To cancel this divergence, we have to introduce the one-loop counterterm
expressed in terms of the regularized parameters E.k, & and superfields @k, Gk. Unfortunately, regularization by dimensional reduction leads to some ambiguities when applied to higher loop supergraph calculations. The problem is the following. To maintain supersymmetry, some objects which arise in the theory under consideration are to be defined directly in four dimensions. The others are to be continued, in the process of regularization, to 9 dimensions. However, there are many supergraphs which involve contractions of four-dimensional objects (spinors or tensors) with 9dimensional ones. These contractions have an absolutely formal character and require special definitions. In principle, several reasonable definitions are available. But making the calculations in different ways may lead to different results. The differences among them disappear only at 9 = 4 , but at 9 = 4 the supergraphs should be regularized. Nevertheless, in spite of possible ambiguities, regularization by dimensional reduction is the basic regularization scheme which is used in practice for supergraph calculations. c. Other possibilities. As we have discussed, only the factor l / ( p 2+ m2) in equation (4.7.22) is required to be regularized in order to make loop momentum integrals convergent. From this point of view, the situation is Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
361
exactly the same as in conventional field theory. Hence we can use any of the standard regularization schemes applied to superfield theories. For example, let us represent the scalar field propagator in the form p2+ i m2 ---
s,'
ds e - i(p?+ m ? ) s ,
To regularize the propagator, we have to modify the behaviour of the integrand at s-0. The regularized propagator can be defined as (4.7.27) f,,, being some smooth function chosen subject to the requirement that f&) and all its derivatives to a finite order vanish at s = 0 also, f&) + 1 when the parameter w tends to some fixed value. The choice fw(s) = (isp2)w,where p is a mass parameter, is often used in practice. The corresponding regularized propagator is given by equation (4.4.57);w -,0 at the end of the calculations. The above choice defines so-called analytic regularization. It can be shown that the divergences of Feynman diagrams appear in the framework of analytic regularization only as poles of the type l/wk, k = 1, 2, . . .. When applied to supergraph calculations, analytic regularization has some features common with regularization by dimensional reduction. However, in analytic regularization all the calculations are performed in four-dimensional space. Unfortunately, analytic regularization can lead to the breakdown of Ward identities in gauge theories'. The example of supergraph calculations with use of analytic regularization has been presented in subsection 4.4.7.
4.8. Examples of counterterm calculations: an alternative technique For given loop order, one must take into account infinitely many divergent supergraphs when trying to compute explicitly the relevant counterterms (4.7.12)and (4.7.15). There exists another approach to finding counterterms which makes it possible to account for all such supergraphs simultaneously. It is based on the Schwinger proper-time technique and its central object is a (super)propagator in an arbitrary external (super)field. This approach will be discussed in detail in Chapter 7. In the present section, the proper-time technique is used to calculate the one-loop counterterms for two supersymmetric models. 4.8.1. One-loop counterterms of matter in an externalsuper Yang-Mills field We consider a theory of matter chiral superfields a= {@'(z)> coupled to an external super Yang-Mills field described by V=gV'(z)T', where g is the 'See, for example,
R. Delbourgo, Rep. Proyr. Phys. 39 345, 1976.
Copyright © 1998 IOP Publishing Ltd
362
Ideas and Methods of Supersymmetry and Superyraoity
coupling constant. The matter multiplet is assumed to transform in a real representation of the gauge group, hence the generators T' are antisymmetric, ( T')T= - T'. Ignoring self-coupling of the matter superfields, the classical action reads S[@,
6;V]=
s
d*zGe*"@+2
is
]
d6z@T@+c.c..
(4.8.1)
Since S [ @ , 6;VI involves no self-coupling of the dynamical superfields, 6;VI of the theory is given only by its classical and the effective action r[@, one-loop parts:
r[o, 6;V]=S[@, 6;V ] + P [ V / I .
VI, defined by the superfunctional integral
The one-loop correction I-(')[
s
e r y ~=l ~ G G ,isC@,QU J ~ can be represented in the form
r(')[V] =-i sTr In H(") 2
(4.8.2)
(4.8.3)
where the operator
determines the Hessian of S [ @ , 6;VI. It should be pointed out that I-(')[ V ] is a part of the one-loop effective action of the general super Yang-Mills theory considered earlier.
Remark. In equation (4.8.3) it is understood that H'") is a linear operator acting on the space of chiral-antichiral columns
More generally, given such an operator d,its supertrace is defined with the help of the matrix superkernel
Copyright © 1998 IOP Publishing Ltd
363
Quantized Superjelds
as follows:
s
sTr d = d6z tr d++(z,z)+
s
d 6 i tr at- -(z, z)=sTr d++ +sTr d--
tr being the ordinary matrix trace over group indices. In order to compute r")[V], we first apply the so-called doubling trick. Let us make in the integral (4.8.2)the change of variables
@+i@
G+-iG
whose Jacobian is unity. This redefinition changes only the sign of the mass term (see equation (4.8.1)), in the exponential. Therefore, if we denote H(m) 5 H("), then
sDet H(m)= sDet H(- m). One now obtains 1
r ( " [ V ]=-sTrln {H(m)H(-m)) =sTrIn 4
(4.8.5) where (4.8.6) ,& !7 (
EH(!\H(J'_ ='D2 16
e2v D 2 e-2v,
Introduce chiral G'J! and antichiral G?'L superpropagators which are solutions under Feynman boundary conditions of the following equations (XY\ -m2)G(J!+(z,z')= - Om6+(z, z')
( X Y L - mZ)G(!"(z, z') = - Oip5-(z, z').
(4.8.7)
Then, the relation (4.8.5)is equivalent to 1
I-(')[ V ] = - - sTr + In 4
i 4
- - sTr - In G(!L.
(4.8.8)
As a result, the one-loop effective action is determined by the (anti)chiral superpropagators GcL ! and G(,Y\.
Remark. The trick applied above is the simplest way to prove equation (4.4.25).
Our next step in the treatment of l-"')[V] will be the introduction of the Copyright © 1998 IOP Publishing Ltd
364
Ideas and Methods of Supersymmetry and Superyracity
Schwinger proper-time representation for the superpropagators C(:)+(z, z')=i
l:
ds UY',(z, z ' I ~ ) e - ' " ' ~
(4.8.9)
and similarly for Cc!L. The superfield kernel U(,"!(z,z'Is) is chiral in both arguments and satisfies the equation
i -+ YP:,"! (is and the initial condition
1
U(:', (s)= o
(4.8.10)
(4.8.11) +(z,z'). With the help of purely formal manipulations described in detail in Chapter 7, one can show that the relation (4.8.8) is equivalent to U',"\(z, z' I s) I
=0
= 0*6
Here the integral over s turns out to be divergent at s+O. We regularize P [ Y ] as follows:
i
jo
* d(is)
r::)[v]= -4- p (is) 1 -io
(sTr+ U(:\(s)+(
+ + -)}
(4.8.13)
p, w being the normalization point and regularization parameter, respectively; w + + O at the end of the calculations. Now, let us note the identity
sTr + U(,")+(s)= sTr - U(!', (s)
(4.8.14)
which can be formally proved, using equations (4.8.6, 10, 1l), as
sTr + U(:\(s) = sTr + {exp(isH',VLH'!!')) = sTr - (exp(isH?+HYL)} = sTr -
U(!'L(s).
Hence we can write
It should be remarked that r;:)[V']is invariant under the super Yang-Mills gauge transformations (3.5.26); i.e. equations (4.8.6) and (3.5.26) show that
-
++-e
++e
~ ( i " ) ih % ( V )
-iA
Hence
sTr + U',"?(s)= sTr + (eih U(,")+e -
= sTr + U(,"!+(s).
The crucial observation is that the fourth-order differential operator %(:)+ Copyright © 1998 IOP Publishing Ltd
365
Quantized SuperJelds
(4.8.6) is equivalent to a second-order one when acting on chiral superfields (by construction, X y ! should act on chiral superfields only). In terms of the gauge covariant derivatives (3.6.40~)satisfying the algebra (3.6.45,46~), we can write -@+)a ( + ) - wggp)--(@+)awz), 1 (4.8.16) ++gL? 2 To simplify following expressions, we shall omit the label (+) which the derivatives 9 y )carry. Now, the structure of &\:(' implies that the solution of equations (4.8.10, 11) can be looked for in the form (compare with equation (4.4.21))
U?!+ (z,Z' Is) = -
i
(471s)
[is +
exp - (x(+
- xi +
xp+, =x a i80aB.
5
1 a;"'(z, z')(is)" (4.8.17)
l n = o
Here the coefficients u',L"(z,z')are chiral in both arguments and satisfy the equations
(x,+ ) - xi+ ,)"[ga- iBk(8a)kawg] a f " =O (n+ l)aiY)l + ( x ( + ) - x i + ) ) " [ ~ ~ - i B 8 ~ w=&'(+"!+U:") ]a~~~
(4.8.18~) n=O, 1, . . ..
(4.8.18b) These equations represent recurrence relations allowing us to determine the coefficients step by step. Note that the initial condition (4.8.11) and the equation (4.8.18~)lead to
(e
z') = - e y q z , z')
(4.8.19)
R being the chiral two-point function subjected to equation (4.8.18a) and the boundary condition
Q(Z,z ) = U*.
(4.8.20)
In accordance with the relations (4.8.15) and (4.8.17), we can now write
So, the effective action is determined by the coefficients U:) at coincident points. As for the divergent part of r g ) [ V ] ,it is easy to show that
lim 0-0
r$)[V ] =- -Jd6z o2 ( 4 ~ ) ~
tr
{$
ub"'(z, z) - m2a(,Y)(z,z ) + u$')(z, z)
+finite terms. On the other hand, it follows immediately from equations (4.8.18-20) that Copyright © 1998 IOP Publishing Ltd
366
Ideas und Methods of Supersjtmmetry and Supergravity
abV)(z,z) = a\"'(z, z) = 0. Therefore, the divergent part of the effective action reads
rg;[VI = - -
d6z tr aiv)(z,z).
(4.8.22)
As a consequence, all information about divergences of the theory is encoded
in a\"). Let us calculate ah") at coincident points. Setting in equation (4.8.18b)n = 1 and taking the limit z + z', one obtains
(4.8.23) where the identity a(,v)(z,z) = 0 has been used. Other simple consequences of equations 4.8.18-20) are:
ga,. . . 9 a , ~ b " ( ~z'), 1
=zI
=0
k = 0, 1, . . .
9&,, ,..9aLa~"'(z,z')/.=,.=0 (4.8.24)
ah"'(z,z)= WUW@.
(4.8.25)
Due to equation (4.8.22),the divergent part of the effective action is given by
(4.8.26) and determines the relevant counterterm. The results described were originally obtained by J. Honerkamp, M. Schlindwein, F. Krause and M. Scheunert in the case of supersymmetric electrodynamics and by I. Buchbinder in the case of general super Yang-Mills theories. 4.8.2. One-loop counterterms of the general Wess-Zumino model Let us consider the general Wess-Zumino model (3.2.8) and determine the one-loop divergences of the corresponding effective action. In accordance with equation (4.3.15), the one-loop contribution to the effective action of the theory (3.2.8) is given as
(4.8.27)
Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
367
Here the linearized action S[x,2; Y , T]is defined by equation (4.4.6).In terms of the operator (4.43,which determine the Hessian of S[x,2; Y ,Y], and its Green’s function (4.4.7,8)under the Feynman boundary conditions, T(’)[@,451 is of the form 1 i r(”[@, $1 =- sTr In H(y)= - - sTr In G(y)
2
2
(4.8.28)
where the operation of supertrace is defined in the manner:
s
s
sTr Giyl= d6z G + +(z, z) + d6f G - -(z, z).
(4.8.29)
An important role in our subsequent consideration will be played by the superpropagator GLy) of real scalar superfield obeying the equation
-1
(4.8.30)
1 -- Y(z)D2 T(z)D2 4 4
A=
Its significance follows from the fact that G(y)can be expressed through GLy) by the rule:
D;DftGhy)(z,z‘) D;D;.G&*)(z, z’) . D;D;,G~(z,zi) D ; D ~ , G ~zi)( ~ ,
(4.8.31)
In this connection, let us act with the operator
on both sides of equation (4.4.8)from the left. Then one arrives at the equations 1 u,G+ +(z, z’)--Df(T(z)G- +(z,z’))=O
4 (4.8.32a) 1 1 o,G- +(z,z’)-- D;(Y(z)G++(z, zf))=-- DfD;,S8(z, z’) 4 16 I
U ,G - -(z,z’) --D:(Y(z)G + -(z, z’))=0 4 (4.8.32b) 1UzG+-(z, z’)--D;(T(z)G- -(z, z’))=’ D:Df.S8(z, zf) 4 16 which the components of G(y),defined by equation (4.4.Qsatisfy. On the other hand, let us introduce one more Ciy’ defined now by equation (4.8.31). Copyright © 1998 IOP Publishing Ltd
368
Ideas and Methods of Supersymmetry and Supergravity
Its components turn out to satisfy the same equation (4.8.32). For example, acting with &D;D:.,on both sides of equation (4.8.30)and using the chirality of Y, one obtains the first equation (4.8.32a), since
D t D y y z , 2') = D;D;ss(z, z') =o.
The relation (4.8.31) leads to the following remarkable identity
sTr In G(y)= sTr In GL?')
(4.8.33)
where the operation of supertrace in the right-hand side is defined as
s
sTr CL?') = d8z GL?')(z, z)
(4.8.34)
(compare with equation (4.8.29)).To prove equation (4.8.33),we consider the variation of sTr In G(y)with respect to arbitrary infinitesimal displacement Y + Y 6Y keeping, for simplicity, T' ' unchanged. We have
+
= Jd6z d6z' 6Y(z)G++(z,z')6+(z,z')
= j d " z 6 v A z GL~)(z,z')/,=,,=61rsTrlnG',YI). Since at Y = O the expressions in both sides of identity (4.8.33) vanish, the above analysis confirms equation (4.8.33). Using equation (4.8.33), one can rewrite equation (4.8.28) in the form PI)[@,
i @] = -- sTr In GLv), 2
(4.8.35 )
This relation will be basic to our further investigations being, in fact, analogous to that described in the previous subsection. First, we introduce the proper-time representation for GL?: GLy)(z,z') = i
[:
ds UL?')(z, z' Is)
(4.8.36)
3Ly) being the unique solution of the equation
(4.8.37)
Copyright © 1998 IOP Publishing Ltd
Quantized Super8elds
369
with the initial condition
uy(z, z'Is-+
+O)=P(z,z').
(4.8.38)
Second, we replace the ill-defined expression for I-(')
T(')[@,
::1
a] = --2
-sTr ULy)(s)
which is equivalent to expression (4.8.35), by the regularized version
(4.8.39) It now remains to analyse the kernel UL?. We look for the solution of equations (4.8.37,38)using the ansatz
z0
(
i U"( Y )(z,z'Is)= -exp)):;i-( ' (477s)Z
aiv)(z,z')(is)".
(4.8.40)
To fulfil equation (4.8.37),o and a(y) should satisfy the equations
20 = 3.0 d,o
1 - -. 1 - - $D~cD"o- - $D"oD,o
4
(4.8.41~)
4
1
= - ( ~ - ( A O ) ) U ~ ? )+aiy) ~ 2
n=O, 1,
,
. ..
(4.8.41~)
To guarantee the initial condition (4.8.38),it proves sufficient to impose the following boundary conditions: O(Z, Z ) = 0
D,o(z, z') 1
=
=0
D,Da~(z,z') I
0
~ ' ) l ~ = ~=O,
(4.8.424
k = O , 1 , 2, 3
(4.8.42b)
D'~,O(Z,z ' ) I ~ = ~ ~ = O D,a,a,o(z, D,, . . . DAlaO( W(z, z ' ) / ~ = ~ , = O
=z , =
D2D2aby)(z, Z ' ) I ~ = ~ , = 16.
(4.8.42~)
Let us comment on these restrictions. At Y=O we have A = 0 , hence
(4.8.43) Copyright © 1998 IOP Publishing Ltd
370
Ideas und Meliiods of Supersymmetry and Supergravity
This implies that in the case Y # O the two-point functions from equation (4.8.40)read 1 2
g ( 2 , z') = - (x - X')Z
a ! y ( z ,z') = S"(0 a T ' ( 2 , z')=O(Y)
+ O(Y)
e')+ O(Y)
n = 1, 2, 3, * . ..
It can also be seen that any object such as
D,! D,,
. . . DAAg(z,2') 1
. . . D , , U ~ ~ )z')\ ( Z ,z = z ,
k = 0, 1, 2, . .
I
is a tensor superfield possibly constructed from the Y-independent Lorentz-invariant tensors cXp, (gJexdr, and so on, as well as depending analytically on Y, 'P and their covariant derivatives to a finite order. The last observation, supplemented by considerations of dimension, tells us that all the expressions in the left-hand sides of equations (4.8.42a,b)should vanish. The boundary conditions (4.8.42)and the identity daabg(2,
z')I z = zf = 4 a b
which follows from equations (4.8.41~)and (4.8.42~)show that as s --t +O the kernel U f ' behaves like the free one (4.8.43). As can be easily checked, direct consequences of the equations (4.8.41)and of the boundary conditions (4.8.42) are the following relations: (4.8.44) Now, we are in a position to determine the divergent part of the one-loop effective action (4.8.39).By analogy with equation (4.8.22),in the present case we have
ri;;[o, 01 - =--/d8.za$")(z,z)=- 20(4n)' /d8z'P'€'. 2w (4742
Therefore, the corresponding counterterm cancelling written as .
I
ri;?[o,a]
(4.8.45) can be
-
(4.8.46) where we have taken into account that Y =Y:(@). As is seen from equation (4.8.46),the theory under consideration proves @ + const. The final to be multiplicatively renormalizable only if Y;p%(@) requirement means that TC(@) is a third-order polynomial in 0. N
Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
371
4.9. Superfield effective potential In conventional field theory the effective potential is defined to be the effective Lagrangian evaluated at constant values of the scalar fields, all other fields being taken to be zero. The effective potential Veff(cp)is an important tool for studying the questions of symmetry breaking and vacuum stability. In the present section we shall introduce a supersymmetric generalization of the effective potential leading to the so-called superfield effective potential.
4.9.1. Efective potential in quantum field theory (brief survey) Let r[q]be the renormalized effective action of some scalar field theory. In general, r[q]is a nonlinear and non-local functional the calculation of which is usually performed in the framework of the loop expansion. Suppose that cp is a slowly varying field. Then we can represent the effective action as a power series in the derivatives of field, that is
).
1
- Veff(cp)-- Z(cp)~""8,cpdncp+ . . . 2
(4.9.1)
The function Veff(cp)is called the effective potential. To calculate the effective potential, it is sufficient to find the form of r[q] at cp = const. Let cp=const be a solution of the effective equation K[cp]/6cp=O. Obviously, this is equivalent to
(4.9.2) The last equation is exact and allows one to determine those constant values of cp providing a minimum of the effective action. It is clear that the existence of such non-zero scalar fields can be associated with symmetry breaking. According to the loop expansion, Vefdcp) is of the form
c m
Veff(cp)
= Vcp)+
fi"V'"(cp)
(4.9.3)
n= 1
where V(cp)is the classical potential of the theory under consideration, while V(")(cp)is the n-loop quantum correction to the effective potential. The quantum corrections are usually evaluated using the standard diagram technique. As an example, let us calculate the one-loop effective potential for a theory with the classical action
(4.9.4) The one-loop contribution to the effective action is defined by the functional Copyright © 1998 IOP Publishing Ltd
312
Ideas und Melliods ?f Supersymmetry and Supergravity
integral
and can be written in the form
where G"p)(x,x') is a Green's function obeying the equation (
- V"(~~))G'"(X, x') = - h4(x - x').
(4.9.6)
Notice, we prefer here to use the notation Tr for the operation of functional supertrace (4.1.8), because the theory under consideration involves no fermionic fields. Introducing the proper-time representation G"p'(x,x') = i
[:
ds Uiq)(x, x' Is)
(4.9.7)
where kernel U('p'satisfies the equation (4.9.8) and the initial condition Lr"p'(x, x' Is +
we regularize
+ 0)= 64(x - x')
(4.9.9)
r(l)[cp]as follows
o + O at the end of the calculations (see the details in Chapter 7).
In order to find the effective potential, it is sufficient to solve the system (4.9.8,9) only for cp =const. The relevant solution reads
u(q= const)(x, x'ls)=-
1
4s
1
isV"(cp) .
(4.9.11)
Now, from equation (4.9.10) we deduce the regularized one-loop correction to Ve,,(cp):
=-
1 2 ( 4 ~ )-~l )( (~-~2)
Copyright © 1998 IOP Publishing Ltd
. (4.9.12)
Quantized Superjelds In the limit w -+ 0, we have T(w) x l / w
373
+ y, y being the Euler constant,
Taking the same limit in equation (4.9.12)gives
vZ'= vi;;+ V(1)(cp) where
(4.9.13~)
Vi:; is the divergent part of the effective potential. It should be cancelled by the counterterm SCO""t*
= -Id'.
64n20
(V"(V))~.
Hence, the one-loop contribution to Veff(cp) is given by equation (4.9.136). As a result, the one-loop effective potential reads Veff(cp)= V(cp)+V(')(cp)=V(cp)+7(V(cp))2( h
64n (4.9.14) The effective potential obtained depends on arbitrary mass parameter p which reflects some inherent arbitrariness in the choice of renormalization scheme. This parameter should be fixed in practice by imposing so-called normalization conditions corresponding to a correct choice for observable parameters of the theory (masses, couplings). Let us consider, for instance, the theory with the following ciassical potential
where E,, is a bare coupling. One can choose suitable normalization conditions in the form
(4.9.15) where cpo is some non-zero constant. The quantity E. can be called the observable coupling. Now, from expressions (4.9.14) and (4.9.15) one finally Copyright © 1998 IOP Publishing Ltd
374
Ideiis irnii Methods of' Supersymmetry and Supergravity
obtains (4.9.16) This result is known as the Coleman-Weinberg effective potential. 4.9.2. Superfield e f e c t ive potential Let T[@, G] be the renormalized effective action of some supersymmetric model described by chiral @ and antichiral @ scalar superfields. Suppose that @ ( ~ ) = e ~ , ~ 0@) (and x , G ( z )=e-iXG(x, 8), where 2 = d d d , , vary slowly in space-time. Then we can represent the effective action as follows
I-[@,
61=
s
d8z 2&(@, DA@,DAD,@, . . ., G, DAG),DAD,@,. . .) d6z 9(,cf(@, a,@, a,8,@, . . .) + C.C.
(4.9.17)
Here Yeffis a power series in the covariant derivatives DA@,DAD,@, . . ., 6,DAG,DAD,@,. . ., while 9$i is a power series in the partial derivatives a,@, dad,@, . . .. It seems natural to call geff an effective super Langrangian and 2'$\an effective chiral super Lagrangian. Before we proceed further, it is worth saying some words about the presented structure of the effective action (4.9.17). The effective action superfunctional is calculated in practice with the help of the supergraph technique described previously. In accordance with the non-renormalization theorem, the contribution to r[@, a] coming from an arbitrary supergraph can be represented as a single integral over d48, but not over d28 or d28. On these grounds, this is only the first term in equation (4.9.17) which is admissible on the basis of general principles. .However, one can expect contributions to the effective action which really exist, of the type {d8z( - g ) G = { d 6 z
(4.9.18)
G
where G is a chiral superfield. It is such corrections that form 6p$\. The relation (4.9.17) defines the exact effective action as an expansion in superfield covariant derivatives. Now, let us look for a supersymmetric extension of the conventional effective potential. The theory under consideration is described by scalar and spinor fields, as is seen from the component expansion
qz) = e i 0 a u 8 ~(u~ ( x+)ea+Dl(x) +e
2 w .
To obtain the effective potential, we evaluate the effective Lagrangian at constant scalar fields, the remaining fields being switched off, that is, under Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
375
the conditions A = const
F = const
I,!Iz= 0.
(4.9.19)
Then, the effective potential is given as - Veff=
s
d48
is
zff+
d29 9’$\+ C.C.
A,F=const
(4.9.20)
Note, however, that the conditions (4.9.19) are not supersymmetric, since applying a supersymmetry transformation makes non-zero. Therefore, it is worth replacing equation (4.9.19)by the supersymmetric requirement
da@= 0
(4.9.21)
and equation (4.9.20)by
(4.9.22) The final object can be called a supersymmetric effective potential. As is seen, Veff and peffcoincide when the component spinor field is zero. Certainly, such coincidence can always be achieved by applying a special supersymmetry transformation which switches off the spinor. However, the supersymmetric effective potential has one essential quality, namely, it can be calculated by purely superfield methods. By virtue of equation (4.9.22), the supersymmetric effective potential is determined by the following superfield objects
“Kff = XffI d,*=O
V%(@) = =ail a,*=o
(4.9.23)
which will be called the real and chiral parts, respectively, of the superfield effective potential. We will also refer to V$h as an effective chiral superpotential. It is easy to see that the most general form of Kff reads
“Ktf = K(@,@) +9(D,O, D2@,D&, D2&;0,@) ‘
(4.9.24)
where
F I D,*= D*@=o = 0. The function K(@,a)is said to be the effective Kahlerian potential, following the terminology of supersymmetric a-models (see Section 3.3). As for the second term in equation (4.9.24),it seems reasonable to call 9 the auxiliary field effective potential. The point is that its contribution to the effective Lagrangian, J d48% evaluated under the supersymmetric condition (4.9.21), turns out to be at least of third order in the auxiliary fields of @ and a. Within the framework of the loop expansion, K and 9can be represented Copyright © 1998 IOP Publishing Ltd
376
Ideas and Methods of Supersymmetry and Supergrmity
as
+
m
K(@,6)= K O ( @6) ,
h"K,(O, 6)
(4.9.25a)
n= 1
(4.9.25b) Here KO(@,@) is the classical contribution. For example, for the Wess-Zumino model we have Ko(@,6)=@@,The h-dependent terms in equation (4.9.25)mean quantum corrections to K and 5Further, the effective chiral superpotential reads (4.9.26) where Pc(@) is the classical chiral superpotential, and Y f f )are quantum corrections. In the case of the Wess-Zumino model, the one-loop correction proves to be zero. to .l"$'f 4.9.3. Superfield eflective potential in the Wess-Zumino model Let us consider the effective action of the Wess-Zumino model (3.2.11). After performing standard manipulations, one obtains
where
f[@,@] = r[@, a] - s[@,61
Y =Y:(@), and the action S[x,;C;Y,T] is given by equation (4.4.6). The representation (4.9.27) is the basis for the calculation of the effective action in the framework of the loop expansion. In order to find the superfield effective potential, we should determine f[@, under the condition (4.9.21). Use of equation (4.9.27) allows us to search for the superfield effective potential on the basis of supergraph techniques with a superpropagator of @ being subject to condition (4.9.21). the type G(yI)(4.4.7),where Y = Yg(@), According to equation (4.8.3l), G(yIy' can be expressed through the Green's function GL,' obeying equation (4.8.30). Hence to develop a perturbation theory for evaluating the superfield effective potential, we should find the Green's function GLw)in an external chiral superfield Y under the condition d a y = 0. In turn GLw)is determined, via the proper-time representation, by the corresponding kernel ULyI)(4.8.37, 38). As a result, in order to find the superfield effective potential, we should
a]
Copyright © 1998 IOP Publishing Ltd
Quantized SuperJelds
377
solve the system (4.8.37,38) for U:?') in an external chiral superfield Y under the condition a,Y=O and then construct the propagator GL?'). The final object is to be used in the loop calculations of K and ,R The solution of the system (4.9.37,38) under the condition aaY=O can be found explicitly'. Calculation of the contributions to V$',demands additional comment. Because of equation (4.9.18),we can impose condition (4.9.21) only at the final stages of transformations. However, since VS',depends only on 0 for its calculation in the massless case, '€'=,IO,it is necessary to keep in the kernel UL?') and the propagator CL?') terms depending only on Y, but not on T.Taking into account the relations DkY= O and DZD2=0, one notices from equations(4.8.30, 36,37) that the "-independent parts of UL?') and CL,?'' are at most linear in Y. Then one finds 1
GL,?"(z,z') = - - d8(2, z') 0
(4.9.28)
40
This ansatz can be used for loop calculations of V$',. In the rest of the present section we shall discuss the superfield effective potential in the one-loop approximation. It is worth beginning with the consideration of some general properties of the one-loop effective action. Recall that the one-loop correction to r[@,G] is given by equations (4.8.27,35). The starting point of our analysis is the observation that the superfunctional integral (4.8.27) proves to be invariant under rigid transformations of the dynamical superfields X+eiuX
j+e-'"j
(4.9.29~)
supplemented by the following displacements of the background superfields Y+e-ziz'+J
T+eZiuT
(4.9.29b)
being a real constant. As a consequence, the unregularized one-loop correction r(l), defined by equation (4.8.35)and regarded as a functional of the superfields Y and \ii, is unchanged under the transformations (4.9.29b). Remarkably, this invariance turns out to be unbroken after introducing the regularization (4.8.39).Let us argue this assertion. According to equations (4.8.37,38),we can write Ub?') in the form Q
uL,?')(s) =
(4.9.30)
A being defined in expression (4.8.30).We represent the operator A in the manner
A=o+g?
1 1 g?=--YDz--YD2 4 4
'I.L. Buchbinder, S.M. Kuzenko and J.V. Yarevskaya, Nucl. Phys. B411 665, 1994. Copyright © 1998 IOP Publishing Ltd
378
Ideas and Methods of Supersymmetry and Supergravity
and expand the exponential (4.9.30)in a power series in 9 One finds
Vl?(s)=
P
1 U&) n=O
~ , ( s= ) eisC
(4.9.31)
dte-i'oPUn-l(t)
n>l.
As is seen, for n > 1 we have
U,(s) = i"
{: :j dt,
dt,-
:j
...
dt, A&, t,,
A,(% t,, . . .) t,)= U,(s-t,)~U,(t,-t,-,)B..
*
. . ., t,)
(4.9.32)
Uo(t2-t1)YUo(t,).
Because of the identities
D&Y=o
D2D2=o
[D,, U&)]
=o
and owing to the explicit form of % the As can be expressed as follows: At,(& t,, . . ., t 2 n1--(')2n-
U,(s-t2,)YD2UO(t2n-t2n-1)
+(YD2-'T'D2).
(4.9.33b)
On the same grounds and due to the cyclic property of sTr, it follows from equation (4.9.33b) that all As carrying odd labels have zero supertrace, sTr(A2n+l(s,ti, ..., t 2 n + l ) } = 0
In the even case, one has sTr{AZn(s,ti, x
. . ., t 2 , ) } = 2
cyn -
n = O , 1, ....
(4.9.34)
~T~(U,(S)YD~U~(~~~-~~~-,)
TD2UO(t2n-1-t2n-2)YD2... Uo(t2-tt,)TD2UO(tl-t2n)}.(4.9.35)
Obviously, this expression is invariant under the transformations (4.9.296). It is worth also noting that
sTr Uo(s)= 0 Copyright © 1998 IOP Publishing Ltd
Quantized Superjelds
379
as a consequence of the explicit form of U,@)= V~(s)(4.8.43). Therefore, the integrand in equation (4.8.39) is of the form m
sTr Uly)(s)=
1 sTr U,,($
(4.9.36)
n=l
and each term on the right contains an equal number of Y - and T-factors. We observe that rlj)possesses invariance with respect to the transformations (4.9.296).The same is true for the one-loop corrections to &@, $) and 9 7 In particular, all these objects are even functionals of the variables Y and 'P. Another important consequence of the above analysis is that the one-loop correction to V$\ is zero.
4.9.4. Calculation of the one-loop Kt'ihlerian ejjective potential In conclusion we would like to illustrate the general approach which has been described above, by evaluating one-loop Kahlerian effective potential in the Wess-Zumino model. As follows from the previous discussion, one must compute explicitly the kernel
uLy)(z,z' 1 s) = eisAJ8(z, z')
(4.9.37)
the external chiral scalar Y being subject to the condition aaY= o
(4.9.38)
in order to find the one-loop correction to Veff. Then, the one-loop correction reads (4.9.39) where we have used the regularization prescription (4.8.39). O n the other hand, if we are interested only in finding the one-loop correction to K(@, we can replace the condition (4.9.38) by the stronger one
a),
Y = const
(4.9.40)
and calculate the corresponding kernel (4.9.37). Then, the regularized reads correction Kl(@,
Let us find the kernel Uby) corresponding to the choice (4.9.40). We can now write
Copyright © 1998 IOP Publishing Ltd
380
Ideas and Methods of Supersymmetry and Supergravity
where
This leads to U(V V = const)
1){DZ7D2}
+
+
1
s i n h ( i s , / m ) ( Y D 2 YD’) U&).
4 J m
(4.9.43)
The correction (4.9.41)is determined by the kernel at coincident points. From equations (4.9.42b) and (4.9.43) one immediately obtains
2 ( c o s h ( i s d m ) - 1]U(x7 x’ls) U(”’Y=const)(z, V z’ls)=x=xI (4.9.44) 0
where
To compute the right-hand side of expression (4.9.44), we need to evaluate expressions of the form O ~ U ( X , X I S ) I This ~ = ~ ,can . easily be done with the help of the equation
which U satisfies, thus obtaining
As a result, the one-loop correction (4.9.41)takes the form
Copyright © 1998 IOP Publishing Ltd
Quantized Superfields
381
or, after introducing the new integration variable z = i#Y:
.
(4.9.45)
Let us use the well-known series representation for the so-called erfi-function
where 2
erfi(x) =
[" ef2dt.
Then we obtain
Rewriting the integrals in the final expression as
we obtain
It is easy to see that tu = C
+ O(o),where
is some finite constant, as follows from the asymptotics
Jol
dt e-I/4(1 - l')
Now, setting
p*"(YY)-W 0
O0
,T
1
=-0
TY In? P
+ O(o)
we obtain
a ) w = K1(@*shiv + K1(@, @I
K1(@,
Copyright © 1998 IOP Publishing Ltd
382
Ideas and Methods of Supersymmetry and Supergravity
where
(4.9.47) (4.9.48) Here equation (4.9.47)defines the divergent part of the superfield effective action. This divergence has been calculated in subsection 4.8.2. The term K l d i v is cancelled by the counterterm (4.8.46).The relation (4.9.48)represents the one-loop correction to K(@,@). Thus the one-loop Kahlerian potential has the form
To complete our work, we should fix p by imposing some suitable renonnalization condition. Consider, for simplicity, the massless Wess-Zumino model with YJ@)= (i/3!)O3 and choose the following renormaliation condition:
(4.9.50) where C+, is a non-zero constant with dimensions of mass. Then one obtains
hi2 K(@,G) = G@- -@@ 32n2
(4.9.51)
The one-loop superfield effective potential Veffincludes not only the Kahlerian part K , but also the auxiliary field effective potential 9. Calculation of the latter is a more tedious task, and we will not discuss it here, referring the reader to the paper by I. Buchbinder, S. Kuzenko and J. Yarevskaya.
4.9.5. Calculation of the tn,o-loop effective chiral superpotential As we know the one-loop contribution to the effective chiral superpotential
is absent in the Wess-Zumino model (see for instance equation (4.9.35) at = 0). Therefore one can expect that the quantum corrections to the chiral superpotential will arise starting at least with the two-loop approximation. A possibility of chiral quantum corrections to the effective potential was first noticed in the book by M.T. Grisaru, S.J. Gates, M. RoEek and W. Siegel, and the relevant supergraphs producing such corrections were found by P. West2. 2P.West, Phys. Lett.258B 375, 1991. Copyright © 1998 IOP Publishing Ltd
383
Quantized Superjelds
For the first time the explicit two-loop correction to the chiral effective superpotential was found by I. Jack, D.R. T. Jones and P. West3 in component form. We give here a completely superfield approach4. We start with the general equation (4.9.27)for the massless Wess-Zumino ' = 0. model. To find the chiral effective superpotential it is sufficient to set P In this case the two-loop contribution to effective action depending only on Y is of the form (4.9.52) The Green function GJz,,
z2)can be found from equation (4.8.31),where
Gr)is given by equation (4.9.28).Then
where Y = -A@, Converting here the integrals over d62 into integrals over d8z, in accordance with the rule d62(- aD2)= d'z, and setting
(4.9.54)
From the viewpoint of the supergraph technique, this expression corresponds to the supergraph given below, the external lines being chiral. As the next step, we rewrite F2)in a form that is suitable for extracting the correction to $'-$KO)contained within it. Note first, that it is convenient to work in the chiral representation where
since equation (4.9.54) involves only chiral superfields. Further, we apply standard D-algebra transformations (see subsection 4.7.1) to in order to reduce the number of integrals over d48 to a single one. Making use also of 31. Jack, D.R.T. Jones and P. West, Phys. Lett. 258B 382, 1991 41.L. Buchbinder, S.M. Kuzenko and A.Yu. Petrov. Phys. Lett. 321B 372, 1994, Copyright © 1998 IOP Publishing Ltd
384
Ideus und Methods of Supersymmetry and SupergravitJ
the obvious identities
we arrive at
where
Copyright © 1998 IOP Publishing Ltd
Quantized SuperJields
385
where J(P1,
Pz) =
d4kl d4kz k:p:
+ kip:
- 2(kI k )(
fak, P)
plpz).
(4.9.58)
Up to now, the chiral suprfield @(x,e) was completely arbitrary. When computing the effective superpotential, we can assume @ to be slowly varying in space-time. This assumption implies ep(y1, e
) ~ ,e), x WX, e).
(4.9.59)
Then equation (4.9.57) takes the form (4.9.60) It remains to use the known integral relation (4.9.61)
i ( x ) being the dzeta-function. As a result, the two-loop chiral effective superpotential read
We note that the chiral effective superpotential is automatically finite. No renormalization is required for its calculation.
Copyright © 1998 IOP Publishing Ltd
5 Superspaee Geometry of Supergravity To make a prairie it takes a clover and one bee, One clover, and a bee, And revery. The revery alone will do, If bees are few. Emily Dickinson
5.1. Gauge group of supergravity and supergravity superfields 5.1.1. Curved superspace
In previous chapters, the supersymmetric generalization of Minkowski space as well as that of matter field theories in Minkowski space was given. Our next goal is to look for a supersymmetric generalization of Einstein gravity - for a supergravity. Ordinary gravity is treated as the theory of a curved space-time. Therefore it is tempting to think of supergravity as being theory of a curved superspace. But what is a curved superspace? A space-time is said to be curved if the curvature tensor Wmnp,,constructed from the metric gmn(x),is non-vanishing. It is a flat space-time when 9%"npr= 0. This condition proves to be equivalent to the statement that there exist preferable coordinate systems in which the metric has the Minkowskian form, gmn(X)
= Vmn.
Any two such coordinate systems are related by some Poincare transformation. It is the Poincare transformations which leave invariant the Minkowski metric. 386 Copyright © 1998 IOP Publishing Ltd
387
Superspace Geometry of Supergravity
In the case of a curved space-time, there is no natural way to choose preferable coordinate systems; all coordinate systems are on the same footing. Hence, a physical theory should be invariant under the group of general coordinate transformations X"
-+
X"
=f"(x)
or, in infinitesimal form,
x m + X" = X"
-
(5.1.1)
K"(x)
where K"(x) is an arbitrary infinitesimal vector field. The general coordinate transformation group is the gravity gauge group, and the metric, with the transformation law dSmn(X) =
VmKn + VnKm
(5.1.2)
plays the role of the gravity gauge field. Now, keeping in mind our space-time intuition, we return to superspace. Clearly, in order to define a notion of curved superspace, one should find a superfield 'metric' object. It was V. Ogievetsky and E. Sokatchev who found such an object. In Chapter 2, subsection 2.5.3, we introduced the family of real surfaces R414(2)in complex superspace C4I2which were defined by equation (2.5.11). Every such surface can be understood as a real superspace R4I4 supplied by a real vector superfield %"(x, Recall that the family {R414(&')} contains a unique super-Poincare invariant surface - R414(00@,and that this surface was identified with flat global superspace iw4I4. Obviously, we are faced with a situation analogous to the space-time one. We can look at space-time as a real space R4 provided with a second-rank tensor g,(x). The family of all space-times { R4(gmn)}contains a unique representative-flat global spacetime R4(qm,), which is characterized by the Poincare invariant metric. Therefore, the family of superspaces { R4i4(%")} is a supersymmetric generalization of the family of space-times { R4(g,,)>. let us Starting with a given complex coordinate system (y",O") on consider holomorphic coordinate transformations
e,@.
y m -+ y'm
=f y y ,
e)
e"
+ 6" = f y y ,
e) (5.1.3)
Infinitesimal holomorphic coordinate transformations are of the form (5.1.4) Here E." and 2%are infinitesimal vector and spinor holomorphic superfields. Copyright © 1998 IOP Publishing Ltd
388
Ideas and Methods of Supersymmetry and Supergravity
The set of all holomorphic transformations (5.1.3) forms a supergroup (the 'holomorphic coordinate transformation supergroup'). Commuting two infinitesimal holomorphic transformations, generated by parameters A;l(y, e), j.;(y, 0) and j.T(y, e), j.?j(y, e),respectively, one obtains a transformation of the same type. The corresponding parameters are
e)
j w y ( y , = i.;8J.y/dyn j.;(y,
6) = j.;a,i:/ayn
+ n$ai,y/aeB - ( 1 ++ + n&?n;/aeB - (1
c-)
2)
(5.1.5)
2).
Given a superspace rW4l4(X), every holomorphic transformation (5.1.4) induces a coordinate displacement on R414(2) as well as changing the superfield H"' in a rather complicated nonlinear way. Explicitly, in accordance with equations (2.5.11) and (2.5.12), we have 1 1 xm-+ xtm= x m - - i m ( x n + ixfl, e/') - - P ( x n - i s " , @) 2 2
(5.1.6) and
(5.1.7)
Introducing the superfield displacement 6 Z " ( x , e, e) = Z " ( X ,
e, 0) - Z m ( X , e, 0)
equation (5.1.7) leads to
axyx,e, 8) = -2i
(nm
- P)+
(: + - (A"
Am = /Zm(xn+ i x " , ep)
xnp, + n ~ a + ,
A~ = P(X"
1
s y x , e, B)
+ i s n ,eb).
(5.1.8)
This transformation law defines a nonlinear superiield representation of the holomorphic coordinate transformation supergroup. It is natural to treat the holomorphic coordinate transformation supergroup as a gauge group and &"(x, 0, e) as the corresponding gauge superfield. Geometrically, the superfields X"' and S m+ d&" are equivalent. They describe the same surface in C4I2 but written in two different complex coordinate systems on C4I2.Therefore, the space of all gauge superfields is decomposed into orbits of the holomorphic coordinate transformation supergroup. Gauge superfields from the same orbit determine equivalent geometries on R4I4. Gauge superfields from different orbits determine non-equivalent geometries on [w4I4. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
309
In accordance with the above consideration, we can make the following preliminary definitions. A superspace rW4I4(isH said ) to be 'flat' if there exists a holomorphic coordinate transformation (5.1.3) which deforms 2"to the form
zyXf, o , P)= 8 , m e w v .
(5.1.9)
said to be a 'curved superspace'. It is the superfield Otherwise, rW4I4(isZ) Z"(X, 8,8)which plays the role of a superfield 'metric' object or 'gravitational superfield'. The transformation law (5.1.8) is the superspace analogue of equation (5.1.2). A t first sight, the holomorphic coordinate transformation supergroup of C4I2 is a candidate for the role of the superspace analogue of the general coordination transformation group of space-time. However, this is not quite correct. The point is that this supergroup includes only non-localizable transformations (a transformation is said to be localized near some point if it coincides with the identity mapping everywhere except for in a small region around this point). Indeed, whan taking the superfield parameters A"@, 0) and E,"(y, e) in equation (5.1.4) to be holomorphic on the whole superspace C4I2, one obtains non-localizable (in y") coordinate transformations only. Recall that one cannot make a holomorphic superfield on C4I2non-vanishing in a small neighbourhood of some point y ; (see subsection 2.5.2). On the other hand, the general coordinate transformation group of space-time includes arbitrary, in particular localized, transformations. To resolve this difficulty, one must consider some 'extension' of the holomorphic coordinate transformation supergroup. Recall, flat global superspace R414(00@is embedded in the truncated superspace C:12 defined by equation (2.5.9). The space C:12 is mapped onto itself by every super-Poincare transformation. Finally, the requirement of holomorphicity on does not give any global restrictions on superfields. In particular, there are localized holomorphic superfields on Therefore, we can proceed as follows. We restrict our consideration to the case of superspaces iW4I4 embedded (2) in C:12 (by analogy with the flat global superspace). This means that Xm(x, 8,8) has vanishing body,
e, Q), = 0.
(~"(x,
(5.1.10)
Further we will consider only holomorphic coordinate transformations on
C:12. This restricts the superfield parameters in equation (5.1.4) in the manner
( ~ m ( ye) , - ~ ( j , = 0.
(5.1.11)
The requirements (5.1.10, 11) make subsequent results physically correct. Let rW414 be(2 a)flat superspace. We choose a preferable coordinate system on R4I4in which &" = 8,"8aa8. Copyright © 1998 IOP Publishing Ltd
(5.1.12)
390
Ideas and Methods of Supersymmetry and Supergravity
There exists a subset of transformations (5.1.4) preserving this superfield, d#"
= 0.
Many of them are given by the superfunctions - P ( y , 0) =
ba + 2iOo"F + Ka,yb
- ~ ( y e) , = b" = B
(5.1.13)
- Kapeo
K a b = ( ~ " ~ ) , p K-~ (Zob)&aB p
K,p = Kpa.
Here b", E' are constant parameters. Obviously, transformations (5.1.4), induced by the given superfields, represent infinitesimal super-Poincare transformations of C4I2(see subsection 2.5.1).Therefore, every super-Poincare transformation leaves invariant the flat gravitational superfield (5.1.12). 5.1.2. Conformal supergravity Above we have interpreted X"(X, 8,B) as a gravitational superfield. In order for this interpretation to be non-contradictory, X" should describe a multiplet of ordinary fields containing a gravitational field. Hence, it is necessary to investigate the component structure of #" and to analyse the action of gauge transformations (5.1.8) on the component fields. Decomposition of #" in a power series in 0 and Bleads to (5.1.14) Constraint (5.1.10) is equivalent to (C"(X))B = 0. Among the component fields, there is a natural candidate for the role of the gravitational field. The 8a"B-coefficient e,"(x) may be identified with the vierbein. Before clarifying this, we show that the component fields C", xm,, pi', S" and S" are purely gauge degrees of freedom. The transformation law (5.1.8) can be rewritten in the form ti^^(^, e, 8) =
1
2
e) - -i ~ 2
( xR), + O ( X )
(5.1.15)
where O ( X ) denotes all terms depending on 2 and its derivatives. Further, decomposition of the superfield parameters in expression (5.1.4) in a power series in 8 gives AY^, e) = ~
Y Y+ ) e v u )+~
W Y )
,qy, e) = E ~ ( J J )+ oaB(y)ep+ e 2 y x ( y ) . Copyright © 1998 IOP Publishing Ltd
(5.1.16)
Superspace Geometry of Supergravity
391
Constraint (5.1.1 I ) is equivalent to
The remaining component fields in expression (5.1.16) are arbitrary. Then, equation (5.1.15) takes the form GYP(X,
1 1 i i e, g) = -(am(x) - ayx)) + - eaqyx)- - B&P(x) + - e2sm(x)
2
2
1
- -BLs"(X) 2
2
2
+ O(H).
We see that component fields C", and S" of Z mhave arbitrary variations. Using gauge freedom (5.1.8),one can make these fields have any given values. In particular, there exists a gauge in which these fields are gauged out ('Wess-Zumino gauge'), P y x , 8,8, = 8oa8e,"(x)
+ i&Wm,(x)
- ie28iTmk(x)+ S282Am(x). (5.1.17)
In the Wess-Zumino gauge, the transformation law (5.1.8) becomes polynomial. Imposing the Wess-Zumino gauge does not fix the gauge freedom completely. There is a subset of transformations (5.1.4) preserving the Wess-Zumino gauge. Every such transformation is characterized by (field dependent) parameters: Am(x,0) = bm(x)+ 2ieoaC(x)e,"'(x) - 2O2C(x)Ym(x) 1 P ( x , e) = P(x)+ -(c(x) 2
+ isZ(x))P + Ka&)@ + e2ua(x)
(5.1.18)
K*p = Kp..
Here bm(x) and o(x), zZ(x) are arbitrary real vector and scalar functions, respectively, E"(x) and ~ " ( x )are arbitrary spinor functions. Comparing expressions (5.1.13) and (5.1.18), we see that parameters b"(x) induce general coordinate transformations of a space-time, parameters K"B(xkloca1 Lorentz transformations and parameters P(x)-local supersymmetry transformations. Now we proceed to obtain the transformation laws of fields e,", Y",, Am. We start with the general coordinate transformations which correspond to the choice e) = byx) A"(x,e) = 0. jVm(x, In this case equation (5.1.8) takes the form G b s m
Copyright © 1998 IOP Publishing Ltd
= b n a n x "- Z n a n b m
392
Itleus uiid Merhods of Supersymmetry and Supergraoity
At the component level, one finds dbeam= bndneam - (dnbm)ean dbv",
=
b'dnYm, - (dnbm)YnE
(5.1.19)
dbA" = b"2,A" - (2,b")A". Therefore, each of the component fields in expression (5.1.17) transforms as a world vector under the general coordinate transformations; hence the field index 'ni' is to be understood as a curved-space one. Let us consider the local Lorentz transformations. They are generated by the parameters jWm(x, 0) = 0
j.'(x, 0) = K1&x)OP.
One can readily deduce from equations (5.1.8) that 6Keam= K;ebm (5.1.20) 4
where
Kab= ( G " ~ ) , ~ K-@( ~ a b ) & u f i , It is seen that eamtransforms as a covector and 'I-", as an undotted spinor under the local Lorentz rotations; hence the field indices 'a' and 'x' are to be understood as tangent space (or flat) ones. In accordance with equations (5.1.19,20), the Xm-component field eam(x) has the same behaviour under general coordinate and local Lorentz transformations as the vierbein. On these grounds, it is natural to identify eam(x)with the vierbein. The last relation (5.1.20) shows that A"(x) changes non-trivially under local Lorentz transformations. It would be more desirable to work with a Lorentz scalar field. For this purpose, we redefine Am as (5.1.21) where q c d is the torsion-free spin connection (see equations (1.6.23)). A" proves to be a world vector and Lorentz scalar field,
6Jm
bndnAm- (anbm)An
6x2'"
= 0.
(5.1.22)
Before considering the local supersymmetry transformations, it is worth concentrating on one technical matter. Let us take two sets of the supergroup Copyright © 1998 IOP Publishing Ltd
393
Superspace Geometry of Supergravity parameters:
AT = 0
= KUp@
and calculate the corresponding bracket parameters (5.1.5). In the spinor case, one has =
+ -ERbCde K 4
- KapEp - O2
b
cd)
(Zd,)'
]
(5.1.24)
where eb = ebmd,. It is seen that the local Lorentz transformations treat P ( x ) as an undotted spinor, but the transformation law of qa(x) contains inhomogeneous ?-dependent terms. To avoid this complication, it is useful to make in expressions (5.1.23) the replacement
Owing to the local Lorentz transformation law of mbcd, one now has
1; = -K"p(Ep
+ 1Q2)
instead of equation (5.1.24). On this basis, we change the parametrization (5.1.1 8) to A"(x, 0) = b" + 2i00RCe," - 202C'P'" A"(x, 8) = E'
1 + -(a + in)0" + Kap@ 2
(5.1.25)
Now, let us find the local supersymmetry transformations. Setting to zero all the parameters in expression (5.1.25), except E" and Fb, and using equations (5.1.8) and (5.1.21), one obtains after long (requiring, maybe, some days) calculations &eom= im"F'"- iY'"o"?
6 , ~ = " ~(aRdbV,&ebm- 2i2", (5.1.26)
Copyright © 1998 IOP Publishing Ltd
394
Ideas and Methods of Supersymmetry and Supergravity
Here Vn and V, = e,"Vn are the standard (torsion-free) covariant derivatives (see Section 1.6). As is seen, the Majorana spin-vector field
);:(
I" =
(5.1.27)
is a superpartner of the gravitational field. That is why I" is called the 'gravitino field'. It follows from the second relation (5.1.26) that the gravitino is a gauge field for the local supersymmetry transformations. Our only remaining task is to consider transformations induced by the parameters o, R and 7%in expression (5.1.25). Making the choice A"(X,
1 2
e) = 0
jla(x, e) = -o(x)eU
in equation (5.1.8), one finds 8oeom= oe," 3 2
&Y "% = -CY m u
(5.1.28)
S U P = 2o2".
Comparing doeamwith equation (1.6.32) shows that the parameter ~ ( x ) corresponds to the Weyl transformations. Conformal weights of Y mu and A"" are equal to 4 and 2, respectively. Further, making the choice P(X,
e) = o
j/(x,
i e) = -n(x)ez 2
in equation (5.1.8), one arrives at the transformation BReam= 0
i 6nY'", = - -nYmu 2
6Jm
(5.1.29)
1 2
= -gmna,n.
This is a local chiral or y,-transformation. When rewritten in terms of the four-component spin-vector 'I",it has the form 6,Y"
=
1
- -Ry,\Y". 2
In accordance with the last relation (5.1.29),2" is a gauge field for the local y5-transformations. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
395
Finally, we set
e)
im(x, = 0
P(x,e) = e2rli(x)
in equation (5.1.8). A straightforward calculation gives
dqeam= 0
6,Ym, = -i((r'Vj)cceam
(5.1.30)
8,Am = iv]ym - i p .
Evidently, this is a supersymmetry transformation mixing bosonic and fermionic fields. Therefore, we have two types of supersymmetry transformations generated by the parameters (E', Ch) and (v]", t&), respectively. To distinguish them, the former will be called Q-supersymmetry transformations and the latter - S-supersymmetry transformations. The S-supersymmetry invariance can be used to make Ymatraceless. Introducing the gravitino field with a tangent-space index Yoa= emeYmZ, and converting the vector index into a pair of spinor indices, YBb, = (d,)&"%', the second relation (5.1.30) takes the form
So, one can choose the gauge Y", = (C?,Y')~ = 0. We have spent some time making exhaustive calculations, and now it is necessary to pause and think over the results. Our wish was to find a supersymmetric generalization of Einstein gravity. Such a generalization should contain three types of gauge transformations: general coordinate, local Lorentz and local Q-supersymmetry ones. However, the supergroup of holomorphic coordination transformations on @:I2, which were chosen in the role of gauge group, turned out to be too large. It includes, besides the mentioned transformations, local scale, local chiral and local S-supersymmetry transformations. Recall that the gauge group of Einstein gravity did not contain Weyl transformations and the vierbein transformation law was given by equation (1.6.48). Nevertheless, Weyl transformations were presented in the gauge group of conformal gravity, where the vierbein transformation law was given by equation (1.6.54). So, the formulation we have constructed is a 'conformal supergravity' - that is, a supersymmetric generalization of conformal gravity. The transformation law (5.1.Q with Am and 'A arbitrary, represents a supersymmetric generalization of equation (1.6.54), but not of equation (1.6.48). In order to obtain a generalization of Einstein gravity, one must exclude from the superfield parameters 2"' and ;I"their independent components a(x), iz(x) and ~ ' ( x(in ) a sense, Q(x) and $(x) are supersymmetric partners of ~(x)).In other words, it is necessary to constrain the supergroup parameters or, equivalently, choose a subgroup in the supergroup of holomorphic coordinate transformations on @:I2. Copyright © 1998 IOP Publishing Ltd
396
Ideas and Methods of Supersymmetry and Supergrucity
5.1.3. Einstein supergravity The supergroup of holomorphic coordinate transformations on @:I2 has a very natural subgroup - the supergroup of unimodular holomorphic transformations
Be(-)
= 1.
(5.1.31)
Infinitesimal holomorphic transformations of the form (5.1.4) are unimodular, if the parameters are restricted as follows (5.1.32)
It should be noted that any super Poincark transformation (5.5.13) satisfies this equation. Rather beautifully, representing A m and A@ as in equation (5.1. IS), equation (5.1.32) can be uniquely resolved by expressing the parameters U: (note, U; = CT + iS2) and q@in terms of other components. So, the supergroup of unimodular holomorphic transformations on C:l2 may be taken as a candidate for the role of gauge group in Einstein supergravity. Now, following V. Ogievetsky and E. Sokatchev, Einstein supergravity is postulated to be a gauge theory with the gravitational superjield &“(x, 8,8, being the dynamical object and the supergroup of unimodular holomorphic transformations being the gauge group. The transformation law for &“ is given by equation (5.1.8), where the superfields 2” and iz obey, in contrast to conformal supergravity, the constraint (5.1.32). At this point, in principle, the descriptive part is over; one has all that is needed to proceed to extracting the consequences. However, it will be more convenient for us to employ a different, but equivalent, formulation of Einstein supergravity. The point is that usually working with fields under constraints is a more complicated technical problem then working with unconstrained fields. For example, it is a rather difficult task to define the path integral on the space of transverse vector fields A k ( x ) , P”; = 0. Fortunately, very often, it is possible to reformulate a theory of constrained fields as a theory of unconstrained fields with an additional gauge invariance. For example, electrodynamics may be treated as the theory S[A’] = $jd4xAL”UAk of a transverse vector field AA(x) without gauge invariance or as the theory S [ A ] = S[AL] of an unconstrained vector field A , = A ; + A!!,,where A!!,is the longitudinal component of A,, armA$= 0, but with the gauge invariance A, + A, d,t. The auxiliary field Aa(x) is a compensating field for the gauge freedom.
+
Copyright © 1998 IOP Publishing Ltd
397
Superspace Geometry of Supergravity
5.1.4. Einstein supergravity (secondformulation) Our wish is to remove the constraint (5.1.32) and to handle arbitrary holomorphic transformations of C:12 at the cost of the appearance of an auxiliary compensating superfield. Following W. Siege1 and S. J. Gates, let us consider, along with #", one more superfield q ( y , 6) holomorphic on and possessing the transformation law
[ (
CP'(Y',@) = Ber -))]1'3q(y,
6)
(5.1.33)
under the supergroup of holomorphic coordinate transformations. Infinitesimally, this transformation law is written
or W y , 6) = PYY, 6) - V(Y,6) = ( A m a m + A%)cp
1
+ j(am2" - ad")cp. (5.1.34b)
@:I2
In accordance with equation (5.1.33), one can find a coordinate system on in which d Y , 6) = 1.
(5.1.35)
Evidently, all holomorphic coordinate transformations preserving this gauge choice are unimodular. As a result, we recover the gauge group of Einstein supergravity. It is the superfield q ( y ,6) which compensates for the additional gauge freedom. q ( y , 6) is called the 'chiral compensator'. Now, one can take the following point of view. Einstein supergravity is a gauge theory of two dynamical objects-the gravitational superfield Xm(x,6,8)and the chiral compensator q ( y , 6 ) transforming by the law (5.1 -8)and (S.1.346), respectively, under the supergroup of holomorphic coordinate transformations-the supergravity gauge group. Of course, for physical applications it is necessary to restrict q ( y ,6) from @' to the real resulting in the complex superfield superspace W4I4(#) embedded in q ( x , 40')= q ( x m+ i#", Om). The chiral compensator has a simple geometrical interpretation. One can = @:lo x Ci in complete analogy with develop integration theory over @:I2 the integration theory over RP14 discussed in Section 1.10(the only distinction between @:ol and R4I0is that variables used for parametrizing C:lo and R4I0 have imaginary souls in the former case and have no souls in the latter case). Similarly to the rule for change of variables (1.1 1.22), one can prove that the measure d4y d26 on C;'' transforms under the change of variables (5.1.3) as Copyright © 1998 IOP Publishing Ltd
398
Ideas und Methods of Supersymmetry and Supergravity
follows
:;)
:;:(
d4y' d 2 0 = Ber - d4y d28. Then, the transformation law (5.1.33) leads to d4y' d28'(cp'(y',
= d4y d28(cp(y,
(5.1.36)
Therefore, the chiral compensator defines an invariant measure on @:I2. The field cp3 is the superspace analogue of the density field e - ' , e = det (eam),used in general relativity. 5.1.5. Einstein supergravity multiplet In subsection 5.1.2, the conformal supergravity multiplet was obtained. It included the following fields: eam(x),Y m a ( x ) ,P ' ( x ) and A"'"(x).Now we proceed by finding a multiplet of fields corresponding to Einstein supergravity. Clearly, it should differ from the conformal supergravity multiplet, since Einstein supergravity is described by the gravitational superfield and the chiral compensator while all the dynamical content of conformal supergravity is encoded in iPmonly. A possible gauge choice in Einstein supergravity is q ( y , 8) = 1. Instead of choosing this gauge, it is more convenient for us to impose the Wess-Zumino gauge (5.1.17) in which the gravitational superfield has the simplest form (in the gauge cp = 1, 2"'will contain 02- and B2-terms as well). Every holomorphic transformation (5.1.4), preserving the Wess-Zumino gauge, is given by parameters (5.1.25). Let us expand the chiral compensator in a power series in 6:
+
+
(p3(x,8) = e - ' ( x ) { F ( x ) Baxa(x) 02B(x)).
(5.1.37)
Here the factor e - ' ( x ) is introduced to make F(x) and B(x) complex scalar fields and xa(x)an undotted spinor field with respect to the general coordinate and local Lorentz transformations, (5.1.38) in accordance with equation (5.1.34b). Making use once more of the transformation law (5.1.34b3, one finds how the Weyl and local chiral transformations act on the fields: 6,
+ *F
= (30 - iR)F
(5.1.39)
Copyright © 1998 IOP Publishing Ltd
399
Superspace Geometry of Supergravity As for the local S-supersymmetry transformations, they are of the form
(5.1.40) As may be seen, the Weyl and local chiral transformations can be used to gauge away F . The most useful gauge choice is F = 1. After this, the local S-supersymmetry transformations can be used to gauge away xu. A useful gauge fixing proves to be 2%= - 2i(oa),&P'. Then, one results, with the gauge fixed chiral compensator $(x,
e) = e-'@){ 1 - 2i60aP(x)+ 02B(x)}.
(5.1.41)
Finally, it remains to analyse the Q-supersymmetry transformations. Setting in expression (5.1.25) all the parameters, except E, and C', to vanish, one obtains a transformation which breaks the gauge (5.1.41). To preserve our gauge choice, any Q-supersymmetry transformation must be accompanied by some €-dependent local chiral + S-supersymmetry transformation. Namely, instead of parameters (5.1.25), one has to consider the set of parameters
Am(x,e) = 2ie&eam
- 282FFm
(5.1.42)
a(€) = -€o,T" - Y'"a,?
1 + -(oaTJa(EabP - Y'"ab?). 3 It should be pointed that we have o ( ~ = ) 0 due to the adopted gauge fixing for xu. Let us denotz the variation of any field under parameters (5.1.42) symbolically as 6,. Clearly, this can be represented in the form 6, = 6, I
+
+ a,(€).
The variations d,, dn and 6, of &"-component fields are given by equations (5.1.26,29,30). The variation 6,B can easily be found using the transformation law (5.1.34b). Then, making use of equations (5.1.39,40) gives &B = i(Eo,P - Y"o,Z)B
+ 2 V , ( p ) + 2Va(Tbd,oa?).
We do not present here expressions for &eam,&Ymland Copyright © 1998 IOP Publishing Ltd
(5.1.43)
&A'", Instead, we
400
Ideas and Methods of Supersymmetry and Supergravity
consider the set of fields with lower curved-space indices
ema
Y,
= g,,Yn,
A,
= g,,A"
and write down the transformation laws of these fields under the transformation (5.1.42) supplemented by the €-dependent Lorentz rotation
The corresponding field variations will be denoted by
6 = & + bK(E). The gravitational field and the gravitino transform by the rule dema = iY,aaC - icoaTm
2i 1 (5.1.45) dy,, = - 26,€, - 2i€,A, - - ema(fJa6bE),Ab - - ema(aa?)$ 3 3 where we have introduced the fields B=B
+ !Pda"bqb + -1Y- a-y a 2 2
1 A, = A, - - ( Y a o a ~ , Y,aaT") 4
+
6,
1
c-dl
Yboaqb- - E ~ a~Y ~ .~ Y 4 (5.1.46)
denote covariant derivatives with torsion: = VmE,
Qmc,
+ 21
-@mab(cabc),
(5.1.47)
The transformation laws of the fields just introduced are 6B = - 2 c 8 b q a b
- iYaaaCB+ 2iFYmA,
1 dA, = - - e:(EabVa, 2 - i€a"Y,Aa
Copyright © 1998 IOP Publishing Ltd
+
i&abcdcdpd)
1
- - €Y,B
4
+ -2i €o,TaA, + -41 ema&abc&bb~cAd + C.C.
(5.1.48)
Superspace Geometry of Supergravity
where
Tabkdenotes
401
the gravitino field strength (5.1.49)
We summarize the results. The Einstein supergravity multiplet is given by the set of fields {ema,Ym,,Tmh, A,,,, B, B}. The component gauge (quasi) group includes the space-time general coordinate, local Lorentz and local supersymmetry transformation. The local supersymmetry transformation laws of the supergravity fields are given by equations (5.1.45,48). 5.1.6. Flat superspace (’nal
definition) and conformally flat superspace In conclusion, we would like to correct some notation and definitions introduced in this section. The point is that conformal supergravity and Einstein supergravity, being gauge theories with the same gauge group, have different dynamical content. Conformal supergravity is described in terms of the gravitational superfield only, while Einstein supergravity needs one more dynamical superfield - the chiral compensator. The gravitational superfield defines the embedding of physical real superspace in while the chiral compensator determines the invariant measure dp = d4yd28q3 on the physical surface. As a consequence, the more accurate symbol for denoting physical superspace is iW4I4(X, q), and not R4I4(X). The symbol lR4l4(%) is reserved for denoting physical superspace in conformal supergravity. In fact, conformal supergravity may also be treated as a theory of the gravitational superfield and the chiral compensator, but with the additional gauge invariance (5.1.50)
where CJ is an arbitrary superfield on @:I2. Equation (5.1.50) is a superspace analogue of space-time Weyl transformations (1.6.32). This is why the above transformations are called ‘super Weyl transformations’. Furthermore, the definition of flat superspace given in subsection 5.1.1 needs correction. In the literal sense of the word, we have defined a conformally flat superspace. To make this precise, let us remind ourselves of the usual meaning of conformally flat space. A space-time is said to be conformally flat if there exists a coordinate system in which the metric is, up to a factor, the Minkowski one, gmn(x) = 9(x)Vmn*
(5.1.51)
All transformations from the conformal group (see Section 1,7), and only these, preserve the conformally flat form of metric (5.1.51). Copyright © 1998 IOP Publishing Ltd
402
Ideas and Methods of Supersymmetry und Supergravity
Now, we formulate the final definition of flat superspace. A superspace said to be a 'flat superspace' if there exists a holomorphic coordinate transformation (5.1.3) such that [w4I4(X, cp) is
x"(X',
e', 8 ) = 6 , m e o " ~ '
cpyX',
e') = 1.
(5.1.52)
The super Poincare transformations (5.1.13),and only these, leave invariant flat superfields (5.1.52). A superspace iw4l4(X,cp) is said to be a 'conformally flat superspace' if there exists a coordinate system in which X mhas the flat form (5.1.12).It is not difficult to find the family of holomorphic coordination transformations of E:'' preserving the flat gravitational superfield (5.1.12). Such transformations are generated by the superfield parameters
- n"(y, 6) = b"
+ KUbyb+ Ay" + f"y2 - 2y"(f, y)
+ 2ieo"c - 26a"a"bqyb
where A, a, b", f" and Kab are constant real c-number parameters and ,E, q, are constant complex a-number parameters. Transformations generated by the above parameters form a supergroup known as the 'superconformal group', which was studied in Chapter 2.
5.2. Superspace differential geometry The supergravity gauge group has been realized in real superspace R4I4in a rather unusual way, as the set of nonlinear transformations of special form (5.1.6). For mathematical beauty and to observe direct contact with standard general relativity, it would be desirable to have a supergravity formulation in which the gauge group is extended to include the full supergroup of general coordinate transformations on iw4l4: Xm
-,Xfm
= pyx,
Copyright © 1998 IOP Publishing Ltd
e, B)
Superspace Geometry of Supergravity
403
or, in infinitesimal form,
xm+ 8’
XI”
= X” - Km(x,8,Q)
+ 6”’ = -+
0’ - K’(x, 8,8)
~ ;=i gfi - R,(X,
(5.2.2)
e, 8)
where K” is an arbitrary real vector superfield and K’ is an arbitrary undotted spinor superfield. In principle, such a formulation may be extracted from that constructed above by means of the introduction of superfluous compensating superfields. We take another course. For the time being we shall forget the results of the previous section and proceed to the development of a geometrical formalism in real superspace trying to keep, as far as possible, the analogy with ordinary differential geometry. 5.2.1. Superjeld representations of the general coordinate transformation supergroup We are going to describe some superfield representations of the supergroup of general coordinate transformations on lR4I4. As in previous chapters, we adopt the compact notation zM = (xm,P, 8,) which makes it possible to rewrite (5.2.1) as
(5.2.3)
Starting from this point, always when dealing with the general coordinate transformation supergroup, superspace indices will be denoted by letters from the middle of the Latin and Greek alphabets (letters from the beginning of each alphabet will be reserved for tangent-space indices). The simplest representations of the general coordinate transformation supergroup are scalar and scalar densities. A ‘scalar superfield’ $(z) is defined by the transformation law
4 ‘ ( 4 = 4(4.
(5.2.4)
A ‘scalar density superfield ~ & ~ , ( zof) weight w’ is characterized by the transformation law
(5 -2.5) Given a scalar density of weight one, $(lj(z), the integral over R4I4of does not depend on the choice of coordinate system on lR4I4,
I$(~)(Z)
(5.2.6)
as a consequence of equation (1.1 1 -32). Copyright © 1998 IOP Publishing Ltd
404
Ideas and Methods of Supersymmetry and Supergruvity
An invariant first-order differential operator on R4I4
v = VM(z)d,
=
vmam+ vh3, + v,ab
(5.2.7)
where V M ( z )are smooth superfunctions, is called a 'supervector field'. The supervector transformation law follows from the invariance requirement v'M(z')a;,
= VM(z)d,
(5.2.8a)
vrM(Z') =
T/N(z)aNz'M
(5.2.8b)
and it reads The value of the supervector field V ( p ) = VM8,1, at a given point p of superspace is said to be a 'supervector' at this point. In accordance with the results of subsection 2.1.4,the set of all supervector fields on R4I4, denoted by SW(4,4), forms an infinite-dimensional Berezin superalgebra with respect to the grading (2.1.46,47)and the super Lie bracket (2.1.51). Based on equation (2.1.50),we introduce the operation of complex conjugation in SW(4,4) as follows (see also equation (2.4.28)):
v-, p =
pMaM p" j
i
(vy)*= ( - l)Hv)Hy')py* = (T/")*
VP
= ( - l)E(")&yvJ)*
p,
= (-
(5.2.9)
l)E(")&ciJ(v")*.
where "(z) is a pure (bosonic or fermionic) superfield. One can explicitly check that BM transforms according to the supervector law (5.2.8). A real c-type supervector field K = h' is characterized by components of the structure K M = r f M = ( K m ,KP, R,) &(KM) =E ~ . (5.2.10) The set of complex (real) c-type supervector fields in SW(4,4), denoted by OSW(4,4)('SWR(4,4)),forms a complex (real) super Lie algebra with respect to the ordinary Lie bracket. A 'supercovector field' (or super 1-form) U&) is defined to transform under the general coordinate transformation supergroup in the same way as a partial derivative of scalar a field, i.e.
(5.2.11)
U p ) = (ahzN)UN(Z). Given a supervector field V M ( z ) the , object V M U , = V"U,
+vw,+V
p
but not U M V M ,is a scalar superfield. So, when working with supertensors, the relative order of factors turns out to be essential. Higher-rank supertensor fields may be defined similarly. For example, a second-rank supertensor field \ y M N ( zis) characterized by the transformation Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
405
law
YL ~ ( z f=) ( a h z P y p R a R z ' N
(5.2.12)
Due to this transformation law, the superfield (- l)""\rMM(z)
is a scalar if ( Y M N is )a C-type supermatrix.
5.2.2. The general coordinate transformation supergroup in exponentialform As in Section 1 . 1 1, here we restrict ourselves to the consideration ofcoordinate transformations on R4I4 expressible in the exponential form ZM
-,z'M
= e-KZM
K =p
a M= R
(5.2.13)
with K being a real C-type supervector field. The set of all such transformations forms a connected subgroup of the general coordinate transformation supergroup. Recall that the Berezinian of transformation (5.2.13)is given by equation (1.1 1.17). The exponential form for superspace reparametrizations is useful, since it makes it possible to write down superfield transformation laws in a similar manner to ordinary Yang-Mills transformations. Substituting equations (5.2.13)in the scalar transformation law (5.2.4) gives e-K4'(z) = 4(z) therefore we have
&(z) = eK4(z).
(5.2.14)
In the supervector case, making use of equation (1.11.19)leads to
v'M(zf)a;,= (e-'vlM(z))e-'aMeK
= e-K(VfM(z)aM)eK
therefore the transformation law (5.2.8)can be rewritten as V' = V f M ( z ) a M = eKl/e-K.
(5.2.15)
It is instructive also to consider the transformation law (5.2.5)in the special case w = 1. From equation (1.1 1.17), we have c
e-V;,,(z) = (1 * e - K)-l&l)(z) =
4-
*eK~)+(l)(z)
where we have used equation (1.1 1.14).Recalling equation (1.1 1.13), we have c
4{1)(4= (1 *eE)eK4c1,(z)= (4(l,eK).
(5.2.16) It is obvious now that the integral of $(,,(z) over R4I4is invariant with respect to general coordinate transformations. Copyright © 1998 IOP Publishing Ltd
406
Ideas and Methods
of^ Supersymmetry and Supergravity
5.2.3. Tangent and cotangent supervector spaces The tangent space T,(lW4l4)at a point p E R4I4is defined as the supervector space of left first-order differential operators (supervectors) at this point, (5.2.17)
T,(IW4l4)= { V = V M a M l pV, M € A m } .
The dimension of T,([W4I4)is equal to (4,4).Clearly, the set of supervectors { d M l p } forms a pure basis of T,(R4I4) which will be called the ‘holonomic basis’. In accordance with equation (5.2.9), this basis is characterized by the following conjugation properties -
a,
-
aP = &..,ac PI = -a,
-
= a,
P=&PVay
=
-aP
(5.2.18)
Every pure basis of TP([W4I4) satisfying the same conjugation properties will be called a ‘standard basis’. These are real c-type supervectors from T,(R4I4) which can be defined as directional derivatives at point p E R4I4. The ‘cotangent space’ T,*(R4I4)at a point PE R4I4is defined to be the right dual of T,([W4I4)(see subsection 1.9.5).Its element will be called super 1-forms at point p . Every super 1-form W E T,*(iW4I4)is a mapping W : T,([W4I4)-+
Am
possessing the property (@,VI
+ @ 2 V 2 ) W = @l(V& + L % W 2 b
for arbitrary Vl, V , E T,([W4I4)and al, u2 €Aa. The value of a super 1-form o on a supervector V reads
( V ) o = (I/Md,)w
= VM(B,)o.
(5.2.19)
The dimension of T,*(R4I4)is (4,4). Given a scalar superfield Y(z) on iW4l4, one can construct a super 1-form dY 1 E T,*(rW4l4)defined by
,
(V)dYI, = ( V Y ) / ,
VV€T,(R4I4)
(5.2.20)
and called the differential of Y at p . Choosing here Y = zN gives (aN)dzM= d N ~ . Therefore, the set {dzMI,} forms a pure basis of T,*(iw4l4),which is the right dual of the basis { aMI on T,(iW4I4).For every superfield on R4I4, one has dYlp = dzMIp.aMYl/,.
(5.2.21)
Then, introducing the tangent bundle T(R4I4) = u p E R 4 1 4 T,(R4I4) and the R4l4 T,*(R4I4), every supervector field cotangent bundle T*(iW4I4)= V = VM(z)aMdetermines section V : R4I44 T(R4I4)and every supercovector field w = dzMWM(z)determines section W : [W4I4 4 T*(R4I4).
ups
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
407
5.2.4. Supervierbein Supervector fields on R4I4
E, = (Ea,E,, EAM=
(;I
F)= EAM(z)aM EaP Eab) EaP E,,
Eam
Eap
(5.2.22)
pfi
are said to form a 'supervierbein', if for any point p€[W4I4 the set (EAlp) constitutes a standard basis in T,(R4l4).Equivalently, the C-type supermatrix EAM@) must be non-singular at each point of the superspace, hence E
Ber (EAM)# 0.
(5.2.23)
Note that the superfield E(z) is real, E = E. Since EAM(z)is non-singular, it possesses the inverse supermatrix EMA@) ('inverse supervierbein') defined by EMAEAN = 6MN
EAMEMB = 6AB.
(5.2.24)
A general coordinate transformation on R4I4acts on EY and E& according to the laws EL M ( ~ ' )= EAN(z)dNz'M
(5.2.25a)
ELA(.?) = (&zN)ENA(z).
(5.2.25b)
It is clear that EA = dzMEMA(z) are supercovector fields and, due to the non-singularity of EMA,the set (EAI,} forms a basis in T,*(R4I4).Equation ( 5 . 2 . 2 5 ~leads ) to E'(z') = Ber
(3
- E(z)
therefore E-'(z) is a scalar density superfield of weight one. In the case of the coordinate transformations (5.2.13),equation (5.2.25a)is equivalent to EL = EL M(Z)aM = eKEAe-K.
(5.2.26)
Using equation (5.2.16),the density superfield E - '(z) transforms according to the law (E-')' = E-'$.
c
(5.2.27)
Remark. We take the convention that letters from the beginning of the Latin or Greek alphabet denote tangent-space indices. Any supervector field V = VM(z)aMcan be decomposed with respect to the supervierbein, V = VA(z)E, Copyright © 1998 IOP Publishing Ltd
V A = VMEMA.
408
Idxis und Methods of Supersymmetry and Supergruvity
can be decomposed with respect Next, any supercovector field o = dzMoM(z) to the inverse supervierbein, w = EAW,(Z)
0, =
E,Mo,.
It is clear that VA(z) and oA(z)are scalar superfields with respect to general coordinate transformations. More generally, using the supervierbein and its inverse, all superspace indices of a supertensor field can be converted into tangent-space indices. For example, in the case of the second-rank supertensor field YMN(z), the conversion reads as follows:
YMN -+ T A B = E, MYMNENB. Supertensor fields with tangent-space indices are very useful in practice, since they change as scalar superfields under general coordinate transformations. In what follows, we will work only with supertensor fields carrying tangent-space indices. 5.2.5. Superlocal Lorentz group
To breathe life into our theory, it is necessary to choose a proper structure (super)group G of the superspace. By its very origion, the structure group does not act on points of the superspace. It merely determines, for each point P E [w4I4, physically equivalent bases (or frames) in Tp(R4I4).Two standard of Tp([W4l4) are said to be equivalent if and only if they bases {a,} and are connected by a structure group transformation:
{&a}
&a = AAB(g)&B
gE G.
Here A is some linear representation of G. Next, two supervierbeins E , and E:4are equivalent if they define equivalent frames at each superspace point:
Ea I
p
= AAB(g(P))ESI p
vP E R4'4
or, expanding EA and EL with respect to the holonomic basis, M ( z ) = AAB(Q(z))EBM(z)*
(5.2.28)
The corresponding inverse supervierbeins are connected by the rule
EL
= E,B(z)ABA((g(Z))-'1.
(5.2.29)
The above expressions involve a supersmooth mapping f : R4I4-,G. The set of all such mappings forms an infinite-dimensional supergroup, with respect to the multiplication (fif2 )(z) =fi(z)fi(z), which will be called the 'superlocal structure group'. Any superlocal structure group transformation moves the supervierbein into a physically equivalent one. Therefore, the symmetry group of a physical dynamical system should include the product of the general coordinate transformation supergroup and the superlocal structure group.
-
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
409
Remark. Changing the supervierbein to an equivalent one induces superlocal structure group transformation of supertensor fields. In particular, the supervector transformation law is VA(Z) VIA(Z)= vB(Z)ABA((g(Z))') (5.2.30) --+
and the supercovector transformation law is OA(4
+
=~AB(S(m430).
(5.2.31)
Now, how do we choose the superspace structure group? In the space-time case, there was the physical principle requiring us to identify the structure group with the Lorentz group. Namely, two vierbeins are equivalent if they determine the same space-time metric. Unfortunately, there is no physically natural superspace generalization of the concept of metric. The most reasonable assumption one can make is to identify the zeroth-order terms in the O,&expansion of the supervierbein's component E," with the space-time vierbein, E,"[ = e,"@). This requires the Lorentz group to be contained in a possible superspace structure group. We simply postulate the Lorentz group to be the superspace structure group and choose the representation (i,3)0 (t,0) 0 (0,i) to constitute the equivalence relation between tangent space frames. Then, two physically equivalent supervierbeins are connected by a 'superlocal Lorentz transformation' of the form (5.2.29),where 0
AAB= (eK)AB K,'
=
KUb(z)= (oab),pKaB(z) - (Fb)&b(z)
(5.2.32)
with K,,&) = K ~ , ( z )being arbitrary. Note that Ber (AA? = 1. Therefore, the superlocal Lorentz transformations do not change the Berezinian of the supervierbein, E = E.
(5.2.33)
As is seen, the superlocal Lorentz group acts reducibly on the supervierbein. Each of the supervector fields E,, E , and 6 transforms independently. Then, any supervector field or any supertensor field can be invariantly decomposed into Lorentz irreducible components. On these grounds, it is possible to speak of Lorentz vector superfields, Lorentz spinor superfields (dotted and undotted) and, more generally, one can define arbitrary Lorentz tensor superfields. A tensor superfield of Lorentz type (n/2,m/2) is defined to transform under the superlocal Lorentz group according to the law V',] . . . ,"&,
I..
B,(z)= etKoh(Z)M.b VI, , . . , , B ] . . .d,(Z).
Copyright © 1998 IOP Publishing Ltd
(5.2.34)
410
Ideas and Methods of Supersymmetry and Supergravity
Here the Lorentz generators Mab act, as usual, on the external superfield indices. When the superspace general coordinate transformations are also taken into consideration, the above transformation law must be substituted by = e X L ...a &
Val...U , d l ... e&)
. I .
Ci&)
(5.2.35)
where
x
1 + -KabMab = KMaM+ KaPn/i,p + @ f l h g . 2
= KMaM
(5.2.36)
Remark. As in Chapter 2, we will assume every tensor superfield with an even (odd) number of spinor indices to be bosonic (fermionic). 5.2.6. Superconnection and covariant derivatives Let us introduce a 'Lorentz superconnection' - that is, a real C-type super 1-form taking its values in the Lorentz algebra,
0 = dZMQM= EARA 1
RA = - n A b C ( z ) M b c 2 QA
bc
= (a be a
Q
)
=R
A ~ ~ ~+/ f~i B ~ ", M &
(5.2.37)
bc n k b c ) CO
and transforming according to the law
R'
= dzMRh =
-dg.g-'
+ gi2g-l (5.2.38)
under the superlocal Lorentz group. Then, the operators 9 A
= (ga, ga,B') E
EA
+
RA
(5.2.39)
change covariantly with respect to the superspace general coordinate and superlocal Lorentz transformations,
9;= exgAe-X. In the infinitesimal case we have 9;= B
A
+
6 g A
6gA=
(5.2.40)
[ c f ,
a A ]
and, using equation (5.2.39), this transformation law is equivalent to the relations 6EAM= xEAM - EAKM
(5.2.4l a )
6RAbc= X C ~ AEAKbc ~'-
(5.2.41b)
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
41 1
which represent the supervierbein and superconnection transformation laws, respectively. Remark. Equation (5.2.41~)shows that EAM is a gauge field for the general coordinate transformation supergroup. Similarly, equation (5.2.41b) means that RAbeis a gauge feld for the superlocal Lorentz group. Let V x l , ,a,d,, &,,,(z)be a tensor superfield. In accordance with equations (5.2.35,36,40),the superfield BBVor, , , , z n k ,, , . J z ) transforms covariantly, ,
,,
(9Bvz~
... a,dl ..
.. ,%.d1
= eX9BVal
Here the Lorentz generators involved in
Yt act on all the indices of
9 B V z l , , , a ~ , ~,,,. l , Therefore, ,,d BA moves each tensor superfield into a tensor
one. On these grounds, the operators g Aare said to be ‘covariant derivatives’. Covariant derivatives act on super(co)vectors according to the rule
where =
(7
0
-RAPt
)=
(5.2.43)
-RAC,.
(Anti)commuting the covariant derivatives gives [BA,
gB] = TABCgc+ RA,
1 R A B= -RABCdMcd = RAi“M,* 2 T,,C
= -(
- l)EapeTBAC
+ RAB’ My6 46
R A , = -(
(5.2.44)
- l)EAEBRBA.
Here TABC is the ‘supertorsion’, and RABcdis the ‘supercurvature’. As a result of equations (5.2.40),TABC and RABCd are supertensor fields. The supertorsion and supercurvatures are expressed in terms of the supervierbein and the superconnection as follows TABC= C A B c
+ RABC - (-
l)EaseRBAC
(5.2.45~)
RABCd = EARBed + R A “ W B ? - ( - l)CaEB(A ++ B ) - C A B L R L C d . (5.2.45b) Here
CABc
are the ‘anholonomy supercoefficients’:
Taking into account the block-diagonal structure of Copyright © 1998 IOP Publishing Ltd
RA,‘,
the expression
4 12
Ideas and Methods of Supersymmetry and Supergravity
(5.2.47)
It is seen that the anholonomy coefficients Lorentz tensor superfields.
C,;,
C,,, and
Cab3
are
5.2.7. Bianchi identities and the Dragon theorem The super-Jacobi identies (2.1.8) are satisfied for any set of operators, in particular, for the covariant derivatives: [3A,
[gB, gc}}
+ (-
1)ca("
+
") C9B,
+ ( - 1)edea + '8)E9C,
EgC,
9A))
= O*
CgA,
Using equation (5.2.44), the left-hand side can be rewritten symbolically as 1
SAKEBE
and each of explicitly as
and
YABcdf
must vanish. The condition
RABcD+ ( - lp('BRC)R BCA +
+ -2y A B C d f M d l
+ (-
l)Ed"
+
'B'RCABD
SABCE
= 0 reads
= AABCD(5.2.48)
where we have introduced the notation =
{ a A T B C-D TABETEcD) + ( - 1)EA("+")
{BBTcAD
- TBcETEAD> (5.2.49)
+ ( - l)sd&a + & B ) { -9cR A B
+ TcA"RDB)
=0
The relations (5.2.48,50)are said to be the 'Bianchi identities'. Given arbitrary supervierbein and superconnection, the corresponding supertorsion and supercurvature satisfy the Bianchi identities. One of the most important consequences of the Bianchi identities is the fact that the supercurvature can be expressed completely in terms of the supertorsion (this statement is known as the Dragon theorem). The point Copyright © 1998 IOP Publishing Ltd
41 3
Superspace Geometry of Supergravity
is that the supercurvature takes its values in the Lorentz algebra, RABed
(5.2.51) =(
- R OAB'i8)
0
and the components RAE;, RA,,' and R,,j$ are connected as follows R~~~~= ( O ~ ~ ) , ~ R , , Y' (dcd),$~,,YJ.
(5.2.52)
Now, let us choose A = a, E = p, C = c and D = d in equation (5.2.48). Then, by virtue of expressions (5.2.51), we have (5.2.53)
= A a p c d*
Similarly, one obtains Ra#cd = A d B c d
&#Cd
= A.&%d .
(5.2.54)
Then, we choose A = E, E = b, C = c and D = d. The Bianchi identities (5.2.48) give
+ Rcab" Since Rabcd =
-Rub&,
d
= Rabc -
= Aabed*
this equation can be resolved as follows Rctbcd
1 2
= -(Aabcd
- Aacdb + Aadbc)*
(5.2.55)
R&bcd can be found analogously. Finally, we choose A = a, E = b, C = y and D = 6 (or C = 3 and D = 8), obtaining
= Aab)6
Rab$
= Aabf$-
(5.2.56)
It only remains to apply equation (5.2.52). Therefore,we have proved that the supercurvature is completely expressible in terms of the supertorsion, and hence, in a sense, the supercurvature is a redundant object. This fact had no place in general relativity, where the curvature and the torsion were independent, and one could switch off the torsion leaving a non-vanishing curvature. In superspace, the supertorsion is the main object determining superspace geometry. 5.2.8. Integration by parts Theorem. Given a supervector field V A ( z ) under proper boundary conditions, the following identity
is fulfilled.
s
dsZE-'(-1p9AVA
Copyright © 1998 IOP Publishing Ltd
=
s
dazE-'(-l)EBVATABB
(5.2.57)
414
ideas and Methods of Supersymmetry and Supergravity
Proof. Making use of equations (5.2.42,43)gives
1)"'9AVA
(-
= ( - l)E"EAVA - VARBAB.
(5.2.58)
Further, equation (5.2.45~) leads to (-
l ) E B T A B B= (-
l)BBCABB -
(5.2.59)
since ( - l)EBRABB = a,,' - R A j+ R A t = 0. Finally, recalling the definition of the anholonomy coefficients (5.2.46),one readily obtains
(-l)EBCABB
= -EAInE-'
+
-(l*EA).
(5.2.60)
The relations (5.2.58-60) show that ( - l)""E-'BAVA= ( E - ' V A ) k A
+ (-
1)EBE-'VATABB.
Here the first term in the r.h.s. is a total derivative. This completes the proof.
Remark. In the cases of vector or undotted spinor superfields equation (5.2.57) reads
Remark. The relation (5.2.57) takes the most simple form in the case when
( - l)EBTABB vanishes, (-
l)EBTABB
s
= 0 * d8zE-'9AVA = 0.
(5.2.62)
5.2.9. Flat superspace geometry The approach developed above enables us to introduce the notion of 'superspace geometry' (or 'supergeometry') in literal analogy with the notion of space-time geometry. A supergeometry is determined by the set of superfields (EAM(z),RAbC(z)},where EAM is a supervierbein and RAbcis a Lorentz superconnection. The E A M and RAbC will be called the 'supergeometry potentials'. Two supergeometries are said to be equivalent if the corresponding potentials are connected by some superspace general coordinate and superlocal Lorentz transformation. A 'flat superspace geometry' may be described as foilows. A supergeometry is said to be flat if these exists a gauge choice with respect to the general coordinate transformation supergroup and the superlocal Lorentz group such that the covariant derivatives g A coincide with the flat covariant Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
4 15
derivatives D,, 9, = D, =
nAbc= 0
(5.2.63)
d,M =
These derivatives satisfy the algebra where the only non-vanishing supertorsion components are
T a t = Tb& = -2i(o'),p.
(5.2.65)
It is not difficult to find all gauge transformations (5.2.41) preserving the gauge (5.2.63).First, choosing X = RMaMand demanding M A M= 0, one arrives at the equation
[D,,
RMaM]= 0.
(5.2.66)
After the redefinition
RMaM -bmBm+ i(PQp + ELQp)
(5.2.67)
where (Qp,Qfl)are the supersymmetry generators (2.4.37),the above equation leads to
D,b" = D,P
=0
=. b"'
= const
e" = const.
(5.2.68)
Thus, we recover the space-time translations and the supersymmetry transformations. Further, we consider X = P a M + +KabMabwith K,b # 0. In accordance with equation (5.2.41), the requirements ~38,~ = 0, 0; = RAk = 0 give two equations
(5.2.69) The general solution of the second equation is K M = zNdNAKABdEM + fZM
(5.2.70)
and we come to the standard Lorentz transformations of flat global superspace. After this, the superspace and tanget-space incides can be identified. Copyright © 1998 IOP Publishing Ltd
4 16
Ideas and Methods of Supersymmetry and Supergravity
5.3. Supergeometry with conformal supergravity constraints The elegant supergeometrical formalism developed in previous section, turns out to be too general to correspond literally to superfield supergravity. To describe a superspace geometry, one must specify the superpotentials EAM@) and OAbC(z), which are arbitrary superfields in the general case. On the other hand, to describe a curved supergravity superspace, it is enough to introduce far lesser superfields: Yfm(z),cp(x + i&', 6) and y5(x - i s , 8). However it is not time to despair. We may hope that superfield supergravity corresponds to a subspace in the space of all supergeometries, where supervierbeins and superconnections satisfy some covariant constraints.:Since the only covariant objects constructed from EAM and OAbC are the supertorsion and supercurvature, any invariant subspace in the space of supergeometries (transforming into itself under the action of the general coordinate transformation supergroup and the superlocal Lorentz group) is selected out by imposing some constraints on TABCand RABCd. In principle, the situation is the same as in general relativity. Gravity is a gauge theory of vierbeins eam(x).From the other side, in order to specify a geometry it is necessary to accompany the vierbein by a Lorentz connection wabC(x).It is the torsion-free condition Fa;= 0, which, expressing w,bC uniquely through eam,determines geometries corresponding to gravitation. If we try to keep a direct analogy with general relativity, it is worth imposing the supertorsion-free condition TABC= 0.
(5.3.1)
This determines, due to equation (5.2.45a), the superconnection in terms of the supervierbein. However, this constraint is unsuitable for two reasons. First, flat global superspace is characterized by the non-vanishing supertorsion (5.2.65). Hence, the constraint (5.3.1) does not admit flat superspace as a particular solution. Secondly, it has been shown in subsection 5.2.7 that supercurvature is expressed via supertorsion. Then, the constraint (5.3.1) leads to the requirement RA, =
0
and as a consequence, to a trivial supergeometry. We have reached the point where our space-time intuition is unable to help us. Now, in order to find reasonable restrictions on superspace geometry, some purely superspace arguments must be taken into account. 5.3.1. Conformal supergravity constraints In global supersymmetry chiral scalar superfields, defined by the equation D,# = 0, were of primary importance. Clearly, it would be very desirable to transfer this supersymmetry representation to the local case - that is, to Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravitv
417
introduce 'covariantly chiral scalar superfields' defined by
G&@ = 0.
(5.3.2)
But this equation proves to be consistent only under non-trivial restrictions on the supertorsion. Indeed, demanding equation (5.3.2) and using the covariant derivative algebra (5.2.44), one obtains 0 = {g?,G,j)@=
=
Tk,jc9C@
T&p'g?,@ + Tk$9.,@
and hence T k I = T,$ = 0 5 Tu$ = T,pj = 0.
(5.3.3)
Recalling that TZ; = C,$and TZ??= C,, (see equation (5.2.47)), we can rewrite equation (5.3.3) in the equivalent form
{Ea,Eg} = C,$";,.
(5.3.4)
We take the requirements (5.3.3) as constraints on superspace geometry. In global supersymmetry, spinor covariant derivatives D, and Ddrgenerated the full algebra of covariant derivatives D, = {D, = d,, D,, Da) since the vector covariant derivative was expressed as an anticommutator of spinor ones, {D,, Da} =
- 2i(oa),&D,=
- 2iDah.
It is very natural to conserve this basic property in the local case also, i.e. to consider only such supergeometries in which spinor derivatives g Eand 9,generate the algebra 9,= (go, 9,,@). This means that 9, can be Bk}: extracted from {gZ,
(23,)gk} = T a t 9 b
+ T,$gg + Tu&@ -I-Rub
det((iiU)auTahh) # 0.
Now, one can make the redefinition
9,3 BA= (Go,B,,@)
-
=
( -;-(ao)bs{9,,
@}, 9&, @
changing only the vector supervierbein 1
EaM -P SUM= - - ( d , ) " T , ~ E B M 4
and the vector superconnection R)
+
Oabc=
Copyright © 1998 IOP Publishing Ltd
-
i -(a,)"{
4
T,/RBbC+ Rudd}.
4 18
Ideas and Methods of Supersymmetry and Supergravity
The operators
BA are characterized by the transformation laws:
a'A -- e.x^a,e-X
x
1 2
= KMdM -b -KabM,b
therefore they define a set of covariant derivatives. It is important that, after redefinition, the anticommutator of dotted and undotted spinor derivatives takes the same form as in global supersymmetry. As a result, the flat supergeometry arises as a particular solution. Motivated by the above discussion, we postulate the constraints
{9,, a,}= -2i(oajab9,
Tu%= -2i(o"),, -2i9,,-
T a t = T,&b= 0 Radd = 0.
(5.3.5)
This means that the vector supervierbein E , and superconnection R, are expressed in terms of the spinor supervierbeins E, and and superconnections 0, and Os.In particular, we have
i E,& E (oa),&E, = - { E , , &} 2
{E,, E&} = C,:EB
1
- - R,&@fi
2
C,:
+ -i n,:EB 2
=
(5.3.6)
After imposing constraints (5.3.9, the only independent supergeometry variables are the spinor supervierbeins and superconnections. Similar to general relativity, it would be desirable to completely express the superconnection via the supervierbein. This requires additional constraints. Let us impose the following constraints Tap,= 0 * T,& = 0
(5.3.7)
T,bc(bbc)bi = 0 * T&bc(obc)p, =0
(5.3.8)
and and show that they are sufficient to express R, and a, via E , and by virtue of equation (5.2.47), the constraint Tap?= 0 leads to 1 2
%si. = - -(Cup,
+ c,,, - Cpy,).
&. First, (5.3.9)
Next, to resolve the constraint (5.3.8), we introduce the semi-covariant supervierbein (changing covariantly under the superspace general coordinate transformations and non-covariantly under the superlocal Lorentz Copyright © 1998 IOP Publishing Ltd
419
Superspace Geometry of Supergravity transformations)
E l } , E,, 6 (5.3.10) [EA, E B )
=
C,BCEC
constructed in terms of E , and Eh only. As opposed to l?,, the covariant vector supervierbein E, depends, due to equation (5.3.6), on the spinor superconnections:
(5.3.1 1)
where we have introduced the notation C,,pb,fi = (ab)p&gC)y’>C@*bc.
Now, demanding (5.3.8) and recalling (5.2.47), one obtains 1
= -(C,,pb.Bi.
4
As a result, we have expressed
Introducing the notation
a, via E,
T,,/3fi,y.)
+ C,,pP.p#). and
(5.3.1 2)
E,. (5.3.13)
= (ab)/31)(ac)y.jTorbc
the constraint (5.3.8)can be rewritten in the form
T,,,p,P.j
=
b
.
(5.3.14)
We can summarize all the constraints on a superspace geometry:
representation-preserving constraints
T4 . = T h i )= T,; = T
=0
(5.3.15~)
conventional constraints ( I ) T,~P
= T J = R,;~ = 0;
T
= -2i(ac),8
(5.3.156)
conventional constraints ( I I ) Tu$’ = T J = T,,p,p,Pj,, = T,,,p~,~,l = 0.
(5.3.154
Representation-preserving constraints make possible the existence of chiral scalar superfields; conventional constraints (I) determine the vector covariant Copyright © 1998 IOP Publishing Ltd
420
Idem and Methods of Supersjimmelry and Supergravity
derivative in terms of the spinor ones; conventional constraints (11) determine the spinor superconnections R, and a, in terms of the spinor supervierbeins E , and E,. Constraints (5.3.1 5a) and (5.3.1 5c) mean that
Ta,;= T,/lC = 0.
(5.3.16)
As will be shown later, constraints (5.3.15) correspond to conformal supergravity.
5.3.2. The Bianchi identities For a given set of covariant derivatives g A ,the Bianchi identities (5.2.48,50) are satisfied identically. The Bianchi identities become non-trivial when the covariant derivatives are restricted by constraints. In this case the Bianchi identities play the role of consistency conditions and may be used to determine non-vanishing components of supertorsion and supercurvature. Now, our goal is to investigate the consequences to which the Bianchi identities lead when the constraints (5.3.15) are chosen. We organize the analysis of the Bianchi identities according to their (mass) dimensions (recall that the spinor derivatives 9, have dimension 3, while the dimension of gais 1). Possible dimensions of superfield expressions in (5.2.48,50) run from to 3. We reproduce below only the identities needed for our analysis, ordering them by dimension. First of all, to simplify the analysis, it is useful to convert any vector index into a pair of spinor indices (dotted and undotted) in the standard fashion: Ta,,fi,i. = (ob)pfiTab,
Tct,Bfi,g = (Ob)pfiTab,j
T,i,pfi,y = (o")ai("b)B#Taby
(5.3.17)
and analogously for complex conjugate quantities (see also equation (5.3.13)) and supercurvature. It is also necessary to keep in mind relation (5.2.51) along with the identity RAB,,q,h6
E (°C)y&od)6dRABcd
= 2Ey$RAB@
+ 2EyhR.4B@*
(5*3.18)
All identities of dimension 3 and 1 are contained in (5.2.48). Taking into account constraints (5.3.15), they can be written as: Dim 112
Choose A
= N,
B = D, C = d, D = c. Then one obtains
(5.3.19)
and its complex conjugate. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
421
Dim I
Choose A = a, B = fi, C = y, D = 6. Then one obtains
+ Rpyd + Rpps = 0
(5.3-20)
and its complex conjugate. C = j , D = 8. Then one obtains Choose A = 2, B = /l,
Rzpjd = 2iL,fi,S + 2iTp,zy,B
(5.3.2 1)
and its complex conjugate. Choose A = a, B = 1,C = p, D = y. Then one obtains TasB;, + TD&B,Y= 0 as well as its complex conjugate.
(5.3.22)
Dim 312 Choose A = a, B = p, C = y in the second Bianchi identities (5.2.50). This gives 9,Rp7
+ 9 p R y m + 9yRap = 0.
(5.3.25)
Choose A = a, B = 8, C = j in (5.2.50). Then one obtains $B,Rg? = 2iR,,,~ + 2iRg,,? and its complex conjugate. Copyright © 1998 IOP Publishing Ltd
(5.3.26)
422
Idecis urd Merhods of Supersymmetry and Supergravity
Choose A
=
5,B
=
b, C = y, D = 6 in expression (5.2.48).Then one obtains
1 1 1.1. -Q+T7,/{fi,d- g;.T*,/j/j.6- - T*,pb, T;,;.~,B - T?,pb,'".T*,;.j.,6 2 (5.3.27) - 2iT?*,/j/Lj .
R,./,/I,;,<j
I
5
1 .
and its complex conjugate. Finally, we choose A = 4,B = b, C = Ij, D = d in expression (5.2.48). Such choice leads to the identity
(5.3.28) and its complex conjugate. Dim 2 Choose A = a, B = b, C = y,D = y in(5.2.48).Since Rub,? = 0, one obtains
5'aT;h;'- 9bT;,a;'-
Ta;T7,;' + T y u E T Eg T,bET~i z g'Taby = 0
(5.3.29)
and its complex conjugate. Choose A = a, B = b, C = j , D = 6 in (5.2.48).One obtains
-
g';:Tab6- gaT9bs + gbTjas + TabET;:E6 - TpaETEb6+ TfbETEas= 0 (5.3.30) and its complex conjugate. The Bianchi identities of dimension 3 and 3 will not be of help to us. To solve the Bianchi identities, we shall decompose supertorsion and supercurvature into their Lorentz irreducible components and then apply the equations (5.3.19-5.3.30). The simplest situation occurs with the tensor Tz,pb,s.j,of dimension $. In accordance with equation (5.3.19), it is antisymmetric in ct and p. In accordance with constraint (5.3.14), it is antisymmetric in /? and j . Hence, the result is -
T2,pp;?j= E ~ ~ E ~=>JTd,pb;,g T . , = ~kbep,?
(5.3.31)
where T, is a spinor superfield. Recalling that T,BY = TasY = 0, equation (5.3.31) leads to ( - l)""TzBB= T,. Copyright © 1998 IOP Publishing Ltd
(5.3.32)
Superspace Geometry of Supergravity
423
5.3.3. Solution to the dim = I Bianchi identities We begin with consideration of the curvature R,p.)a = R,apXya) and decompose it into irreducible pieces: R,p7s = f ' ( ~ , ~ & p a+ ~ ~ s & g ,+ . ) fZ(zp.)a, + &,(7f3a)p + &p(J3a)u where . f Z z p ; 6 and f 3 , p are completely symmetric tensors. Substituting this decomposition into equation (5.3.20), one finds thatf2,pvs = 0 and f3,p= 0. If we denote f 1 -2R, then our result is
R,p)a = -2R(~+pa
+
+ &,6&py) *
R u. / ).i . ~- ~ R ( E ~ E~ Bd d ~~ p , ; ) .
(5.3.33)
Next, let us analyse equations (5.3.21) and (5.3.23). By virtue of equation (5.3.31), we can rewrite equation (5.3.23) as 1 2~;;dR,p;a + 2~~sR,p.jj = -- E ~ B ( E , ~ E ~ E ~ ,SE~))T~ 4
+
+ & ? d ~ @ u T a + &u;,gpTa)+ 4i&ctsTp,y?,~ + 4iqa Tu,yy,s
(5.3.34)
T Z = T"T,.
R,pyd is symmetric in indices
and
EraR,p?S = iE,a(Tp,gy,S
d, therefore equation (5.3.34) leads to
+ Tp&) + i&pa(T,yy,S + T,yS;j).
Comparing this result with equation (5.3.21), it is not difficult to deduce that
R,p.;.S = 0 *Rib,* = 0. Then, equation (5.3.21) is re-solved by
(5.3.35)
TepB,.;.= E z p f s ' i
for some tensor fbi. Now, the relations (5.3.33)and (5.3.34)are consistent only under the conditions:
+
9@Tp 3pTu= 0
(5.3.36)
as well as
(5.3.37) It remains only to study equations (5.3.22)and (5.3.24).The first implies that
(5.3.38)
Copyright © 1998 IOP Publishing Ltd
424
Ideus und Methods of Supersymmetry and Supergravity
where $,p is a complex vector superfield. Let us substitute equations (5.3.31) and (5.3.38) into equation (513.24), resulting in
Supertorsion Tabcis antisymmetric in indices a and b, therefore Txe,,/iy.;. = - T/?/],,B,yp
For consistency of this equation with equation (5.3.39), one demands that (5.3.40) Defining a real vector superfield G, by the rule $a
+ $a
E
2Ga
(5.3.41)
we obtain
(5.3.42)
To summarize, we have resolved all the Bianchi identities of dimensions 3 and 1 in terms of tensors T,, G, R, and G,. In fact, we have completely determined the anticommutators of spinor covariant derivatives, since the relations (5.3.16), (5.3.33) and (5.3.35) mean {9,,9 p ] = -4RM,p
{a&,G j ) = 4RRib.
(5.3.43)
5.3.4. Solution to the dim = 4 Bianchi identities We proceed by resolving the identities (5.3.25-28). Taking into account expressions (5.3.33), equation (5.3.25) means simply that
9,R
= O*
GkR = 0.
(5.3.44)
Therefore, R is a chiral scalar superfield. Further, since Rprjj = 0, equation (5.3.26) means that 9&yaB Copyright © 1998 IOP Publishing Ltd
= 2q,,fi,as
+ 2iRfi,ay,aB
(5.3.454
Superspace Geometry of Supergravity
425
and (5.3.45b) Rh.B,BY + RB&i..B.,= 0. Recalling equation (5.3.33),we can rewrite the first relation (5.3.45),in the form 3Qil,R = iRg.,i.,BI.
(5.3-46)
The identity (5.3.28)may be rewritten, by virtue of expression (5.3.31)and (5.3.38), as
Since Re,ab,ysis symmetric in f and
8, this relation is easily resolved:
(5.3.47) Now, equation (5.3.46)gives (5.3.48) This formula relates the tensors G, and R . Finally, we must consider the identity (5.3.27).First of all, we decompose the supertorsion Tud,pj,y,arising in the commutator of vector derivatives, into its irreducible components: Tua,pB,y = EbBW(,Bu) + &aP(E.yf;
+ E p y f : ) + Eapf&$y
(5.3.49)
where W,py is a totally symmetric tensor and f i b y is symmetric in its dotted indices. Inserting this decomposition into identity (5.3.27) and bearing in ~ ssymmetric in y and 6, one can express f: and f&? in mind that R a , p ~ , is terms of T,, R and G,. After doing this, one obtains
+
R a , p ~ ,= ~ s - iedP(EpjX6 ~paX,)- 2 i ~ ~ b W p ~ 6
Evidently, equation (5.3.45b)is satisfied. Copyright © 1998 IOP Publishing Ltd
(5.3.50)
426
Ideas and Methods of Supersymmetry and Supergravity
5.3.5. Solution to the dim = 2 Bianchi identities
One can check that equation (5.3.29)is consistent under the requirement
(5.3.51) Next, equation (5.3.30) is satisfied if and only if the identity (5.3.52) holds. It may be shown that the other Bianchi identities, not considered above, are satisfied identically or play the role of relations determining the supercurvature Rob.
5.3.6. Algebra of covariant derivatives Now we are in a position to write down the algebra ofcovariant derivatives:
{a,, a,}= - 2igab
(9,)9 p ) = -4RM,p,
{Bi,a,} = 4RRbg
- 2ieapRyMgi.+ 2ie,PWijbR4 where X , is defined by equation (5.3.50). The supertorsions T,, R, Gadand Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
421
W,, satisfy the following Bianchi identities:
(5.3.54)
We did not give above the commutator of two-vector covariant derivatives. In accordance with equation (5.3.53), it can be calculated in accordance with the rule i 1 1 c g u i , 9,bl = -j C{%, Si}, 9,/d = 2 {[%, 9pa1, %I + - {% [Sa,9,aI). 2 5.3.7. Covuriuntly chirul tensor superfields The set of representation-preserving constraints (5.3.15~)was sufficient to ensure the existence of covariantly chiral scalar superfields. The full set of constraints (5.3.15) makes possible the existence of covariantly chiral tensor superfields n-ith undorted indices only. Indeed, the equation %@U
I...U"
=0
(5.3.55)
is consistent with the anticommutator {Si, Bg} = 4RMdg. As opposed to the flat-superspace case, in a curved superspace there are no covariantly chiral superfields with dotted spinor indices unless R # 0. Note that, given a tensor superfield U u l , , , u nthe , tensor superfield - 4R)U,, is covariantly chiral
(a2
,,,
B & ( P- 4R)UR1,,.@" = 0.
(5.3.56)
5.3.8. Generalized super Weyl transformations By construction, the algebra (5.3.53) is covariant under the general coordinate transformation supergroup and the superlocal Lorentz group acting by
B A+ 3; = exgAe-X
+ 21
%- = K M a M - K~'M,,.
Copyright © 1998 IOP Publishing Ltd
(5.3.5 7)
428
Ideas and Methods of’ Supersymmetry and Supergravity
In fact, the algebra admits a richer symmetry group. It is not difficult to check that constraints (5.3.15) are invariant under the transformations
(5.3.58) where L(z) is an arbitrary complex scalar superfield (non-vanishing everywhere). In particular, one finds
- 9; In (L4L2)
Tk =
(5.3.59) Let us introduce real superfields A(z) and ~ ( zaccording ) to the rule eA
LE-’
=
(5.3.60)
Then, by virtue of equation (5.3.58), the supervierbein transforms according to the law
(5.3.61) E,
--$
EA = eAEa+ ... .
Therefore, the parameter A induces superlocal scale transformations (or superspace Weyl transformations) and the parameter K induces superlocal chiral or y,-transformations. We will refer to the transformations (5.3.58)as the ‘generalized super Weyl transformations’. The set of all such transformations forms a supergroup which represents a superspace extension of the Weyl group appearing in general relativity (see Section 1.6).
5.4. Prepotentials After imposing constraints (5.3.15), the supergeometry potentials E A M and QAbC are expressed in terms of the spinor vierbein superfields EmMand EiM.The latter are constrained superfields, in accordance with equation (5.3.154. In practice, operating with unconstrained objects is usually more convenient than with objects subject to constraints. On these grounds, it is worth resolving constraints (5.3.15 4 in terms of unconstrained superfields (‘supergeometry prepotentials’). The term ‘prepotentials’ is not incidental. Indeed, following the terminology of gauge theories, TABCand RABcdplay Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
429
the role of field strengths, being gauge covariant superfields, while the gauge superfields E A M and izAbcplay the role of potentials. So, it is perfectly reasonable to consider objects determining potentials as prepotentials. The solution to the constraints described below was first given by W. Siegel. 5.4.1. Solution to constraints (5.3.15a)
Equation (5.3.4) says that the supervector fields E, form a closed algebra. What usefulness does this fact contain for us? To answer this question, it is helpful to remember Frobenius' theorem from differential geometry (see, for example, Warner's book*). It may be formulated as follows. Let 4 be a p-dimensional manifold with local coordinates x", and {Vi}, i = 1,. . .,4, be a set of vector fields on 4, Vi = Vim(x)(a/2xm),linearly independent at each point of the manifold and forming a closed algebra: [v ,V j ] = CijyX)v,.
Then, there exists a set of independent scalar functions f '(x), . . . ,f "-4(x) such that the one-forms d f a vanish on vector fields Vi, df"(Vi) = 0 Equivalently, if one considers submanifolds (surfaces) in A, defined by the system of equations f " ( x ) = const c( = 1,. .. ,p - q then V iare tangent to each surface:
a
vi = A i ( y )ay' -
det ( A i ) # 0
where y' are local coordinates on the surface. The variables 2" = (y', . . .,y4, f . . . , f P - 4 )define a local coordinate system on the manifold, and we have
',
for some vector field U . Finally, we obtain
i = 1, ...,q. Returning to superspace, we deduce from equation (5.3.4)that E , = A$(Z)&
Ep = eWa,eww
{RP,Ey}= 0
w = wMaMz W
(5.4.1)
*Warner, F.W., Foundations of Dijierentiable Manifolds and Lie groups. - Graduate Texts in Mathematics, no. 94. New York, Berlin, Heidelberg, Tokyo: Springer-Verlag, 1983. Copyright © 1998 IOP Publishing Ltd
430
Ideas and Methods of Supersymmetry and Supergravity
Here W M ( zare ) unconstrained complex superfields such that the operator W , being considered as a supervector field, is of C-type, &(WM) = cy. The bosonic superfields Aap(z)are subjected only to the requirement det ( A , p ) # 0. W is not restricted to be real because the submanifolds generated by E , can be seen to be defined in a complex superspace, where dotted and undotted fermionic coordinates are not related by conjugation, in which our superspace is embedded as a real subspace. Taking the complex conjugate of relations (5.4. l), one obtains =
-
A I
-
= -eWabe-T
(5.4.2)
Recall that the superlocal Lorentz group acts on E , as follows: E; = (exp K):Es, det (exp K ) = 1. Therefore, it is useful to factor Aap into a Lorentz scalar and unimodular matrix, - .-
A
Ea = FNZl?,
E6 = FNJE,
det ( N a p ) = 1
(5.4.3)
Then, the superlocal Lorentz transformations act on F, N Z and W according to the rule: F' = F
W' = W
Nip = (e")tN{.
(5.4.4)
With respect to the general coordinate transformation supergroup, the objects under consideration change in the manner: E & = eXEae-"
K = KMa, = R
(5.4.5)
To summarize, after imposing constraints (5.3.15), all geometrical objects are expressed in terms of unconstrained complex superfields Nu", F and W M and their conjugates, which will be called the 'supergeometry prepotentials'.
5.4.2. Useful gauges on the superlocal Lorentz group The prepotentials N,P prove to be compensating superfields for the superlocal Lorentz group. As is seen from rule (5.4.4), the superlocal Lorentz N,p into the unit matrix. Therefore, transformation (e")! = 6,g(N- )'!moves the simplest gauge choice on the superlocal Lorentz group is Nap = Jap
In this gauge we have E , = F P =. C,i' Copyright © 1998 IOP Publishing Ltd
= E,
In FGpY+ Es In F6,'.
(5.4.6)
Superspace Geometry of Supergravity
43 1
Then, equation (5.3.9)gives
Rapy= E.?E/~In F
+ E ~ ~InEF,
neb, = ~ & f i In F + E&,&
(5.4.7)
In F.
Another useful Lorentz gauge is of the form
n,B, = ndpy= 0.
(5.4.8)
Such a gauge choice is possible since the supercurvature RUpfi9vanishes identically, in accordance with constraints (5.3.15). In fact, we have 1
8
Rap&) = 0 * E,,Rp,fiy = -j C$Ryfiy - R,,fi Rp,~p
This equation means that the SL(2, C ) connection Rabj is trivial:
ZZFA~ = .g)- ' ~ , g g = exp ( ~ f i i ~ B ~ ) for some symmetric tensor superfield Lfi?.Therefore, one can find a superlocal
Lorentz transformation leading to the choice of gauge (5.4.8). Gauge fixing (5.4.8)is helpful when working with covariantly chiral tensor superfields. To make our discussion clear, let us start with the scalar case. Consider a covariantly chiral scalar superfield @(z).By definition, it satisfies the equation $ 6 0 = Ea@ = 0.
By virtue of equations (5.4.2),we can represent @ in the form @=ew&
J,&=o.
(5.4.9)
Hence, every covariantly chiral scalar superfield is determined by a 'flat' chiral superfield &(x,O), depending on xm and ep only. The exponential ew plays the role of a picture-changing operator, since it transforms any 'flat' chiral superfield into a covariantly chiral scalar one. Note that & is invariant under general coordinate and superlocal Lorentz transformations. Next, consider a covariantly chiral tensor superfield x%,, &). We have ,,
-
0 = $ i X s c , ...un = ~
d
...u.~
-
+ unfl~~,~cc, ~ .. . a, = ~ d ~ p r..,a,. + (g-'E&)xc(, ... a.
where g = exp (LpyMp,).On these grounds, xu,,, , a ncan be represented in the form (5.4.10) q i i a , ,... U"
=0
which looks to be much more complicated than equation (5.4.9). However, in the gauge (5.4.8)both these representations take the same form. Copyright © 1998 IOP Publishing Ltd
432
Ideas and Methods of Supersymmetry and Supergravity
5.4.3. The /I-supergroup Often, when solving constraints in terms of unconstrained fields, we are faced with the appearance of a new gauge invariance. The most familiar example is electrodynamics. The first Maxwell equation reads &abcddbFed = 0, F c d is an antisymmetric tensor unifying the electric and magnetic fields. Solving this equation using Fa, = aaAb- dbAa introduces the gauge arbitrariness 6 A a = da$. So, we should scrutinize our theory for the appearance of an additional gauge invariance after solving constraints (5.3.15~)in the form (5.4.1). We shall look for transformations of the prepotentials N a p , F and W Mleaving invariant the spinor supervierbein E , and hence, the covariant derivatives. The general coordinate transformation supergroup acts on eW by the left shifts (5.4.5).Now, let us try to consider a right shift eW
= ,W e-T
A =Pa,
eW =
(5.4.11)
with i. being a complex C-type supervector field. In the infinitesimal case, we have
Eh = ew’d,e-”’
=
E,
+ eW(d,X)o‘,e-W.
To preserve the ,!?,-form, one must demand
resulting in
a?, = a,Ai,r =-a$.’’ =
=
o
(5.4.12)
=0
sE, = (eWa,,x)E,= (eWd,,X)),!?,, + 21 (eWa,,X),!?,,. -
If one supplements this transformation with 1 6F = - -(eWd v X ) F 2 6N,P = - N$(eW 8(,,P)
(5.4.13)
then E , does not change. Rather beautifully, we do not obtain any restrictions on the parameter while the parameters Am and I! satisfy chirality constraint (5.4.12). Transformations (5.4.11) with the parameters restricted according to rule (5.4.12) form a supergroup (the ‘A-supergroup’). This follows from the fact that two supervector fields of the type (5.4.12) commute on a supervector field of the same type. The A-supergroup can be realized as a group of triangular transformations Y’” = Y”(Y,
e)
e , = e y y , e) p; Copyright © 1998 IOP Publishing Ltd
= &(Y,
4p)
Superspace Geometry of Supergravity
433
acting in complex superspace C4I4 with c-number coordinates y” and a-number coordinates &’ and pb. This is the geometrical interpretation of the A-supergroup. The covariant derivatives and tensor superfields are unaffected by the R-supergroup. It acts on the prepotentials only. In addition, the prepotentials transform non-trivially with respect to the general coordinate transformation supergroup (the ‘K-supergroup’) and the superlocal Lorentz group. So, the full gauge group of the prepotentials is the product of three groups: the i-supergroup, the K-supergroup and the superlocal Lorentz group. In accordance with equations (5.4.4,5,11, 13), the infinitesimal transformation laws of the prepotentials are Jew = K e W - e W X 1
SF = K F - -(eW dyA*Y)F 2 K = KMaM= 17
A = AMaM
abj!’=
(5.4.14)
abAV= 0.
There is an interesting connection between the K- and A-supergroups. Let 0 be a covariantly chiral scalar superfield. Then 0 is transformed by the K-supergroup and is inert with respect to the A-supergroup. Next, we represent 0 in the form (5.4.9). Now, 6 is inert with respect to the K-supergroup, but is transformed non-trivially by the A-supergroup. In accordance with equation (5.4.1l), we have w.. ~ = e w c $ = e Q’ therefore
6‘= e%
= exp
(m,+ Aq6
ab&’ = 0.
(5.4.15)
Note that constraints (5.4.12) could be obtained from the requirement ab& = 0. Remark. In the Lorentz gauge (5.4.6) every ,?,-transformation should be supplemented by a Ldependent Lorentz transformation with the parameters
K , , = eW ?(,Z/{,
~ , p= e*a[*X,jh
(see equation (5.4.14)) where
a,
= 6,pa,
=
(5.4.16)
s,v;r,.
5.4.4. Expressions for E, T, and R This subsection is devoted to obtaining expressions for the Berezinian E = Ber ( E a M )and the supertorsion T , and R in terms of the prepotentials. These results will be used later. Copyright © 1998 IOP Publishing Ltd
434
Ideas and Methods of Supersymmetry and Supergravity
We begin by determining E. For this purpose we choose the Lorentz gauge (5.4.6) and introduce a semi-covariant supervierbein E, (different from 8, (5.3.10))according to the rule:
E,
E,
= (E,, E,,
1
= - -(6,)””{E,,
4
A
Ea) = & M a M = Ber
&}
(5.4.17)
(EAM).
Note that superfields EAM,in particular Berezinian E, are constructed in terms of the prepotentials W and JT’ only. By virtue of equations (5.3.6) and (5.4.3), E , and E, are connected as follows: A
p=Fp
E,=FE,
(5.4.18) 1 -
- - F(~,)~,(CI,! 4
+ BE, In F&.
Then, one obtains (5.4.19)
E = FZF2E.
Since E, E and F are Lorentz scalars, this result holds in the general case. To find T,, we again choose the Lorentz gauge N a p = 8;. Recalling equations (5.2.59,60) and (5.3.32) leads to T , = E, In E
- (1
c
E,) - Oflap,
Then, making use of relation (5.4.7) reduces this expression to T , = E , In E - (1
k,) + 3E, In F = E , In (EF’) - (1
c
&)F
As a result, we have c
T , = E , in (EF2(1*eW)).
(5.4.20)
Since T , and E, have the same Lorentz transformation laws, equation (5.4.20) holds in the general case. From equation (5.4.19), we read off the final Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
435
expression
(5.4.21)
and use equation (5.4.7).As a result, one obtains
(5.4.22) Given a scalar superfield U , the superfield
(g2- 4R)U is covariantly chiral and, by virtue of relation (5.4.22),we have
(a2- 4R)U = i&&F2U).
(5.4.23)
The supertorsions Go and Wmpycan also be expressed in terms of the prepotentials. However, these expressions turn out to be highly complicated and we do not reproduce them.
5.4.5. Gaugejxing f o r the K- and A-supergroups Let us analyse the W-transformation law (5.4.14).It can be rewritten in the form
I + o(w) = (K" - Xm)dm + (KP - X ) a , + (R, - Xp)P + O(W). Recall that K" is an arbitrary real vector superfield, while I p and R, are J W = 6wMaM= K
-
arbitrary spinor superfields. Using our freedom in the choice of K", we can kill the real part of W". On the other hand, the parameters X and R, can be used to gauge away the spinor components Wp and W,. After this, we obtain a purely imaginary vector superfield W= -iH
H = H"8, = R.
(5.4.24)
Therefore, the prepotentials Re W", W p and W , are compensating superfields for the K M -and P-transformations. Copyright © 1998 IOP Publishing Ltd
436
Ideas und Methods of Supersymmetry und Supergruvity
Remark. Considering the K-transformations only, we can gauge away the real part of W resulting in a purely imaginary supervector field
Rernark. After imposing the gauge (5.4.24), a covariantly chiral scalar superfield takes the form mix, 8 , Q = elH@x,
e) = 6(elHx,8).
(5.4.26)
In the Lorentz gauge N $ = 6$, the spinor supervierbeins are of the form E , = Fe-lHa,eIH
& = FelH(-8&-lH,
(5.4.27)
It is instructive to compare these expressions with the 'flat' ones (see (2.5.18-21)). There are some residual K- and A-transformations preserving the gauge choice (5.4.24). Making use of the W-transformation law deW= KMaMe-IH- e-lHr?aM = {KMdM- (e-lHr?)e-'HZMelH}e-lH
and demanding 6H to be real and to have no spinor pieces, 6H = 6Hmam= 6H, one arrives at the following restrictions:
Here Km(jb,X) is a rather complicated function. It proves to be quite difficult to operate with this function in practical calculations. So, the gauge (5.4.24) is not a useful one. It is worth obtaining a gauge-fixed version in which the K-transformations are excluded by construction. As the key to the problem, let us consider the transformation law for exp ( - 2iH): de-2iH = q e - W e W ) = le-2iH - e-i2HX
(5.4.29)
We see that only i.-transformations act on exp (- 2iH). Hence, it is sufficient to find a reformulation in which H arises only in the combination exp (- 2iH). We shall turn to the chiral representation.
5.4.6. Chiral representation For the time being, it is worth forgetting about any gauge fixing and starting from the beginning. Once again, we choose the Lorentz gauge NEfi= dag and, then, perform the following picture-changing transformation: every superfield V is changed to vE
Copyright © 1998 IOP Publishing Ltd
e-wj/
(5.4.30a)
Superspace Geometry of Supergravity
437
the covariant derivatives 58A are transformed into
(5.4.30b) The spinor supervierbeins take the form
-
where we have denoted F simply by F. The representation obtained is said to be chiral, while the original representation is called real. The chiral representation is well suited to the treatment of covariantly chiral scalar superfields:
4j
= @x,
e)
qx,
e)
but
*
= e-2iHG(x,
e).
(5.4.32)
The only complication this representation produces is that it modifies the complex conjugation rules: given a superfield I/ and its conjugate P, for their chiral transforms we have
-
v= ( e - @ p y = In particular, if V is real, then
v- =
e-W
p=
-
e2iHF
e2iHE
(5.4.33 4 (5.4.33b)
Now, let us analyse K - and A-transformation laws. In the real representation, we have 8 9 A
x
= [c%?,gA]
SI/= XI/
+ (eW a p ~ , , ) ~ p+r ( e w J b + W +
=K M ~ ,
where I/ is a tensor superfield (indices are suppressed). Recall that in the Lorentz gauge N , p = S u p every ,?-transformation must be supplemented by the Lorentz transformation (5.4.16). Making the chiral transform, one obtains
In particular, the spinor supervierbeins change as follows:
(5.4.35) As may be seen, the K-transformations never arise in the chiral representation. Of course, there is no mystery in this disappearance: the K-invariance can Copyright © 1998 IOP Publishing Ltd
438
Ideas and Methods of Supersymmetry and Supergravity
be used to gauge away the real part of W . There are no independent K-transformations in the gauge Re W = 0. But in the chiral representation W enters into all quantities in the combination exp (-2iH) = exp (- W ) exp W only, and hence the real part of W is excluded by construction. It should be pointed out that the spinor part of H can be gauged away by proper choice of gauge parameters Afi, resulting in the gauge (5.4.24).After this, we work with the restricted 1.-transformations (5.4.28). There is a simple recipe for transferring each expression in the real representation to the chiral representation. Namely, one is to make the following replacements: + 62, W + -2iH, 0, F + F, F + e-2i"F (and to place tildes on the_ remaining symbols). In particular, the chiral transformed supertorsions Tiand I? read
w+
Nz
- - - T- = In(F2E)
Ti= EiT
(5.4.36)
where we have used equations (5.4.20,22). Let Y be a scalar superfield. Equation (5.4.30b)then says that
s
s
d82E-lY = d 8 z E - l 2
(5.4.37)
anticipating suitable boundary conditions at infinity for 2. Therefore, there is no need to worry about representations (real or chiral) in which integrals over superspace are done. 5.4.7. Gravitational superfield Instead of imposing the gauge (5.4.24) with purely imaginary vectorial W , we can retain some Re W"' if we wish. Then, the arbitrariness in choosing Re W" can be used to make the gauge fixing
exp (Wnan)xm = xm + iX"'(x,
e,@
X m= 2%
(5.4.38~)
Understanding this relation as an equation in W"', its perturbative solution is /
W m= i&" +
1
2
+ 0(2f3).
(5.4.38b)
Now, covariantly chiral scalar superfields have the form
WX,e, e) = ew@(x,e) = @(xm+ i X m ,er).
(5.4.39)
Let us find the residual K - and &transformations preserving the gauge (5.4.38).We have i 6 X m = 8ewxm= KNaNewxm - ewANaNxm = KNaN(X"
+ iXm)- ewAm= K"
Copyright © 1998 IOP Publishing Ltd
- ewIm+ iKNaNXm.
Superspace Geometry of Supergravity
439
This expression should be purely imaginary, hence 1 K" = -ew,?" 2
1 1 + -ew 1"= -Am(xn+ i%,, 2 2
0')
+ -21 P ( x " - iX", By)
(5.4.40)
and 6%" = KNdN%"
i + -(A"(x" + i%", 0") - xm(x"- i X n ,p)). (5.4.41) 2
It is also necessary to require two consistency conditions: 6emOp = o
Jew& = o =-
K p = P ( x " + i P , e'),
R,
= I,(,, - i Z n ,8)
(5.4.42)
e), P(x, e),e- wewI,(x, O).
AM = (nm(x,
Finally, let us substitute the expressions for K",
Kp
and R, into expression
(5.4.41) and compare the result with equation (5.1.8). Good Lord, we have
recovered the transformation law of the gravitational superfield! In the 'gravitational superfield gauge' (5.4.38), all the building blocks which have appeared in the expressions for the covariant derivatives can be readily expressed in terms of %", F and F. The semi-covariant supervierbein 8, is then given by
+ i(d,%n)(l - ia%)-l,V, E&= -ai + i(Jd%")(l + id%)-',"'d, E,
= d,
=
a, + i(E,%")d,
A
8
a
=
1 - -(aa)""([&, 4
-3, - i(Ea%")a, (5.4.43) A
=
&]xm)a,&"a,,,
where we have introduced the notation (1
+ id%),"
Introducing the anholonomy coefficients
[E,,
EB}
+ i-. a%" ax,
= 6,"
(5.4.44)
eABC via
= eABCeC
=
+i
[&a,
= CabY = 0
bb])x" (5.4.45)
Copyright © 1998 IOP Publishing Ltd
(5.4.46a) and ((5.4.46b)
(5.4.47)
t
(1.e5)=det(l +ia2')
( l * e W ) = d e t ( l-idAO)
hold. Let us comment on the derivation of equations (5.4.43-47).Consider the change of variables ,x ;
= X"
+iP" (5.4.48)
The corresponding Berezinian is equal to = (1
*e") = det(1
+ id%)
which coincides with the second relation (5.4.47).Furthermore, we have
A
Now, making use of expressions (5.4.48)leads to the expression for (5.4.43). Finally, the expression for R,,j.,: (5.4.46~) follows from equation (5.3.12). In conclusion, it should be pointed out that the gauge transformations (5.4.40,42) (preserving the gauge choice (5.4.38)) act on the covariant derivatives and tensor superfields in the manner:
(5.4.49)
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity KP = i P ( x
+ i&',
441
0)
+ iX,0)
K,,] = ~!?,,ig,(x
Bz
since in the gauge N Z = every i-transformation must be supplemented by the Lorentz transformation (5.4.16).
5.4.8. GaugeJixing on the generalized super Weyl group
We have nearly reached the aim formulated at the beginning of Section 5.2. Namely, we have found a supergeometrical description for the gravitational superfield. But what about the chiral compensator of Einstein supergravity? The point is that the set of prepotentials does not contain chiral superfields. Notice, however, that among the prepotentials N,", F and W M the superfield F played a rather passive role in the above considerations. By construction, F is a complex superfield transforming as a scalar under the K-supergroup and as a density under the R-supergroup (see equation (5.4.14)). In principle, there exists the possibility of obtaining a chiral superfield, if one manages to restrict F i n a covariant way. Recall that in the supergeometrical approach every covariant constraint is a constraint on the supertorsion. Hence, in order to obtain a supergeometrical description of Einstein supergravity, one must supplement equations (5.3.15)by some additional constraints. One can look on the prepotential F as a compensating superfield for the generalized super Weyl transformations (5.3.58).They act on the prepotentials as follows
F
= LF
WIM= W M .
NraP=
(5.4.50)
Nz
Now all the prepotentials acquire a clear interpretation. F, and Re W are compensators for the generalized super Weyl group, the superlocal Lorentz group and the K-supergroup, respectively. W p and Wp are and Rp-transformations. Finally, compensators for the independent Im W" plays the role of gravitational superfield. It has been shown in subsection 5.3.8 that the maximal symmetry group of constraints (5.3.15)is given by the product of three supergroups: the general coordinate transformation supergroup, the superlocal Lorentz group and the generalized super Weyl group. In a theory with such a symmetry group, the prepotential F can be gauged out by proper choice of super Weyl transformation. Then, in the gauge (5.4.38) we work with the gravitational superfield only, as in conformal supergravity. This is the reason why constraints (5.3.15) are said to be conformal supergravity constraints. Since the prepotential F plays he role of super Weyl compensator, to impose a constraint on F is the same as fixing a gauge on the generalized super Weyl group. The simplest gauge choice is
x-
F Copyright © 1998 IOP Publishing Ltd
=
1.
442
Ideas and Methods of Supersymmetry and Supergrauity
However, this gauge fixing is not invariant under the A-supergroup (see equation (5.4.14)). To obtain a covariant gauge, let us recall the super Weyl transformation law for T, (5.3.59).By virtue of equation (5.4.20),this can be rewritten in the form T:, = 9 ; ~T’ = T + i n ( ~ 4 ~ 2 ) .
(5.4.51)
Hence, one can take the super Weyl gauge T , = 0. Let us proceed to an analysis of this constraint.
5.5. Einstein supergravity 5.5.1. Einstein supergravity constraints
In the space of supergeometries (5.3.15) we consider the subspace characterized by the additional constraint
-
T , = Ta = 0.
(5.5.1)
Now, the full set of constraints is T,;=O
T,pC=O
T a t + 2i6:(03,~ = 0
Raid= 0
(5.5.2)
T,: = 0. Tat= 0 These constraints (in a slightly different form) were suggested by J. Wess and B. Zumino. We are going to show that the set of constraints (5.5.2) corresponds to Einstein supergravity. 5.5.2. Chiral compensator
is expressed through the prepotentials in accordance The supertorsion with the rule (5.4.21). Setting Ti= 0 gives ~ 2 ~ 4 8.e@) ( 1 = (p-3
E&
=0
(5.5.3)
where @(x,e)is a flat chiral superfield. Adding the complex conjugate formula, one obtains
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
443
As a result, all geometrical objects are expressed in terms of the old prepotentials N,p and W Mand the chiral superfield cp (and their conjugates). To find the transformation law of cp, we represent it in the form c p3 -- E-'F-2(1.erv)-l
using equation (5.4.19). E - ' changes only under the general coordinate transformations, 6E-' = K M a M E - '
+ (-l)""(d,KM)E-'
(see equation (5.2.27)).The transformation law for F-' follows from (5.4.14). Finally, using equations (1.11.21)and (5.4.14), one readily obtains t
6 ( 1 . e C V ) = ( l . e 5 ~ ) - ( l , A-e5 .eW) + (- I P ( ~ , K M ) ( ~ .e') t
= KM~,(I
- (-
lyM(eW ~ , A M ) ( I -e@)
From these results, we deduce or The second relation coincides with the transformation law of chiral compensator (5.1.34b)in Einstein supergravity! Hence, the prepotential cp can be identified with the chiral compensator. As a result, we have found the supergeometrical description for Einstein supergravity. The supergravity formulation constructed in Section 5.1 and here, is usually called the minimal supergravity. Later we shall consider other versions of supergravity characterized by non-chiral compensating multiplets. The superfields Nap, W M and cp are said to be the 'minimal supergravity prepotentials'. 5.5.3. MinimaI algebra of covariant derivatives
Under constraint (5.5. l), the covariant derivative algebra (5.3.53)is simplified drastically:
{a,, gi-d)= {ga,
gp} = -4RM,p
gppl
[ai,
=
{Gd,ab>= 4RRdb
-isd&R9p + GBPG.J- igpRMdb
+ isi@Gpd&l.jd - 2isabWpY6MY8 [ g xB,Q,~] , = is,p(mg + G'gBY) + iBbRM,, - i&,&PG6gMY6+ 2ie,,WdGR,d. Copyright © 1998 IOP Publishing Ltd
(5.5.6)
444
Ideas and Methods of Supersymmetry and Supergravity
The commutation relations for vector covariant derivatives are also quite simple: C 9 @ d ,9 p B 1
= EiB$ap
+ E?p$eB
(5* 5.7)
where
The supertorsions R , G, and W"pY satisfy the Bianchi identities G, = G o
Wap, = W ( z p y )
BiR = 0
Wapy= O
(5.5.8)
BiGui = g a R 1
9yWZpj= - 9 : G p b 2
i + -9{Gua. 2
We point out that in minimal supergravity Wupvis chiral. From the covariant derivative algebra, one can obtain the following useful identities: g Z 3 p=
d@g =
1
- 2RM,p
2 1 --E,@'
2
+ 2RRkb
5BZg2 = 4 R 9 p ( ~ ~+pM a p )
a2gg= - 2 R 9 8 ( ~ ~+p 2M,p)
[B',
(5.5.94
(5.5.9b)
ai]= -4(GaB + iGBaB)5Ba+ 4md- 4(9)'Gsb)ll.1;,6 + 8 W l J M $ 8 (5.5.9d)
[a2,gz]= -4(GUd - iBai)gd+ 4 R a a - 4 ( a ; ' G t ) f i , ~ + 8 W j ' S M , s (5.5.9e) Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
1 -9,(a2 - 4 R ) 9 , - Ig,(gZ- 4 R ) @ 8 8 = {i(9yG6i)Bi- W,6,9a
1
+ i(9tGBd) M’6
445
(5.5.9f)
It is worth noting that the identities (5.5.9a-c) hold for covariant derivatives under the conformal supergravity constraints, while the other identities are correct only in the Einstein supergravity case. Let us give some simple applications of the above relations. If $ a ( ~ )is an undotted spinor superfield, then equations (5.5.9b,c) lead to
(5.5.10)
If qa(z) is a covariantly chiral spinor superfield, then making use of equations (5.5.94 e ) gives (5.5.11 ) For every scalar superfield V(z),we have
ga(G2- 4 R ) g a V =
$&(a2 -4QPV
(5.5.12)
in accordance with equation (5.5.9f).
5.5.4. Super W e y l transformations
It has been pointed out in subsection 5.4.7 that the Einstein supergravity constraint (5.5.1)can be treated as a gauge fixing condition for the generalized super Weyl group acting on the covariant derivatives according to the law (5.3.58). However, the gauge condition (5.5.1) does not completely fix the generalized super Weyl symmetry. There exist some residual transformations (5.3.58) preserving the gauge (5.5.1).In accordance with equation (5.4.5 l), the corresponding superfield parameters must satisfy the equation gaIn (L4L2)= 0 *
Here o(z) is an arbitrary covariantly chiral scalar superfield. We conclude that the minimal supergravity algebra (5.5.6) is covariant Copyright © 1998 IOP Publishing Ltd
446
Ideas arid Methods of Supersymmetry and Supergravity
under the transformations Be+ 9; = LB, - 2(@L)M,,
a; = m, - 2(@L)M#, i B,* -+ 9j*= - {g;,a&} 2
d, -+
(5.5.13)
which will be called the 'super Weyl transformations'. It is not difficult to find how the supertorsions R, G, and Wap,change under the transformations (5.5.1 3). The results are 1 R' = - -e-2a(az - 4R)e5 4
wkPt= e- 3a/2 wa
h
As may be seen, WEBY changes homogeneously. At the prepotential level, equation (5.5.13) means the following WIM= W M
NkP = Nap
q' = eaq.
(5.5.15 )
Therefore, cp is a compensating superfield for the super Weyl transformations. The super Weyl invariance can be used to impose the gauge condition q = 1.
5.5.5. Integration by parts From the algebra (5.5.6,7) one can obtain the relation (-
= 0.
l)EBTABB
(5.5.16)
Therefore, in Einstein supergravity we have the rule for integration by parts (5.2.62). In particular, if +%(z)and V,(z) are spinor and vector superfields under proper boundary conditions, then
s
dsz E - ' 9 u + e = 0
s
d8z E - ' 9 ' V , = 0.
5.5.6. Chiral integration rule
Let U,(z) be a covariantly chiral scalar superfield, y C= e w P ,
Copyright © 1998 IOP Publishing Ltd
a,PC= o
(5.5.17)
Superspace Geometry of Supergravity where
2,is characterized
447
by the transformation law
6 9 , = (Amam + PQ9,. Equation (5.5.5b)tells us that the integral
s
S , = d4x d2843$, e
s
d6z 43$c.
(5.5.18)
is invariant with respect to all supergravity transformations ( K , A and superlocal Lorentz ones). Now, we are going to rewrite expression (5.5.18) as an integral over the superspace [w4I4. In the chiral representation (see subsection 5.4.6),equation (5.5.3)can be rewritten as 43F2 = E-1,
(5.5.19)
Acting on both sides with the operator @,Zfi and recalling equation (5.4.36), one obtains
(5.5.20~)
or (5.5.20b)
Therefore, S , can be represented in the form
s
S , = d4x d28d28(E- '/R)$,.
Finally, due to the identity (5.4.37),we can write
s
s
S , = d6z g3Pc= d8z(E-'/R)9,.
(5.5.21)
This relation, known as the 'chiral integration rule', was obtained by W. Siegel. It should be stressed that the chiral integration rule becomes unacceptable in cases when the body of R vanishes at some superspace points. In particular, this prescription has no flat superspace limit implying R -,0. Hence, the chiral integration rule illustrates itself the peculiarity of curved supergeometry. Using the chiral integration rule, any integral over R4l4can be reduced to an integral over R,4 x Cf. Given a scalar superfield 9 ( z ) ,two integrals Ids, E - ' 9 Copyright © 1998 IOP Publishing Ltd
448
I d e m and Merliods of Supersymmetry und Supergruvitj
and
coincide, owing to property (5.5.17). Then, since R is chiral one can write
= - L s d 6 z @ 3 e - w [ ( l z- 4R)YJ
(5.5.22)
4
5.5.7. Matter dynamical systems in a supergravity background
We have completed the supergeometric description of curved superspace corresponding to Einstein supergravity. Now, one can easily develop matter superfield theory in a given (background) curved superspace. By matter superfields we will understand unconstrained tensor superfields of arbitrary Lorentz types (n/2,m/2), unconstrained chiral tensor superfields of Lorentz types (n/2,0) and unconstrained antichiral tensor superfields of Lorentz types (0,m/2). The main symmetry principle which underlies dynamical systems in a curved superspace is invariance with respect to the general coordinate superspace transformations and the superlocal Lorentz transformations (the general covariance principle): 9A'-- e x 9 A e-," x ' = exx 1 X = K M d M - K"M,,. 2
+
(5.5.23)
Here x is a set of matter superfields, in terms of which some dynamical system is described. By definition, a local dynamical system is characterized by an action superfunctional with the structure: S [ x ; gA] =
s
d'zE-'Y(x; 9,)
+
J
J
d6z@3e-WYc(x; gA) + d 6 2 6 3 e - W 9 c (Q~A;) (5.5.24)
with Y being a real scalar superfield and Y ca covariantly chiral scalar superfield; Y and LfCare assumed to be functions of superfields x and their covariant derivatives to a finite order. In addition, Y and Y cmay depend Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
449
polynomially on the supertorsions R, G, and WaB;in order to have a well-defined flat superspace limit. Clearly, the dynamical superfield equations for the system (5.5.24) should be covariant under the transformations (5.5.23).In order to have an explicitly covariant form for these equations, it is worth obtaining covariant variational rules. We shall now find covariant variational rules in two particular cases: for dynamical systems described by a real scalar superfield V ( z ) or by a covariantly chiral scalar superfield @ ( z )and its conjugate @(z).Tensor cases may be treated similarly. Having a dynamical system with an action superfunctional S [ V ; 91, we represent an infinitesimal variation of S [ V ; 91 in the form: S S [ V ; P ]= S [ V + S V ; 9 ] - S [ V ; . 9 ]
(5.5.25) The SS[ V ;S]/SV ( 2 ) will be called the ‘left superfunctional (or variational) derivative’ of S [ V ; 2 ] with respect to V(z). Obviously, 6S/6V(z) is a real scalar superfield. It follows from (5.5.25) that
The bi-scalar d8(z, z’) will be called the ‘covariant delta-function’. Having a dynamical system with an action superfunctional S [ @ , @; represent an infinitesimal variation of S[@,Q 91 as follows:
6 S [ @ , q 931
91, we
= S[@ + 6 @ , 6 + 6@;91 - S [ @ , 6;91
where SS[@, 5;9]/6@(2) is a covariantly chiral scalar superfield, which will be called the ‘covariant superfunctional (or variational) derivative’ of S[@,%; 91 with respect to @(z). Owing to the identity
s
~ O ( Z= ’ ) d8ZE- ‘6@(~)6’(~, 2’) = -
we have
S@(Z’)
-=
d8z(E-’/R)6@(z)(B2 - 4R)d8(z, 2’)
1 - -(B2- 4R)d8(z,2’)
6@(2) Copyright © 1998 IOP Publishing Ltd
4
d + ( Z , 2’).
(5.5.28)
450
tdeus und Methods of Supersymmetry und Supergruvity
The bi-scalar 6+(z,z‘) is covariantly chiral with respect to each argument,
a$3+(z,2’) = Z@+(z,
(5.5.29)
2’) = 0.
This can be readily seen in the chiral representation:
-(52 - 4R)$ = apa”(F2$)
= 3-364(x - x y 2 ( e
V$
*
- e’)
where we have used equation (5.5.19). S+(z, z f ) will be called the ‘covariant chiral delta-function’.
5.6. Prepotential deformations In this section we would like to discuss a covariant variational technique for the supergravity prepotentials. It will be necessary in order to obtain a supersymmetric generalization of the energy-momentum tensor and to find (in the following chapter) supergravity dynamical equations. To start with, we introduce a slightly modified parametrization of the supergravity prepotentials and the supergravity gauge group. The parametrization given below proves to be most convenient for perturbative calculations. In the main body of subsections 5.7.1 and 5.7.2, the covariant derivatives under conformal supergravity constraints only will be considered. Each occasion that specification to the Einstein supergravity case is needed will be specifically mentioned. 5.6.I . Modified parametrization of prepotentials
In the prepotential parametrization of subsection 5.4.1, the flat superspace limit corresponds to the choice: N,p
=S
g
W = -ie88!,
F=1
-iso.
Hence, when making perturbative calculations with respect to a flat background, one must expand W around its flat non-vanishing value ( - i s o ) . This is not so convenient. It would be preferable to have a parametrization in which the flat superspace limit corresponded to the choice W = 0. Such a parametrization is easily obtained from the old one by merely replacing ,W
Copyright © 1998 IOP Publishing Ltd
+
,W ,-i%o,
(5.6.1)
Superspace Geometry of Supergravity
45 1
Then, the spinor covariant derivatives take the form g X= FN,,eW D,e-W
1 + -fl,b'Mbc 2 1 + -agbCMbC 2
_ .
gi = FNipew
(5.6.2)
where D, and D, are the flat spinor covariant derivatives (see equations (2.5.19,21)).Then, it seems reasonable to write W as follows W = WMDM
D, = (d,,,, D,, Dfi).
(5.6.3)
Further, we exclude from our consideration the unimodular matrix N a p at the cost of introducing a complex superlocal Lorentz transformation. Namely, the covariant derivatives will be represented in the form:
(5.6.4) W = WAD,
+ WaPMap+ W"BRd/) # f l
instead of the form (5.6.2).Here W"p(z)and W"(z) are symmetric bi-spinor superfields. One can easily see that W"p is in one-to-one correspondence with the old prepotential NZ. A novel feature is the appearance of redundant superfields W"l' ( # ( WzB)*= W i b )which we have introduced by hand. In principle, one can set W d bto zero or uniquely fix their values by imposing the requirement
b>M. pj - 0
(5.6.5)
which is possible owing to the results of subsection 5.4.2. However, we prefer to keep arbitrary W", similarly to the other superfields arising in W . It will soon be shown that W @ is a pure gauge degree of freedom associated with some auxiliary gauge invariance. Considering the parametrization (5.6.4),it is convenient to represent 9 j A in the form B A= E,,D, + i f l A b C M b c as well as to write the transformations (5.3.57)according to the rule g A+ 9: = e3Y9Ae-x (5.6.6) L
These act on the prepotentials just introduced according to the law "
=
Copyright © 1998 IOP Publishing Ltd
F' = F
E; bc
= -&bc.
(5.6.7)
452
Ideus and Methods of Supersymmetrj) and Supergravity
To find an analogue of the A-supergroup in the parametrization (5.6.4), we seek a transformation of the form (5.6.8) supplemented by displacements of F and 2: such that the covariant derivatives remain unchanged. In the infinitesimal case, we have
Due to the relation [Dk,
1 A] = - -(D,Abfi)dbp 2
+ 2iAbaPa+ (D,Ab)D,
'
- (@,AB)
+ h))Dj + !(DhAbr)Mbc 2
the above requirement leads to restrictions on the parameters D,Ab = 0
D,I\Pfi = 4iAbdl
Ais = -D(iApl
(5.6.9)
Ad and Aab arbitrary
and also the transformation laws
GF = A F + !(D,AU)F 2
dZ.,b'
=
-DdAbc
a,
+ ([A,
(5.6.10)
Under the relations (5.6.9, lo), is invariant with respect to the transformation (5.6.8). The first equation (5.6.9) is equivalent to the two identities: 8iA" = DdAEd. D i A b ~+ DpApe= 0 They can be solved in terms of a spinor superfield L, as follows: A,& = -2iDdL,
1Aa = - -DZL,. 4
(5.6.11)
Then, the second equation (5.6.9) is satisfied automatically. We see that the A-transformations (5.6.8, 10) are generated by unconstrained parameters L,, Ad and A,, = A,. This is one of the main advantages of the parametrization (5.6.4). Let Q(z) be a covariantly chiral scalar superfield, B,Q = 0. It can be Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
453
represented as
D,&=o
@=e$'&
(5.6.12)
with 6 being a flat chiral superfield. The A-transformations act on 6 according to the law @ = e%$
(5.6.13)
or, in infinitesimal form, d6=A6 =
1-
- -D2(L1D,6)
(5.6.14)
4
where we have used equation (5.6.11). Clearly, 66 is chiral. The X-transformations (5.6.6)can be used to gauge away the real part of W resulting in
w = -ix
2 = H ~ D ,+ H ~ B M , + ~ iWJIj,j = 2.
Such a gauge choice is equivalent to introducing a chiral representation defined by the rule: every tensor superfield I/ is to be changed by
p=
e-KI/
(5.6.15a)
the covariant derivatives are to be transformed to
-
g A= e-*gAeq.
(5.6.15b)
There is one difference between the chiral representation just introduced and that considered in subsection 5.4.6.To obtain the representation of subsection 5.4.6, one has to make the complex coordinate transformation (5.4.30). In the present case, since e-@ = exp[
- WADA-
1
1wabMab= eXp 2
1
- - W'abMab exp [ - WAD,]
[:-
one has to apply not only a complex coordinate transformation, but also to accompany it by some complex superlocal Lorentz transformation. Since Lorentz transformations (real or complex) do not change invariant functionals of the form J d8z E - ' 9 , with 9being a scalar superfield, the identity (5.4.37) holds in the present case also. The covariant derivatives in the chiral representation are
(5.6.16)
Copyright © 1998 IOP Publishing Ltd
454
Ideas and Methods of Supersymmetry and Supergravity
They transform according to the law:
3; = e A g A e - A
e-2iX
- e A e -2iMe-A
(5.6.17)
where the parameters are defined as in equations (5.6.9) and (5.6.11). Since ( - 2i6%) = A,@ - A"D, + A @ M , ~- A"BR,~ + . . . ,we can use A& and Azs to gauge away Ha,Ra, HUB and R"B resulting in % = H"a,.
(5.6.18)
In this gauge, we have a restricted set of gauge transformations: de - 2i.H - - 2% - e - 2iM-A has no spinor and Lorentz part =. I\,= e-2iX&
A aB - e-2iXA
U P
(5.6.19)
Remark. Instead of choosing the gauge (5.6.18), one can impose a weaker gauge condition % = HADA.
(5.6.20)
In this case, only the second restriction (5.6.19) should be imposed. In the chiral representation, one can readily repeat the analysis of subsection 5.4.4obtaining explicit expressions for the geometrical objects:
E = j 7 2 ( e - ~ i ~ F ) 2 ~R- = - ! ~ 2 ( ~ 2 ) 4
where we have denoted:
E = Ber(EAM). In the Einstein supergravity case, we have Tu = T d = 0. Then it follows from the above relations that
Ep
= 4n-3
Dd(p = 0.
(5.6.22)
The chiral compensator cp proves to have the transformation law 1 6(p3 = AADA(p3+ (8,A' - D,Au)(p3 = -D2D,(L?p3). 4
(5.6.23)
5.6.2. Background-quantum splitting We go on by describing the covariant supergravity variational technique developed by M. Grisaru and W. Siegel. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
455
Consider two sets of covariant derivatives under conformal supergravity constraints: 1
9~= E A ~ D + M -QAbcMbc 2 and V A= ( E
1
+ ~ E ) A ~ +D 2-(a M + 6R)AbCMbc
1 (5.6.24) ~ A ~ +D- @M AbcMbc. 2
Using the parametrization of previous subsections for g A ,one can write 1
( F + 6 F ) D , + -(C 2
1
+ SC),bfMbc e-w
- 6w
(5.6.25)
where 6W = 6 W M D M+ $6 Wb'Mbc and 6F are finite deformations (6c,bc is determined in terms of 6W and 6F because of the conformal supergravity constraints). The main problem one is faced with is that dW transforms highly nonlinearly: the X - and A-transformations act on W and (W + SW) according to the laws e7y' + 6 W ' = e3yew + 6We-A ew = exe"e-A (5.6.26)
To circumvent this problem, it is worth substituting the variation 6W by the covariantized one, A W , defined by eW
+ 6W
eAWeW
(5.6.27)
In accordance with equation (5.6.25), A W transforms covariantly: eAW'
= e.WeAWe-X
(5.6.28)
Now, if one represents A W in the form
1 A W = AWMDM + - A W b c M b , 2 1 + -(AWb' 2
=AWMEMAaA
E
- AWME,uAnAbc)Mbc
1 A W A g A+ - A W b C M b c 2
then A W A and A W b c will be Lorentz tensor superfields. There is another argument in defence of variation (5.6.27).Namely, looking at equation (5.6.4),it is seen that W enters as a complex general coordinate and superlocal Lorentz transformation. Therefore, it seems reasonable to represent any change in ew as a (complex) supergroup shift. Similarly, the superfield (e"'F) enters as a super Weyl transformation (5.3.58). Hence, it is Copyright © 1998 IOP Publishing Ltd
456
Ideas and Methods of Supersymmetry and Supergravity
tempting to represent its change as a supergroup shift, F + 6 F = (e-W'9)F. Finally, making a redefinition of GC,bc in equation (5.6.25),one arrives at the V representation:
(5.6.29) 1
Vzi = - { V x , Pc,} 2
AW = A W A 9 A
1 + -AWbCMbc # AW. 2
This representation is known as 'background-quantum splitting' (originally, it was developid for covariant quantization of supergravity). Backgroundquantum splitting determines deformed covariant derivatives V, in terms of The derivatives V A turn out to be the initial (background) derivatives 9,. coupled to the background derivatives g A .All objects AW', A W b c , 9 and A Q P , which specify VA, are tensor superfields. There exists some inherent arbitrariness in the choice of A W , 9and A n E . Let us seek a transformation of the form
A = AA9A
(5.6.30)
+ AZpMzp+ A'bi@&b
supplemented by displacements of 9 and A n $ such that the operators V, stay unchanged. Imposing the requirement
0 = e-'*67&'"'
= -
1
A, =c, 4-h p M b c
2
1 + 6mi + -6(An&'Mbc) 2
and making use of the algebra (5.3.53), one finds the following restrictions on the parameters - -T
16 - b -
TB
)
=
o
Ai and Azp arbitrary.
The first two equations can be solved in terms of an unconstrained spinor Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
457
superfield L,:
Azk = -2i
(9+ - + ;-) -Ti
L,
(5.6.32)
"
4
Here Ti, R and Gzi are the supertorsions corresponding to 9,. Further, the transformation law for P is
The explicit form of 6[AR,b7 is not essential for later applications. In summary, the transformations (5.6.30-33) leave the derivatives VA unchanged. The background-quantum splitting proves to be very powerful when making covariant variations with respect to the supergravity prepotentials of invariant superfunctionals of the general structure
s
S ( 9 ) = S[X(9$91 = dazE- l a X ( 9 ) ;91
(5.6.34)
with 9 being a scalar dependent on a set of matter superfields x(9) coupled to supergravity covariant derivatives .$BA. Here we mark the matter superfields by the label (a),since covariantly defined superfields often depend on supergravity prepotentials. For example, if x(g) is a covariantly chiral scalar superfield, = 0, then one can write x ( ~= ) eqi, with being an independent flat chiral superfield, D,i = 0. Hence, any supergeometry deformation B A+ V, = $BA + 69, will induce some change of the matter superfields, -+ x(v) = x(9) + 6x, as well as of the functional (5.6.34), Sea, -+ S(v)= S ( g ) + 6 S , where
akx,,)
s
S(v)= S[X(V);VI = dszb -
~ ( X ( V ) VI. ;
(5.6.35)
Both functionals S(9) and S(v)are invariant under the X-transformations. In the former case, they take the form
xi9) = e"x(B)
= exBAe-"
(5.6.36)
in the latter case, they read
xiv) = exx(v)
VL = e"V,e-".
(5.6.37)
However, after introducing background-quantum splitting (5.6.29), the transformations (5.6.37) can be realized in two different ways: (1) as Copyright © 1998 IOP Publishing Ltd
458
Ideas and Methods of Supersymmetry and Supergravity
'background' transformations
or (2) as 'quantum' transformations
where X ( Q=) K A g ,
1 + -KbCMb,
=
2
It is worth noting that the existence of two types of X-transformations is brought about by the splitting of a single object (W + SW) into two independent parts W and 8%'" (a similar situation occurs, for example, in electrodynamics, where, after splitting V , = Vf) + V g ) ,gauge transformations SV, = ami can be realized as S V f ) = d,i and SV:) = 0 or as SVf) = 0 and SV?) = dm,l). Note also that S(v)is invariant under the A-transformations (5.6.3Ck33). The real part of AW is a purely gauge degree of freedom for the 'quantum' %-invariance. This invariance can be used to gauge away ReAW. Equivalently, Re AW is excluded automatically when working in the 'quantum' chiral representation introduced by *
&v) * &v) = e
Since 8- = [Ber @ A M ) ] -
- A*
X(v)
gA
gA
c
= (8-
e-A@-),the quantum chiral transform
of S(v)coincides with S(v).In the quantum chiral representation, we have
,-2iH
1
- e-AfleAW =
H = HA9A + -HabMa,= H 2
and the functional S(v)is invariant under the transformations
j&)
= e"j7v) e -2iH
Copyright © 1998 IOP Publishing Ltd
= e"Q,e-" - 2iHe - A
Q; ~
(5.6.41)
459
Superspace Geometry of Supergravity
with A defined by equations (5.6.30-32). Using this invariance, one can impose the gauge
(5.6.42)
H = Haga.
After doing this, we work with a restricted set of gauge transformations (5.6.41) selected by the requirements
+ O(H)
A, = A,
Amp= Aag
+ O(H).
(5.6.43)
is a covariantly chiral scalar superfield, 0,2,,,= 0, then its
Remark. If &) transformation law is
1 6f(v) = A&) = - -(G2 - 4R)(Lm9uf(v)).
4
(5.6.44)
In conclusion, we give explicit expressions for some geometrical objects, constructed on the basis of PA, in terms of H and 9. We will denote the supercurvature and supertorsion tensors of P A in bold type, [PA,
a,> = T;ABcOc + -21R A B C d ~ c k
Demanding the anticommutator
{e,,e h }
one finds
= 4RRaj
- A€i,fiV = &,@pP+ &,@fi9 (5.6.45)
1 R = - -(a2 -4R)P.
4 Next, similarly to the derivation of equation (5.4.20), one can obtain
+F$=
~n[BZd(egF)2(1.e‘)].
On the other hand, the supertorsion
Ti,found from aA,takes the form 4-
T$ = GG~n[E(eYi’F)’(1*e”‘)]. From these expressions we deduce that 4
T, - P T ,
-
= Pdln [
92d~-’]
(5.6.46)
Finally, let us obtain a useful formula for d. Introducing auxilliary derivatives
6
=
A - (
a
- - ( 6 ~ ) ” { Q p , e b } , e-2iHg&2iH,ad = u A B g B + ‘hAbcMbc (5.6.47)
Copyright © 1998 IOP Publishing Ltd
2
460
Ideas und Methods of Supersymmetry and Supergravity
and recalling equation (5.6.40),one readily obtains
2= ( P e - 2 i H F ) 2 E U
(5.6.48)
U = Ber
It is not difficult to check the following conjugation rules e
(2-1)* = (2-1e2iH
(U-')* = (U-'.e2'H).
(5.6.49)
5.6.3. Background-quantum splitting in Einstein supergravity In the Einstein supergravity case, one must impose the constraints
-
--
T i = Ti = 0.
Then, the expressions (5.6.46) and (5.6.47) give
aicp = 0
j Z 4 ( e - 2 i H ~ ) 2 ~= c p - 3
(5.6.50)
where cp is a covariantly chiral scalar (with respect to g A )superfield. Using this relation together with equation (5.6.48), one can easily obtain
(e - 2iH97) = rp 1/2(e-2iHg) Now the Berezinian
- 1U - 1/6( 1
.e -2iH) +
1/3.
(5.6.51)
8-' takes the form
2-1 = ~ - 1 ( ~ ~ - 2 i H @ ) ~ - 1.e-2iH / 3 ( 1 1/3 I * c
(5.6.52)
To complete out discussion, it is necessary to determine the transformation law of cp under the A-transformations (5.6.41). It is an easy task if one represents this superfield in the form: cp3 = P."-'b-'E= , F - 2 ~ - 1 ,where 8 is the Berezinian of the auxiliary supervierbein defined by
0,
=#ABgB
+ ..
*
Here dots denote terms involving the Lorentz generators (superconnection terms). The transformations (5.6.41) change 8; as follows 6 8 A B = A g A B- Q,AB
+ 6ADACTCDB
and hence
&#-'= A 8 - l + (-1)&A#-19AAA - (-l)EAd-lABTBAA = #-'A
t
where we have used the identity ( - l)EATBAA = 0, which is fulfilled in Einstein supergravity. Then, the transformation law for 9is given by equation (5.6.33) Copyright © 1998 IOP Publishing Ltd
461
Superspace Geometry of Supergratlity
(in which one has to set T , = T , = 0). As a result, one obtains 6 q 3 = Aq3
1 4
=-(92
+ (~~(9~11" - 9 J a- iGaAa) (5.6.53)
- 4R)9&(L"fp3).
The reader can compare this result with equation (5.6.44). Note that, when H = 0, the derivatives V A coincide with those obtained from g Aafter applying the super Weyl transformation (5.5.13)with Q = In rp.
5.6.4. First-order expressions Now, we would like to calculate all geometrical objects constructed from P A to first order in the quantum superfields H and Q (q = e"), i.e. considering the approximation
e-2iH x 1
- 2iHQBa
rp = e' x 1
+ Q.
(5.6.54)
An application will be given in the next subsection. First of all, we calculate the Berezinian U of the supermatrix U A Bfound from the derivatives 0, (5.6.47). In the linear-in-H approximation we have U A B= d A B
+ AUAE
U = (Ber U A B )= 1
+ (-
l)E"AUAA.
From equation (5.6.47) we deduce that
6, = Ba - 2i[Hbgb,9,]
+ H,8aa, +
= (St - H,fiGPfi)gP- i(2JaHPfi)gPfi
eau,= -2i {Q~, ed)= gee- fig^, + !2 ( B ~ B , H ~ ~+~.)..B ~ ~ where dots denote superconnection terms and off-diagonal terms. One readily obtains U=1
+ 2GaH, + !@9aH,d. 2
(5.6.55)
Analogously, we have +
(1 .e-2iH) = 1 - 2igQHa.
(5.6.56)
After plugging the upper two identities into expression (5.6.52),one obtains 12 Copyright © 1998 IOP Publishing Ltd
462
Ideas and Methods of Supersymmetry and Supergravity
Further, from equation (5.6.51) one has
P=l+AP
Now, making use of equation (5.6.45) gives A&,qq = E~,@YA.F + ~ @ b A g (5.6.59)
1 R = R - -(a2 - 4R)Ag.
2
To finish the calculations, it is necessary to determine the superconnection A&py and the supertorsions and In order to find Antipy,one must calculate carefully the derivatives (5.6.47) and then impose the Einstein supergravity constraints on the covariant derivatives 0,. After doing this, taking (anti)commutators [o, a,} one may obtain explicit expressions for e, and wupy.It is a trivial but tedious task. Omitting the details, we reproduce the final results:
e,
A
Z =G ~
wup,,.
~AA,@~~= ! B ~ B ~ ( , H+ ~ ,G~, , B H ~ , ~ 2
~~ "py
A Wupi =
= wupy
+ AW@y
3 2
- - cs Wugy- AP6M,6
1 8
Wupr
- -(a2- 4R)[9(,Apy,
+ H(,"9pGyp]
Cud= Gad + AG,d
(5.6.60)
Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
463
5.6.5. Topological invariants
As an application of the technique developed above, we show that the superfunctional
s
8 = d8zE-'
{f
+
W1pyW,pv G"G,
+ 2RR ]
(5.6.61)
is invariant with respect to arbitrary variations of the Einstein supergravity prepotentials. The superfunctional 8 turns out to be a supersymmetric generalization of the four-dimensional topological invariant (P - x), where Pis the Pontrjagin invariant and xis the Euler invariant (see subsection 1.6.6). First, we vary the superfunctional
J
11 = d8z(E- l/R)W1'"Wlpy.
(5.6.62)
For this purpose note the identity
s
s
d8z(c?-'/R)dpc = d8z(E-'/R)(1
+ 3a)dpc
G i p c = 0 (5.6.63)
which follows from the relations (5.6.57-59). Then, making use of equation (5.6.60) gives 611 =
s
s
d8z(b- ' / R ) W W a p y- d8z(E- '/R)W"pyW,py
n
I
1
+ -((@R)gp + (G,&@)G,e 4
(5.6.64) - (aaGb)glGb- 3RRGla
i + -81 Gae(GZR+ g Z R )+ -g1e(BZR - g*R) 4 where we have used the relations (5.5.6,8). It is seen that 61, does not involve variation of the chiral compensator, a. Therefore, the superfunctional (5.6.62) Copyright © 1998 IOP Publishing Ltd
464
Ideas and Methods of Supersymmetry and Supergravity
does not depend on the chiral compensator cp. Equivalently, the above observation means that I is invariant under the super Weyl transformations (5.5.13). It is worth pointing out that the right-hand side of equation (5.6.64) is real. Hence, the superfunctional
s
(5.6.65)
I 2 = d8zE-'G'G,.
(5.6.66)
-s
B - 9 = dsz(E-'/R)WaP'jW,p, - d8z(E-'/R)",bi.wubf
does not change under arbitrary variations of the Einstein supergravity prepotentials. This object represents the supersymmetric extension of the Pontrjagin invariant. Secondly, let us vary the superfunctional
s
Making use of the relations (5.6.57, 60) leads to
I
1
5i 2 -* - -gad(a2R- g 2 R ) + - R9,eR 12
3
1 - + -(9dR)9,R
3
Finally, we consider the superfunctional I , =2
s
dsz E - ' R R .
Since in the quantum chiral representation
Copyright © 1998 IOP Publishing Ltd
(5.6.68)
Superspace Geometry of Supergravity
465
making use of the relations (5.6.57,59) leads to n
n
1 2RRGXi - -GXi(6'R + 9 ' R ) 6
2 3
1 + -%Xi(a2R - 9'R) 6
--
- -Rg,+R
(5.6.69)
1 - - -(9&9,R 3
From equations (5.6.64,67,69) we see that 89' = 0. In conclusion, we point out that the superfunctional Sd". E-'(GaG,
+ 2RB)
(5.6.70)
is invariant under the super Weyl transformations (5.5.13), similarly to the previously considered superfunctional (5.6.62).
5.7. Supercurrent and supertrace In general relativity the energy-momentum tensor of a matter dynamical system coupled to a gravity background satisfies equation (1.6.60), owing to the general covariance. Our goal now is to find a superfield generalization of the energy-momentum tensor as well as to look for a supersymmetric version of equation (1.6.60).
5.7.1. Basic construction Consider a theory of matter superfields curved superspace. Its action S = S[x; 91=
J
living on a given (background)
d'z E-'LZ(x; 9)
(5.7.1)
g Abeing the relevant covariant derivatives, is supposed to be invariant under the superspace general coodinate and superlocal Lorentz transformations. In the present section it will be useful for us to understand the action as a superfunctional of the matter superfields and of the background supergravity prepotentials W = WMD, + WabM,,and cp(Bdcp= 0), SCx; 91 = scx; w,401.
+
Copyright © 1998 IOP Publishing Ltd
466
Ideas and Methods of Supersymmetry and Supergravity
Suppose the prepotentials suffer slight disturbances such that the covariant derivatives change in the manner 9 A +V A
= &AMDM
+ -21@ A b c M b c H
= H a g a= H
(5.7.2)
with 9 being as in equation (5.6.51) and An: being determined by supergravity constraints. This change in the supergeometry is accompanied by some disturbance of the matter superfields
x
+
X(V)
= f(x;H,cp)
(5.7.3)
depending on their superfield types. The action changes as follows: S [ x ; g] + S[x(v,; V]
E
S[x; gIH,~ p ] =
s
dsZ E-'LT(x(V); V).
(5.7.4)
In practice, it is useful to transform S[Q~); V] into the 'quantum' chiral representation:
(5.7.5) Let us introduce two superfields
(5.7.6) which will be called the 'supercurrent' and the 'supertrace' of the system, respectively. The supercurrent turns out to be a real vector superfield, since the same is true for Ha. Similarly, since cp is covariantly chiral and scalar, the supertrace represents a covariantly chiral scalar superfield, GdT = 0. Copyright © 1998 IOP Publishing Ltd
(5.7.7)
Superspace Geometry of Supergravity
461
Remark. For calculating the supercurrent and the supertrace, the following variational rules dHb(z') -- 6,6E68(2 - z') 6H"(z)
(5.7.8)
- z')
) ; is In accordance with the results of subsections 5.6.2 and 5.6.3, S [ X ( ~ V] invariant under the transformations de--2iH
~ - 2 i-H,-2iHA
=~
(5.7.9) where A =AA9A
+ A'pM,, + A i 6 M i ~ 1 A, = - -9 2La, 4
Aui = (cP),~A, = -2iBiL, ~
~ = 1- ~QIB, )
Ai = Ai
+ O(H)
+ -21 A ~ ( & G ~ )A,@ = Aap+ o(H).
(5.7.10)
Here L, is an unconstrained spinor superfield. The transformation law for Had can be rewritten in the form 6H,B = BiL, - 9,Ei
+ O(H)
(5.7.11)
where O(H) denotes all H-dependent terms. Now, let us choose vanishing quantum superfields Ha = 0 In this case,
fp=
1.
xCv) = x and the transformation laws (5.7.9) reduce to
dH,i = GiL, - 9,I& S:(v, = Ax 1 1 6p3= -(a2- 4R)9,La =S 6~ = -(B2 - 4R)9,LU. 4 12
(5.7.12)
Further, the condition that S[qv);V] is invariant under the transformations (5.7.9) now reads
(5.7.13) Copyright © 1998 IOP Publishing Ltd
468
Itleas una' Methods of Supersymmetry and Supergravity
Suppose the matter superfields satisfy their dynamical equations
sscx;91 = 0. 6x Then, equations (5.7.12)and (5.7.13)say
Due to the arbitrariness of L", this relation is equivalent to the equation
@Tori=
2 3
- -9@T.
(5.7.14)
It is this equation which expresses the condition of invariance of the action superfunctional S[x; 91 under superspace general coordinate and superlocal as a supersymmetric Lorentz transformations. One can look on equation (5.7.14) generalization of the conservation law (1.6.60).
Remark. There is a simple prescription for calculating the supertrace: considering the action S [ x ; 91 as a superfunctional of the supergravity 1= S [ x ; W ,(PI,the supertrace proves to be given by prepotentials, S [ x ; 9
6
T=-S[~;~,e'~]Iu=o 60
gda=O
(5.7.15)
with the variational derivative 6/60 being defined as follows
6a(z')
-=
60(z)
1 - -(g2- 4R)Ed8(z - 2'). 4
(5.7.16)
5.7.2. The relation with ordinary currents Here we would like to discuss the component content of the supercurrent and the supertrace. As we know., the invariance with respect to the supergravity gauge group can be fixed by imposing the gravitational superfield gauge (5.4.38).In this gauge, the matter action becomes a superfunctional of the gravitational superfield %"(x, 8, e) and the chiral compensator @(x,e), S[x;91 = S[x; %", 41. Clearly, making a slight disturbance of the gravitational superfield %" + #" + 6%" will induce some change in the covariant derivatives. This change, supplemented by an auxiliary 6#"-dependent general coordinate and superlocal Lorentz transformation, can be represented in the form (5.7.2).The two variations 6%" and Ha are in one-to-one correspondence. However, the former transforms in a nonlinear, complicated way, while the latter possesses the vector superfield transformation law. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravit-v
469
Further, instead of considering Tad,one can introduce
6 S[x; 3rm, 41 (5.7.17) 6s" which will be called the 'non-covariant supercurrent' of the system. The two are uniquely connected to each other but, in contrast to T,, objects T, and .Tm F,,,is characterized by a nonlinear complicated transformation law. We can look on the supercurrent T, as a covariantized form of Fm. Expanding 9,in a power series in 8, 8, we obtain 9"'s-
9,= J i - 2i0'QmU+ 2i&Qmd+ 280aBT,, where
6S 6A"'
6S Qm-6Ym,
J 5 ="'
a--
+ .. .
6S T," = - de,"
(5.7.18)
(5.7.19)
here e,"', Y,"' and A"' are the component fields of &" in the Wess-Zumino gauge (5.1.17). Obviously, T," is the energy-momentum tensor. Next, it was U A"' are gauge fields for local supersymmetry shown in section 5.1 that Y m and and local chiral transformations, respectively. Therefore, Q," is the spinor supersymmetry current and J I is the axial or y,-current. As a result, the supercurrent contains the multiplet of ordinary currents, including the energy-momentum tensor. The leading term in the expansion of the supertrace in a power series in 8 is
T(x,8) = 3G(x) + . . .
G
= 6S/6B
(5.7.20)
where B is the complex scalar field appearing in the power series expansion of (p3 (see equations (5.1.41)).Higher-order component fields in the expansions (5.7.18) and (5.7.20) turn out to be expressed through the currents (5.7.19) and the field G. To argue this statement, let us analyse the conservation law (5.7.14) in a flat superspace. 5.7.3. The supercurrent and the supertrace in flat superspace It is almost obvious that when background superspace is flat the supercurrents T , and F mcoincide. Then, using equation (5.7.18), the leading component ~ T ,given by fields of Tad= ( c T ~ ) " are Tordl= J &
DaTpbl = -2iQp~," = -2i(ab)pbQba
(5.7.21) Equation (5.7.14) takes the form 2 DdTad = - - D , T 3 Copyright © 1998 IOP Publishing Ltd
D,T= 0.
(5.7.22)
470
Ideus und Mrrhods of Supersymmetry and Supergravity
Recalling basic properties of the flat covariant derivatives, one can deduce from equation (5.7.22)some useful consequences: DBDj'T . =
1
- -6P1D2T
3
zz
1
DBD"T,, = -6b6D2T 3
(5.7.23~)
From relations (5.7.22)and (5.7.23~1) we obtain
D,TI = -3i(o"Q,),
Therefore, the supertrace is expanded in a power series in 0 as follows: 1
-T=
3
1 G - i0a(8QJ, - -02(T," - iallJ2). 4
(5.7.25)
We see that the supertrace contains the trace of the energy-momentum tensor, the trace of the supersymmetry current (."Go), and the axial current divergence a'J,5. On these grounds, the supertrace can be treated as the supersymmetric extension of the trace of the energy-momentum tensor. The supercurrent component fields, omitted in the expansion (5.7.18), can be easily found with the help of equations (5.7.23,24).The results are 1 4
- - D2TUB I = 2id,,G 1
- -D
4
~ T~~ DI = ~ 2ia,dr(obQb)p
(5.7.26)
As we see, these components are expressed via those presented in equations (5.7.18, 19). The conservation law (5.7.22) encodes information about the symmetry structure of the energy-momentum tensor as well as about divergences of the currents under consideration. It foliows from equation (5.7.23a)that
D,DBTpb
+ DpDbT,b
=0
(5.7.27)
a"T, = -i(D2T- D2T). 6
(5.7.28)
1
Now, taking the space projection of equation (5.7.27), one readily obtains Copyright © 1998 IOP Publishing Ltd
471
Superspace Geometry of Supergravity
the relations
P f i J j f i = -d",a'J~ (5.7.29)
Tp$B= -dJT:.
The second relation shows that the energy-momentum tensor is symmetric, =
Tab
(5.7.30)
Tba
Further, equation (5.7.28) can be used to find aaQap =
- da(ua6'Q' )B (5.7.31)
Therefore, the currents Qogand Tab,obtained in accordance with rule (5.7.19), are not conserved. But the improved currents (5.7.32) turn out to be conserved.
5.7.4. Super Weyl invariant models Consider a dynamical system such that its action superfunctional S[x; 91 = S[x; W ,cp] is invariant under super Weyl transformations of the general form
x~ = e - d ( + ~-~
x
w"' = W
cp' = e"cp
$&a = 0 (5.7.33)
with d ( + , and d ( - ) being numbers (fixed for the system under consideration) and a(z) being an arbitrary covariantly chiral scalar parameter. In this case the dynamical system is called a 'super Weyl invariant model'. Recalling the definition of supertrace (5.7.15), we see that super Weyl invariance requires the supertrace to vanish when imposing the matter dynamical equations, il1s/sx
=0
(5.7.34)
= 0.
Then, the conservation law (5.7.14) takes the form (5.7.35)
D'Tu, = 0.
At the component level, equation (5.7.34) leads to the requirements (a"Oa),=O
T:=O
in accordance with equations (5.7.26). Copyright © 1998 IOP Publishing Ltd
PJ: = O
(5.7.36)
472
Idetis
tinti
Methods of Supersyinnzetry und Supergruvity
5.7.5 Example To obtain some experience in the calculation of the supercurrent and the supertrace, we are going to evaluate these objects in the case of a theory of a covariantly chiral scalar superfield x, = 0, and its conjugate f The action superfunctional reads
adx
1
S[z, ic; 91= d 8 z
E - 1 7AX
+ { j d 8 ~ ( E - 1 / R ) 2 ' c (+~ C.C.] )
(5.7.37) L2+Yc(%) =0 Y Cbeing a chiral superpotential. This action generalizes the Wess-Zumino model action (3.2.8) to the case of a curved superspace. It is worth pointing out that if one represents the chiral compensator cp and the matter superfield z in the form (5.7.38)
then the chiral superpotential term can be rewritten as
s
s
d8Z ( E - ' / R ) P C ( ~=) d6z @'YC(9.
(5.7.39)
In accordance with the prescription of subsection 5.7.1, to find the supercurrent and the supertrace one must calculate the variation *
where SCi(V,!
6s = S [ i ( O , , i(V$ QI Cv,; QI = [d".
8-1Gv)i(O)
-
+
- S[x,
{
z91
pz(2-1/R)Yc(:d
+c.c.)
CO, = e--ZiHTA. f(v7)= x Using equations (5.6.57,63), to first order in quantum superfields H", B and 0 we have 1 + --oT+ --aT R R where
--I
(5.7.40)
and 1
-
+
(5.7.41) T = - - ~ ( 2-' 4R)F 3YC(x). 4 Remarkably, the supercurrent does not involve YC(x). This follows from the fact that the superpotential term (5.7.39) depends only on the chiral compensator. Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
473
Let us clarify whether or not the supertrace vanishes on-shell. Due to the variational rule
the dynamical equation for
is given by
1 4
- -(g2- 4R)i
+ Y',.(x)=O.
(5.7.42)
The relations (5.7.41) and (5.7.42) show that the supertrace vanishes if and only if
(5.7.43) with g being a constant. In this case the action turns out to be invariant under the super Weyl transformations
x' = e-"%
W' = W
cp' = eacp
B60 = 0.
(5.7.44)
The reader can explicitly check that the supercurrent (5.7.40) and the supertrace (5.7.41)satisfy the conservation law (5.7.14)under imposition of the equation of motion (5.7.42).
5.8. Supergravity in components
In this section we intend to describe the technique of passing from superfields (supergravity or matter) to component fields. Explicit formulae expressing the component fields of the supertorsion tensor R, G, and WzP,via the fields of the Einstein supergravity multiplet (see Section 5.1) will be given. It will be shown how to read off the action functional from the action superfunctional. Local supersymmetry transformation laws of matter component field will be discussed. Throughout this section, we work in the gravitational superfield gauge (5.4.38),in which the covariant derivatives are built from the gravitational ' (x, 6) = @(x + i H , 8) superfield 2'"'and the chiral compensator cp(z) = e@@ (and conjugate @(z)= ew $(x, 8) = $(x - i H , @) by means of equations (5.4.4347) and (5.5.4). As was shown in Section 5.1, the residual gauge invariance can be used to set the Wess-Zumino gauge
Z m= 8@8eam+ i ~ 6 z Y " ' ,- i628iT'"k + O2pA"
O3 = e-'{ 1 - 2i8a,Ya + 0 2 B } G3 = e-'{1 - 2 i 8 5 , ~+~BZB} Copyright © 1998 IOP Publishing Ltd
(5.8.1)
474
Ideas and Methods of Supersymmetry and Supergravity
where y o z
= emu'+",
= emoTmd,
(5,8-2)
We take the Wess-Zumino gauge as the starting point for investigating the supergravity component structure. 5.8.1. Space projections of covariant derivatives In flat global superspace, superfield component fields were defined using the notion of space projection. It seems reasonable to follow the same line when working in a curved superspace. By definition, the space projection VI of a superfield V ( x ,8,Q coincides with the zero-order term in the power series expansion of I/ in e, 8: V I=
v(x,e = 0, B = 0).
one can also define space projections of differential operators. Given a first-order differential operator of the general structure
x = XM(z)dM+ Xab(z)M,b its space projection is taken to be XI
E
xMld,&f + XUbIMa&
It is clear how to extend this definition to the case of operators of second and higher orders. We would like to point out, however, that when calculating space projections one should be careful and remember that the expression
(xv)l= X M I ( a M v ) I + XUbIMabVI does not coincide with
+
X I v I = xmlamv/l xublkf&Vl.
Now, we proceed to find the space projections of the covariant derivatives
in the Wess-Zumino gauges (5.8.1). For this purpose let us obtain some auxiliary relations. Owing to the Wess-Zumino gauge, the semi-covariant supervierbein 8, (5.4.43) is characterized by the projections
A
8,1 =a, 821 = - p a ,
-[E,, E&]I = - a,a, 1 2
Copyright © 1998 IOP Publishing Ltd
-
-ai - -. = -a#
E&l =
-
(5.8.3)
Superspace Geometry of Supergravity
475
and
(5.8.4) We see that Ea/ coincides with the ordinary vierbein. Similarly, considering the superfield anholonomy coefficients ea; appearing in the commutator
ea:/
[Ea, &I = cab%
proves to coincide with the ordinary anholonomy coefficients WO:,
Lea, = %abeec since I?, does not involve spinor partial derivatives. The above identities make it possible to rewrite the expansions (5.8.1) in the manner =E
,PI
. 1 [Ea, &]YPI 2
--
=
=0
= (off)a8eam
(5.8.54 1 32
4
- {E2,E2}3Eaml= Am and
(5.8.5b)
and similarly for @. Among the relations (5.8.5a), only the latter requires comment: due to equations (5.8.3) and (5.4.43), we have A
E 2 E 2 z m (= - & a " E 2 ~ mI = - a,Bfi(a, =
16A"
+ 2i(offB),ea)au~m I
+4i(~')zfiea~u~a~m~
which means
(5.8.6)
Copyright © 1998 IOP Publishing Ltd
476
Ideas and Methods of Supersymmetry und Supergravity
To derive the relations (5.8.5b), it is sufficient to note that in the Wess-Zumino gauge cp is of the form 1 1 + iXmd, - -Xmxna,d,
2
and then to use the expansion of (a3 (5.8.1). Remark. Using equations (5.8.6), one can prove the relations
(5.8-7) Now all the necesary ingredients are at our disposal in order to evaluate the projections 9,l and a61.It is worth recalling that the spinor covariant derivatives read QZ
= FE,
+ R,pI.MP? + R,fiiM#.i
9, = S K + QidrByMB.i + 0. ZBY '.jqS.i Here the connection superfields are built from the F and P and anholonomy (5.4.45) by the rule (5.4.46). In accordance with coefficients (3,;and relations (5.4.47)and (5.5.4),F and P have the form
ed;
F = ( ~ ' ' * qdet-''3(1 -~ - idX)det''6(1
+ idX)det-"6(8,")
det'I6(l - idX)det-"3(1
+ id$P)det-"6(E,")
F= where
(1
+ id#),"
= 6,"
1
Ea" = - -(c?,)"c'[E,, 4
+ id,Xm A
Ed]$P".
.,
Therefore, one must calculate FI, 8,FI, EdFl and eat and their conjugate. Using identities (5.8.4,5) and expressions (5.4.45)one obtains
(5.8.8) Here we have converted the gravitino vector index into a pair of spinor indices, =
ad^",
qad,fi
= (oa)miqafi
(5.8.9)
Further, one readily obtains det (E,")I = e Copyright © 1998 IOP Publishing Ltd
Eadet (8,"')l= ie(gaTm)a.
(5.8.10)
477
Superspace Geometry of Supergravity Along with the relations (5.8.5b) these mean FI = El = 1
(5.8.11)
Now, in accordance with definitions (5.4.46), the space projections of the spinor superconnections are
2
and
1 n& I = -(E#
2
i nep,/ = - -(y!?a,y
Then, since FI = 1 and
I = a,,
(5.8.12a)
+ Tpj,J) )qy + E d j Y
n&I = +Pas,.!
(5.8.12b)
+ Y',B,B).
2 one can immediately arrive at the final results (5.8.13)
It remains to determine dol.Using the explicit form of the supervierbein vector components (5.4.18),it follows from the above identities that (5.8.14) Now, we find it convenient to represent gal as follows
1 2
gal = Va+-Ya@2p1
1 + -Y,&VI 2
(5.8.15)
where V, are ordinary space-time covariant derivatives, 1 va = ea + -WabCMbc 2
Here the Lorentz spin connection Wabc
= nabcl
W,bc
(5.8.16)
is related to Rabc/by the rule
i + -(y,abTc - y,acTb + 2
ybgc\TIa
- \Ycabq,).
We shall determine the connection WQbc(and hence nabcl) in subsection 5.8.3. Copyright © 1998 IOP Publishing Ltd
478
ideas und Methods of Supersymmetry and Supergravity
5.8.2. Space projections of R , R and G, As the next step, we evaluate the projections of the supertorsion tensors R , R and G,. Recall that R is given by 1
R = - -82(F2). 4
Then, making use of the explicit form of F gives
-
RI =
1
- -q-21g2((pdet-"3(g,m))/.
4
After a short calculation, one obtains 1
RI=-B 3 (58.17~)
hence
B=B
1 1 + -Y'"a,c?bYb + -Yaya.
2
(5.8.17b)
2
We see that 3RI and 3R 1 coincide with the scalar field B and B, respectively, from the Einstein supergravity multiplet. To find Gal, we note the relations
-iG, = Tp! = Cpap
(5.8.18)
TpaY being the components of the supertorsion, p p ,
gal= T $ g Y
+ ..
,
and Cp,Y being the supervierbein's anholonomy coefficients, [Ep, Ea1 = Cp,YEy +
In deriving relations (5.8.18), we have used the commutation relations (5.5.6) and the expressions for the supertorsion components (5.2.47). Expanding the commutator Fpp, FF$,
i + -F(l.?$)& 2
i + -Q,iF8, + -L&,YFl?, 2 2 1
Copyright © 1998 IOP Publishing Ltd
1
+ LF(&F)I?p 2
Superspace Geometry of Supergravity
479
one arrives at
All the terms, except the last two, have been calculated above. When evaluating E,&FI and l?&Fn,%/')l,the relations (5.8.6,7) prove to be helpful. After some calculation, one arrives at
4
Gal = - A a
3
1 1 A" = A" + -EabCdWbcd - - ( Y a O b T b 8 4 1 2
- -ybaaqb
+ YbabT',)
(5.8.19)
+ -81& a b c d y b b c q d
We see that ZG, coincides with the vector field from the Einstein supergravity multiplet.
5.8.3.Basic construction The relations (5.8.13,14,17,19) turn out to be sufficient to determine all the other quantities: the connection a&, the projections W,prI, BaRI, gaGaI, @RI and so on. This is done as follows. We are going to evaluate the space projection of commutator [9ad, 9,~] in two different ways: first, using the representation (5.8.15) and secondly, making use of the commutation relations (5.5.7).In the former case, we have
Copyright © 1998 IOP Publishing Ltd
480
Ideas and Methods of' Supersymmetry and Supergravity
Now, introducing the torsion and curvature tensors associated with the derivatives V,,
and applying the (anti)commutation relations (5.5.6), the upper expression takes the form
+ 8'1- and Mi"'-terms. Copyright © 1998 IOP Publishing Ltd
Superspuce Geometry of Supergravity
48 1
(5.8.22)
Now let us analyse the relations (5.8.21) and (5.8.22). Since the V-terms in relations (5.8.21)snd (5.8.22) must coincide, the torsion YabC is determined in terms of the gravitino and the vector field A , as follows:
Therefore, the V-connection reads
where l-abc
= %ubc -
(5.8.25)
It is seen that the connection depends on the gravitino and the vector field A,. Extracting from wabcthe term containing Ad gives the gravitino-dependent 'Y). The corresponding covariant derivatives connection
PQ= e,
1 + -a~bc(e, 'Y)Mb' 2
(5.8.26) will often be used later. To simplify subsequent work, it is convenient to introduce the gravitino Copyright © 1998 IOP Publishing Ltd
482
Ideas and Methods of Supersymmetry and Supergravity
field strengths:
and their spinor analogues
In terms of the derivatives yaby
= vayby
a,, then y a b y reads
- v b y a y - Fabeycy
2i
+ -Ac{(gcayb)y 3
- (bcbya)y} (5.8.29~)
or
1 amyny - anymy + -21 omcd(dd~,)y - - o,,,d(~~~yr~)~ 2
where Qmcd = emauac,,(e,Y).It is worth pointing out that T a b , , and T a b ’ differ from similar quantities which arose in the transformation laws (5.1.47) only by the presence of A-dependent terms. Let us return to the analysis of relations (5.8.21) and (5.8.22). Setting the g>,I-termsin both expressions to be equal, one obtains 1
-~&3,R(dYg 2 = E@,,,
+ E ~ B W , J-( -21 E , ~ ~ ~ G ~ B I
+ E,flY,B.
Copyright © 1998 IOP Publishing Ltd
- :B(T,d,@Yg - YpB,e&J. 6
Superspace Geometry of Supergravity
483
We decompose this identity into irreducible components resulting in 2 3
4
+3
'+ 'BTab! 3
= -y 3 4
(5.8.30a)
G(,.G/'I),I= - 2y.d b + - -
p%b, '
(5.8.30b)
1 $ %'J+
(5.8.30~)
= - -(oabvab)a
W,B;.I =
'B(oaqa)x
y?y,a
+ y,a,p).
Now, one can easily calculate the projection BaGpp(,using the Bianchi = 9,R. Next, conjugating the relations (5.8.30),one obtains identity giGZd the projections 9,,GB!1 and ",p91. At present, it remains to investigate the MY6-terms in expressions (5.8.21) and (5.8.22). First, however, we would like to discuss the properties of the curvature tensor in the case of non-zero torsion. 5.8.4. Algebraic structure of the curvature with torsion As is well known, in the torsion-free case the curvature tensor is characterized by the following algebraic properties:
- gbacd
=
gabcd
+ gbcad + gcabd gabEd
=
- gabdc
gabcd
=
(5.8.31a) (5.8.31b) (5.8.31 c )
= %cdab*
The first property follows directly from the definition of the curvature. The second relation represents the torsion-free Bianchi identities. Finally, the third property is a direct consequence of the second one. In the case of non-vanishing torsion, the Bianchi identities are modified in the manner
therefore neither equation (5.8.31b)nor equation ( 5 . 8 . 3 1 ~hold. ) The curvature components induced by the torsion can easily be determined. Let us decompose the curvature into two parts: gabcd
=
("aabcd + (A)gabcd
1 (",aabcd = Tgabcd (A)&
1 2
= -gabcd
Copyright © 1998 IOP Publishing Ltd
1
+ iWcdab 1 2
- -2cdob
= (S'Wcdab
=
-(")acdab.
484
Iderrs and Methods of Supersymmetry and SupergraGity
Since ( S ) 8 a b c d is symmetric with respect to transposition of the first and second pairs of indices, one can readily see that (S)gabcd
+
(S)c@?bcad
+
("gcabd
is totally antisymmetric, therefore, (5.8.33) being a curvature tensor under the torsion-free constraints (5.8.3la-c). (A).%abcd, antisymmetric with respect to transposition of the first and second pairs of indices, it contains two irreducible components: antisymmetric
egabcd
As for
(A)t%cacb
= 390, -i
g b a
(5.8.34)
where (5.8.35)
g a b E gcacb
and symmetric traceless p d e ( A ) g b + & b c d e ( A ) gcde(l cde
= EacdegCd:
+ EbCdegcdeQ + -21~ ~ ~ ~ ~ ~ " f a ~ d , p ( 5 . 8 . 3 6 )
It should be stressed that the tensors (5.8.34) and (5.8.36) and &"cdgpbcd are generated by the torsion, owing the equation (5.8.32). For further analysis, it is useful to consider the curvature tensor with vector indices converted into spinor indices, gxci,pb,.id,6d
= ('"),d(ob)p~ac)u.i(ad)a6Wabcd = 2E&%a,pfi,ys
+ 2&ys&,pfi,.jd
where Wxd,Bfi,rs proves to have the general form g x a , p # , ; d = Ed&?@
+ &,pE;Isup
+ &a&~$ps + ~ p , S ~+s &,gSpy + E ~ S , ? )
(5.8.37)
+ Ed&&xi.&Ba + &xs&B;,)F with Cxp.ig and S,p being totally symmetric and E y g d ~symmetric in its undotted and, independently, its dotted indices. We would like to recall that in the torsion-free case the scalar F and the symmetric traceless tensor
were real and S x p was absent (see equations 1.6.65,66)). Another situation Copyright © 1998 IOP Publishing Ltd
485
Superspace Geometry of Supergravity
occurs in the general case. One can explicitly check the following relations:
-(
i&abcdgabcd
F= 1 W+ 12 2 i E,, = - Bab + - & 4
{
acde
gcde b + Bba
i
+ -&bcdeBCdea 2 (5.8.39)
where d = vabWabis the scalar curvature. Now, we are ready to complete the investigation of relations (5.8.21) and (5.8.22).
a'$
ace9cyGa,b, and 9(a WBra) Using the Bianchi identities (5.5.8), the MYa-terms in equation (5.8.22) can be rewritten as
5.8.5. Space projections of
where dotted and undotted indices in ace9(,,Ga,b,(and everywhere below) are symmetrized independently. Further, we insert the expressions for 9 z d , ~ b , (5.8.37) pS in (5.8.21).Now, setting the MY'-terms in (5.8.21)and (5.8.22) to be equal leads to >
+ 'BB 9
2 9
- -B(ya(T,dbyb + \ya\y,) (5.8.40~)
Copyright © 1998 IOP Publishing Ltd
486
ldeas and Methods of Supersymmetry and Supergravity
When deriving the last relation, we have used equations (5.8.30b,c). Let us discuss the results obtained. From equation (5.8.40~)we see that the P-component of the supertorsion R (and the 02-component of R) contains the scalar curvature. Therefore, one can look on R and R as supersymmetric extensions of the scalar curvature. Further, it follows from equation (4.8.40b)that the Oab@-componentof the supertorsion G, contains the traceless Ricci tensor
in accordance with equation (5.8.39). Therefore, G, represents the supersymmetric extension of the Ricci tensor. Finally, equation (5.8.40~)shows that the leading term in the 6'-component of the supertorsion Wpy6coincides with the anti-self-dual part of the Weyl tensor (see equation (1.6.66)).On these grounds, WPt6can be called the 'super Weyl tensor'. Above we have evaluated only a special set of supertorsion space projections. Beautifully, these results (combined with the commutation relations (5.5.6,7)and the Bianchi identities (5.5.8))turn out to be sufficient to determine Goi9yG6,yand 9,WgydI and all the other space projections, such as B2G,&(,g2W,ptI and so on. For example, we have B2G,b/ = g g 9 a G g d l - 9,@Gge/ =
{@, ga)Gg&I- 293fiGgdI
+
= 4R(M,gGBel- 2B,G&l = 6RJGaeI 4i9,eRI
where we have used equation (5.8.15). As a result, we have at our disposal the full component description of Einstein supergravity.
5.8.6. Component Jields and local supersymmetry transformation laws
Our next goal is to give a proper definition of the component fields of tensor superfields and to discuss their transformation laws. We begin by recalling that in the gravitational superfield gauge (5.4.38) we stay with the restricted set of gauge transformations of the general structure (5.4.49).The stronger gauge fixing (5.8.1) imposes severe constraints on the superfield parameters Am(x,6) and P ( x , 0). Namely, the only transformations preserving the Wess-Zumino gauge are: the general coordinate transformations described by P(X,
e) = ~
Copyright © 1998 IOP Publishing Ltd
( x=) ~ ( x ) ,I'= o
(5.8.41)
Superspace Geometry of Supergravity
the local Lorentz transformations,
P
=o
A ~ ( xe) , = KyX)ep
481
(5.8.42)
and the local supersymmetry transformations, P(x, 0) = 2iOo"Ce," - 2 O 2 ~ P
i
P(X,
e) = E ~ X +) -iz(qea + Ka&)ea + 02-terms
(5.8.43)
2
with a(€) and K'&E) being given in equations (5.1.42)[email protected], respectively. Consider a tensor superfield V (indices are suppressed). In accordance with relations (5.4.49), its space projection VI transforms as follows 1
6VI = -(%"I 2
+ X"I)a, + n'la,vl + X,laL'vl
+ a,,lp,IM"BVI + a(BAb,(R%q. In the case of the general coordinate and local Lorentz transformations, this law is reduced to 6b+kT/I
= b"(x)d,VI
1 + -Kub(X)M&1/1. 2
(5.8.44)
Therefore, VI is a tensor field. Similarly, since the covariant derivatives g A move every tensor superfield to a tensor one, each of the fields g A V I ,gAgBVJ, . . . , is a tensor. Under the local supersymmetry transformations (5.8.43),VI changes by
Now, comparing the explicit form of K,&E) and If,&€) with the projections of the spinor superconnections (5.8.12)and using (5.8.13), one obtains
+
6,VI = Ea(x).$BUVI &(X)@VI.
(5.8.45)
This marvellous result means that the set of tensor fields {VI,gAVI,9A9'BVl,...} is closed with respect to the supersymmetry transformations, and the relevant representation is linear, owing to the %transformation law (5.4.49). It can be readily seen that among the fields {VI,gAVI,QAgBV1,.. .} only a finite subset is functionally independent. In particular, if V is an urlt,unstrained superfield, such a subset can be chosen as follows
Copyright © 1998 IOP Publishing Ltd
488
k k u s und Methods of' Supersymmetry and Supergravity
Obviously, these fields are in one-to-one correspondence with those arising in the expansion of V in a power series in 8 and B. Therefore, we can identify tensor fields (5.8.46)with the component fields of V . In the case of a covariantly chiral tensor superfield x, Gix = 0, its component fields can be taken as
As an example, let us find how the local supersymmetry transformations act on the component fields of a covariantly chiral scalar superfield x.Defining the component fields by
C ( x )= X I
i,(x) = gXXI
1 F(x) = - - 9 2 x 1 4
we obtain
= 2€,F
+ 2iC'VmiC + iF'Y,i,Pi.a
1 6,F = --(E%%, 4
+ Cbga)&xI= -€'RIBXx(- !?b[92,g5]xl 4
=
1 - -BEE"+ C'GXilWx/ + i&3XiWxj 3
=
1 4 - - BEJ.- - A"Ao,C - i(V"i)o,? - iY'"o,CF
3
3
where equations (5.5.9b, d ) have been used. We hope the reader has already been convinced that finding local supersymmetry transformation laws is a simple procedure. Remark. The transformation laws (5.1.48)can easily be reproduced utilizing the fact that the fields B and A, represent the space projections of tensor superfields R and G,, respectively (and keeping in mind the difference between the definitions of gravitino field strengths (5.1.49) and (5.8.27)).
5.8.7. From superfield action to component action Before we conclude, one more important question must be discussed - how Copyright © 1998 IOP Publishing Ltd
Superspace Geometry of Supergravity
489
to represent an action superfunctional of the general structure S=
s
d8zE-'9
in terms of the relevant component fields. Owing to the well-known identity
it is sufficient to solve the problem for chiral action superfunctionals of the type
s
a&,
S, = d'z-9,
ER1
= 0.
(5.8.47)
Recall that one can equivalently write S , as
s
S, = d 6 z Q 3 P c
where LfCand its chiral transform gauge (5.8.1) in the manner
2,are connected
(5.8.48) in the Wess-Zumino
Next, performing the integral over 8 reduces S , to
's
S , = - d4xdc(da(Q38,). 4
Now, since E2 I = -dud, and &" in the form
s
- 8,
we can rewrite the last expression
S, = d4x1 - !&'fi,(q'UJ}l 4
O n the other hand, we have the simple relations gadPC I = FI E&
Copyright © 1998 IOP Publishing Ltd
I =E,9, 1
490
Ideas and Methods of Supersymmetry and Supergravity
since FI = 1. Using equations (5.8.11, 12), one obtains the relation
which must be inserted in expression (5.8.49). Finally, after using equations (5.8.5b), S , takes the form
(5.8.50)
This is the working formula for calculating the action functional.
Copyright © 1998 IOP Publishing Ltd
6 Dynamics in Supergravity To invest existence with a stately air Needs but to remember That the acorn there Is the egg of forests For the upper air! Emily Dickinson: To Venerate the Simple Days
6.1. Pure supergravity dynamics We have already succeeded in developing a deep understanding of curved superspace geometry. Naturally, the formalism developed inevitably leads those familiar with Einstein general relativity to new hypothetical worlds-supersymmetric universes. By a supersymmetric universe we shall understand a curved superspace, empty or inhabited by some matter tensor superfields denoted Y, , with a joint evolution law (action principle). It is reasonable to take an action superfunctional determining physically admissible supergeometries and dynamical matter superfield histories in the form = sSG[g.41
+ s M k 9.41
where S,, is a pure supergravity action, depending on the Einstein supergravity prepotentials only, and SM is a matter action. Both terms in S are required to be invariant under the general coordinate transformation supergroup and the superlocal Lorentz group. In this section we consider the simplest candidates for the role of pure supergravity action. 49 1 Copyright © 1998 IOP Publishing Ltd
492
ideas and Methods of Supersymmetry and Supergravity
6 . I . I. Einstein supergravity action superfunctional We begin by finding a supergravity action such that the corresponding dynamical equations reduce to ordinary Einstein equations after switching off all the fields, except the vierbein, of the Einstein supergravity multiplet. In other words, the dynamical equation should admit the following particular solution
Ym"=Am=B=O
gab(e)=O
gQb(e)being the Ricci tensor constructed from the torsion-free covariant derivatives. This implies that the purely vierbein-dependent part of supergravity action must coincide with the ordinary gravity action S,=?
2ti
Sd"x e-'92(e)
(6.1.1)
where ti is the graviational coupling constant. It seems natural to look for a supergravity action in the class of superfunctionals of the general structure (6.1.2)
B+Yc= 0 where Yc is a covariantly chiral scalar depending polynomially on the supertorsion and its covariant derivatives up to a finite order.
Remark. Recall that owing to the identity
each integral over R4I4 can be reduced to a chiral-like integral. Let us show, guided by considerations of dimension, that the supergravity action can be restored uniquely modulo a constant. The dimensions of quantities appearing in (6.1.2) are [1/~']=2
[daz]=-2
[E]=O
[R]=l.
Since the action is dimensionless, we must have
[Yc]= 1. Next, we suppose that Zchas no explicit dependence on the gravitational coupling constant. Then, by virtue of the relations [Go] = 1
[WZp,] = 3/2
[gATBCD]
2 3/2
which follow from the commutation relations (5.5.6), Zcmay be at most a linear combination of R and R. Finally, the requirement of chirality means Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
-
493
Yc R. We therefore obtain SS G -- - KZ
j
d'2E-l.
(6.1.3)
The normalization constant ( - 3) is needed to produce the correct vierbein contribution (6.1.1) at the component level (see below). As may be seen, the supergravity action has a simple geometrical interpretation. Namely, SsGis proportional to the supervolume of the curved superspace. Using relations (5.4.19) and (5.5.4), the action can be rewritten in terms of the prepotentials as follows: (6.1.4) In the chiral representation, the action reads (6.1.5) and this form of the supergravity action is just like the gravity action (6.1.1). 6 .I .2. Supergravity dynamical equations To vary the supergravity action, we follow the prescription of background-' quantum splitting described in Section 5.6 (see also subsection 5.7.1). Making use of equation (5.6.57) gives = sSG[vAl = - - j3d s r K2
- sSG[9Al E-1{~+b--G 2 a H a + - [18 u , ~ ' ] H , j - i 9 W a 3 2
where we have omitted total derivative terms. Recalling the variational rules (5.7.Q with rp = 1 + o being the quantum chiral compensator, one obtains (6.1.7) Therefore, the supergravity dynamical equations look like Ga=O
R=O
(6.1.8)
We see that on-shell, only the super Weyl tensor WaP, remains non-vanishing, and it satisfies the equation
g' W.p, = 0 in accordance with the Bianchi indentities (5.5.8). Copyright © 1998 IOP Publishing Ltd
(6.1.9)
494
Ideas and Methods of Supersymmetry and Supergravity
6.1.3. Einstein supergravity action functional
We would like to describe how the action (6.1.3)looks in terms of components. For this purpose we apply the component technique developed in Section 5.8. To convert S,, into components, we make use of the reduction formula (5.8.50):
'l
SS G -- --
E-' d8z-R=
K2
2 Next, recalling that RI =*Band making use ofequation (5.8.40~) one obtains
1- 1- - - B(y'"8,ObT' - 'Yayy,) (P8,YgrRl --
+
9
2
-I 4
6
The last two terms can be written, with the help of equations (5.8.30a,b), in the form 1 -6
2
with Y c d r being the gravitino field strength (5.8.27). Therefore, the action functional reads
I
S ~ ~ = 1? { d ' x e - ' { Z&?-q&abcd&?~bcd+'&"bcdT'"~bycd--BB 1 i . (6.1.10) K 4 3l Note that the second term in S,, is purely imaginary. In addition, the third term also has an imaginary part. These constributions must form a total derivative, since the original action (6.1.3)is real by construction. Hence, we can rewrite SSG in the explicitly real form: 1 Ss~=,[d'x K
I
e-1{jlR-iBB+-&,bcd~~,8bycd-y,0b\T.'cd) 1 11 . (6.1.11) 4
The action has hidden dependence on the vector field Ad via the connection (5.8.24) and the gravitino field strengths (see expressions (5.8.29)). Extracting all A-dependent terms, one obtains
o,bc
SSG=
4 K
Id'.
e-
(6.1.12) Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
495
Here 9 ( e ,Y) = is the scalar curvature generated by the covariant derivatives 0,(5.8.26), and qaby
= PaY',;,- ? b y a y
-T a b c y c ; ' .
(6.1.13)
From equation (6.1.12) we see the part of SSG determined by the vierbein simply coincides with the gravitational action (6.1.1). It is seen also that SsG leads to trivial dynamics for the fields B and A,, which vanish on-shell,
B = A, = 0.
(6.1.14)
On these grounds, B and A, are said to be the 'supergravity auxiliary fields'.
As for the last term in (6.1.12), it turns out to be a curved space nonlinear
generalization of the Rarita-Schwinger action of a massless spin-2 particle (1.8.42). Therefore, we can consider the supergravity theory as describing interacting massless spin-2 and spin-; particles. It should be noted that there exists a wide class of models of interacting spin-2 and spin-3 fields, with Lagrangians of the form 1 2
+ &abcdTa8bVC(e)Yd + 0(Y4)
-a ( e )
where V,(e)are the torsion-free covariant derivatives. But it is the supergravity action which describes consistent interaction in the sense that Ss, possesses = am&, +. . ., invariance under local supersymmetry transformations 6YmZ which generalize the linearized gauge invariance. Let us stress also that a naive curved-space generalization of the Rarita-Schwinger action
S=
s
d4x e - 1&abcd\Tra8bVc(e)Y,j
proves to be contradictory, since the dynamical equations
&abcd8bVc(e)yd =0 are consistent only if
which implies a strong constraint on the curvature of curved space. It is supergravity which gives us the only satisfactory way to describe massless spin-2 particles in curved space. In conclusion, we would like to comment about the component form of the supergravity equations of motion (6.1.8). First, setting RI =Gal = O gives the auxiliary field equations (6.1.14). These mean, in particular, that the connections and uabe(e,Y), defined by equation (5.8.24),coincide on-shell. Then the requirements 9,RI = acbGBg,I =O are equivalent, due to equations (5.8.30a,b),and to the gravitino field equations which can be written in several Copyright © 1998 IOP Publishing Ltd
496
Ideas and Methods of Supersymmetry and Supergravity
(6.1.15~)
(6.2.15 b) (6.1.15~) (6.1.15d) Finally, setting the left-hand sides of the relations (5.8.40~)and (5.8.40b) and their conjugate to vanish, one arrives at
where we have used the second relation (5.8.39). These equations constitute the supersymmetric extension of ordinary Einstein equations. 6.1.4. Supergravity with a cosmological term It has been pointed out that S,, is actually given by the invariant volume of curved superspace R4I4. Besides the supervolume d8z E - ' , we have at our disposal two other invariant 'volumes'
and
corresponding to the superspaces on which chiral and antichiral superfields respectively, reside. Hence, one can consider an action superfunctional which combines all the supervolumes:
(6.1.1 7) with p being a complex constant. What physics does this action lead to? In accordance with the reduction formula (5.8.50) at the component level we have
Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
497
therefore the total action functional reads
As is seen, the fields B, B and A, do not propagate, since their dynamical equations are B=3p
A,=O.
(6.1.19)
Si2 leads to non-zero constant values for the However, in contrast to ,,S, scalar fields. Eliminating B and B with the help of their equations of motion produces the following contribution to the action (6.1.20) which can easily be recognized as an ordinary cosmological term. As a result, we can interpret the superfunctional (6.1.21) as the 'supersymmetric cosmological term'. The action (6.1.17) describes supergravity with a cosmological term. The dynamical equations of supergravity with a cosmological term can be readily found with the help of equations (6.1.7) and (5.6.63): G,=O
R=p.
(6.1.22)
Clearly, their space projections coincide with the auxiliary field equations (6.1.19). Further, setting 9=RI,B(hGBp,Iand B2RI to vanish, one obtains the gravitino field equations 1 2
-, p d 8
y
- p30bp - Pp =0
(6.1.23)
and the expression for the scalar curvature
Equations (6.1.23) show that the gravitino becomes effectively massive in the presence of the cosmological term. As for the scalar curvature, it acquires a constant negative value modulo gravitino excitations. Copyright © 1998 IOP Publishing Ltd
498
Ideas and Methods of Supersjw"try and Supergravity
6.1.5. Conformal supergravity
We have seen that the full covariance group of the commutator algebra (5.5.6,7) also includes, on a level with the superspace general coordinate and superlocal Lorentz transformations, the super Weyl transformations (5.5.13). Clearly, the Einstein supergravity action is not super Weyl invariant. However, it is not difficult to find a supergravity action possessing such an invariance. In accordance with the results of subsection 5.6.5, the superfunctional S,,=l{dsi-
E-' R WzppW1B.t
7
(6.1.25)
does not depend on the chiral compensator cp, and, hence it turns out to be super Weyl invariant. Note, S,, is real modulo total derivative terms (see subsection 5.6.5). In components, the purely vierbein-dependent part of S,, coincides, due to equation (5.8.40c),with the integral from the squared Weyl tensor. Hence, the superfunctional (6.1.25) represents the supersymmetric extension of the conformal gravity action (1.6.53). On these grounds and since the super Weyl invariance is inherent in conformal supergravity (see subsection 5.1.6), the S,, can be identified with the action superfunctional of conformal supergravity. Arbitrary variation of Ssc can be represented in the form (5.6.64).Therefore, conformal supergravity is characterized by the dynamical equations
+
Wap)9?pGYk-F k B ; l ~ g G z p (a6Gb)g,Gb
+-4i 9 , 6 ( g 2 R - a2R)=0.
(6.1.26)
It follows from these equations that the component vector field A,, being auxiliary in Einstein supergravity, becomes dynamical in conformal supergravity. Remark. Owing to the fact that the integrand in (5.6.61) forms a total derivative, the conformal supergravity action can be rewritten in the manner dszE-'(G"G,+2RR).
6.1.6. Renormalizable supergravity models
(6.1.27)
As is well known, the sufficient condition for a quantum field theory to be renormalizable is the appearance of dimensionless coupling constants in the classical action. Since the gravitational coupling constant IC has non-zero Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
499
dimension, Einstein gravity turns out to be non-renormalizable, and this fact has proved to be one of the most serious obstacles on the way to constructing quantum gravity. Renormalizable gravity models exist, but they are described by higher-derivative actions of the general structure i
+ K + 4aub(e)BUb(e) + @'(e)
9(e)
(6.1.28)
with 4 and [ being dimensionless constants. These models are known as Rz-gravity theories. It is worth pointing out that we could add to the expression in braces in (6.1.28) one more term-the squared Weyl tensor. However, there is no necessity to do so because of the topological nature of the functional (1.6.44). In superspace, the most general supergravity action with dimensionless coupling constants reads =
SSR2
s
d8z E - ' ( ( G " G ,+ [ R R + q R 2 +*}
(6.1.29)
and renormalizable supergravity models are described by classical actions of the form S = ss,
+ SS.cosm + SSR?.
(6.1.30)
Using the relations (5.8.40a,b), one can readily see that at the component level the first and second terms in (6.1.29) contain the squared Ricci tensor and the squared scalar curvature, respectively. So, they are supersymmetric analogues of the structures presented in the Rz-gravity action (6.1.28). The last two terms in (6.1.29), however, prove to have no direct space-time analogues in the sense that they do not produce purely vierbein-dependent contributions in components. In conclusion, we note the action (6.1.30) leads to non-trivial dynamics for the component fields B and A,. 6.1.7. Pathological supergrauity model The naive superspace generalization of the gravity action (6.1.1) would be of the form
s
S = d8z E - ' ( t / R + G )
(6.1.31)
where q is a complex constant. But it turns out to be pathological from a physical point of view. To argue, this point we take for simplicity q to be real, q = ij = - 91c-l. In components, one obtains S=-
's
K
d4xe-'(B+B)
Copyright © 1998 IOP Publishing Ltd
500
Ideas and Metkodr of Supersymmetry and Supergravity
where dots mean gravitino-dependent terms. Setting the gravitino to vanish, the B-equation of motion reads 8
+-2 BB +; 1 B2 =0 2
J
J
which has the solution B=B= i J - 9 .
I
For consistency, the scalar curvature should be non-positive. Now, eliminating B from the action gives
We see that the action becomes non-polynomial in the curvature after elimination of the scalar auxiliary fields!
6.2. Linearized supergravity We are going to show that at the linearized level the Einstein supergravity theory, constructed in the previous section, is reduced to describing two massless super Poincare states of superhelicities (-2) and 3 (or, equivalently, four massless Poincare states of helicities +$ and k2). We also consider linearized conformal supergravity. 6.2.1. Linearized Einstein supergravity action
To obtain a linearized form of the supergravity action (6.1.3), we shall use the prepotential parametrization of subsection 5.6.1. It will also be convenient for us to work in the chiral representation defined by equations (5.6.15, 16). After imposing the gauge (5.6.18), the action S,, reads in terms of the prepotentials H a = R", cp and 40, with cp being a flat chiral superfield, Dkcp = 0, as follows (6.2.1) where
with D, being the flat covariant derivatives. Here we have used the relations Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
501
(5.6.2I , 22). The action is invariant under the gauge transformations ae- ZiH -- A ~ -2iH -e- Z i H A (6.2.3) 6(p3= -!D2D"(L,(p3)
4
where 1A = - - D 2L"
A=AADA
"
4
A& = -- ~-2iHD2i. 4 with L, being an arbitrary spinor superfield. Now, let us set
Axi = - 2iDkL,
@=e*
(p=eb
Dko=O
(6.2.4)
and decompose the supergravity action to second order in the superfields H a , o and 6,First, we note that l?=det(l?,b), because H does not involve spinor pieces. From (6.2.2)one readily obtains
Ea = a, + u,ba,,
1 1 U O b= - (i?,Y"D,D,Hb -- (i?,)""Di{(D,H')a,Hb - HCD,8,Hb}+ 0(H3). 2 2 Due to the formula det eA=etrA, we have
E - ' I 3 =det- ' 1 3 ( 0
+ U )= 1 --31 tr U +-61 tr U'+- 181 (tr U ) 2+ 0 ( U 3 )
which leads to - 113 = 1
-A6 D
f
-L
~ D +~ H p,D ~ a ~p ) 2 + 72 24
+ H2-total derivative terms + O(H3). Up to the same accuracy, we have (l.e-2'f'
11 / 3 x 1
c
- -2i a o ~ " + - (4a , ~ " ) Z 3 9
and also
+
1 ( 0 +E)' -2iH"8,~. 2
cp e-2iH@ x 1 o+i?+-
Copyright © 1998 IOP Publishing Ltd
502
idens und Methods of Supersymmetry and Supergravity
Finally, making use of the identities D+D,H"
c
.
1 2
= - - [D,,
Dk]H"
c
dsz (D,D,HPP)(DBD~H"') = - dsz{(D+D,Hlx')2+ HaD"D2D,H,}
one arrives, after the rescaling Ha+ K H , and '2)sSG-j
+ 2idaHa
{i
1 dsz -HaD"D2D,Ha-3E~+-([D,, 48
U -+ K U ,
at the superfunctional
D,]H"')' - (2,H")'
I
+ 2i(0 - i?)d,Ha
(6.2.5)
where we have dropped numerous total derivative terms. It is instructive to compare the first term in equation (6.2.5) with the massless vector multiplet action (3.4.9). Looking at S&, we see that the (antijchiral superfields are characterized by a kinetic term with the wrong sign (compare with equation (3.2.5)). Nevertheless, this is not a defect of the theory, since o and 8 turn out to be purely gauge degrees of freedom with respect to the transformations 6H,+ = DhL, - D,L,
1 60 = - -D2D'L, 12
(6.2.6)
which leave invariant S i 2 and represent the linearized version of the original nonlinear transformations (6.2.3). The above gauge invariance can be used to impose the guage condition 0=5=0.
(6.2.7)
D"H,i = D'H,, = 0.
(6.2.8)
Another useful gauge is In this gauge
$2
proves to take the form d% {Ha H , + 350}
0 = 2'8,.
(6.2.9)
Remark. All we have done above is, in fact, the expansion of S,, around a flat superspace background. To obtain an expansion of SSG around a curved superspace background described by covariant derivatives gA,one starts with the following ansatz for SsG: t
Ss,=
- i KS d s i t - l ( g e - i ' H g ) ( l . e - i ' H ) l i ' l l
Copyright © 1998 IOP Publishing Ltd
(6.2.10)
Dynamics in Supergravity
503
where
H = H % ? ~ = H q=eB
U=Ber(UAB)
B&a=O
with the supermatrix U A Bbeing defined by equation (5.6.47).In accordance with the results of Section 5.6, the action is invariant under the background transformations
6Ha= X H "
6gA= [ X ,gA]
6 q =X q
+ 21
3?= K A D A - KabMOb =2
and the quantum gauge transformations
6H,&=a&,- 9?,&
+ O(H)
6 q 3=
-A4 (a2-4R)ga(L,q3)
with L, being arbitrary. 6.2.2. Linearized superfield strengths and dynamical equations Our next goal is to analyse which supersymmetry representation the theory (6.2.5) describes on-shell. For this purpose, we must investigate relevant superfield strengths-that is, descendants of Ha, 0 and 5 which are invariants of the gauge transformations (6.2.6).We can take in the role of such objects linearlzed versions of the supertorsion tensors R, G, and W.8r. Indeed, since R, G , and WZp;.transform homogeneously with respect to the supergravity gauge group, their linearized counterparts, which will be denoted by R, G, and Was.,, respectively, should be invariants of the linearized gauge group. Explicit expressions for linearized supertorsions follow from the relations (5.6.59,60)by setting the covariant derivatives gAto be flat. This leads to the superfields
+ -81 D P D ~ D ~ H , ,
G , =~ii?,&(5- 0)
1 -- ED,, DJ [DB, DBIH~B 24
+ a,&abH, (6.2.11)
1 -
Wapv= - DZd,,'DBH7)p. 8 It is instructive to explicitly check their invariance under the transformations (6.2.6). The Bianchi identities (5.5.8) are reduced at the linearized level to
D,R = DbW,p? = 0
D'G,'
D"W,p~=ii?(s~G.,)il. Copyright © 1998 IOP Publishing Ltd
= D,R
(6.2.12)
504
Icieus una' Merhocis of Supersymmetry und Supergravity
Rernark. In the gauge (6.2.8), the expressions (6.2.11) are simplified drastically 1 1 -
G,k=id,k(3-a)-nH,k
R = --D23 4
(6.2.13)
The dynamical system (6.2.5)is characterized by the equations of motion
G,=O
(6.2.14)
R=O
which represent the linearized form of equations (6.1.8). We see that only the (anti)chiral superfield strengths W k f iand WuB,'do not vanish on-shell, and they satisfy, due to identities (6.2.12), the equations
D*W,C1;'= D ' W k f i , = 0.
(6.2.15)
In accordance with the results of Section 2.7, these equations define massless on-shell superfields, and the corresponding superhelicities are
3 K(W,B;,)=2
K(W*&.)=
-2.
(6.2.16)
Therefore, the Einstein supergravity theory at the linearized level describes two massless super Poincare states of superhelicities ( - 2) and $. Similar on-shell analysis can be done in components. The component supergravity action (6.1.12) reduces at the linearized level to the sum of the linearized gravity action (1.8.47) and the Rarita-Schwinger action (1.8.41), modulo auxiliary fields. As we have seen, the former theory describes two massless states of helicities ( k2) and the latter, two massless states of helicities ( i:2). 6.2.3. Linearized conformal supergravity The action describing linearized conformal supergravity is given in terms of the linearized super Weyl tensor Wup,(6.2.11) as follows I-
(6.2.16) It is evidently invariant under the gauge transformations 6H,k
= DkL, - D,L,
(6.2.17)
with L, being arbitrary. The dynamical equations are of the form 1
G , +- ~ , ( D ~ DR2 R )= o 4
(6.2.18)
and represent the linearized form of the conformal supergravity equations (6.1.26). Copyright © 1998 IOP Publishing Ltd
505
Dynamics in Supergravity
Let us rewrite S @ in terms of Ha. Using equation (6.2.1l), we obtain
i 1 +(a',HU)D2D2a',Hb+ - (DaH,,) 0dP')DbHpb 12 6 In the gauge (6.2.8), Si$ reduces to
s
= - d8z H " 0 OH,.
(6.2.20)
We see that conformal supergravity leads to higher derivative dynamical equations.
6.3. Supergravity-matter dynamical systems Einstein supergravity action S,, is suitable only for describing empty supersymmetric universes. In the presence of matter, it must be substituted by an action superfunctional of the general structure
s = SSG + SMk 91
(6.3.1)
where SM is a matter action and x denote matter superfields under consideration. Such a dynamical system is characterized by equations of motion which naturally fall into two groups: supergravity equations and matter ones. The matter dynamical equations read simply as SSMIdX = 0. Introducing the supercurrent T, = dSM/dH"
(6.3-2)
-
(6.3.3)
G,T=O
(6.3.4)
Ta= Ta
and the supertrace T=dSM/dq Copyright © 1998 IOP Publishing Ltd
506
Ideas and Methods of Supersymmetry and Supergravity
of the matter superfields (see Section 5.7), the supergravity equations of motion are given by L
-GG,+T,=O
3
-.-R+T=O
(6.3.5)
ti’
ti’
where we have used the relations (6.1.7). These equations generalize the ordinary Einstein equations with matter. They show that the supercurrent and supertrace characterize the coupling of matter to supergravity. Since the supertorsions G, and R satisfy the Bianchi identity @Gbh=gmR, the system (6.3.5) is consistent under the supercurrent conservation law (6.3.6) which, at the same time, expresses the fact that SM is gauge invariant, in accordance with the results of Section 5.7. Now, we consider some specific matter models in a curved superspace. 6.3.1. Chiral scalar models We commence by discussing simple models of a covariantly chiral scalar superfield @, B$D =0, and its conjugate embraced by the common action
a,
s
S = d8z E - ’ @ @
where
2%(@, R ) = m0’
+
(6.3.7)
+ lR@’ + -1 ,la3 3
2%(@)+ (R@’
with m, 5 and i.being coupling constants. The system with this action can be considered as a curved-superspace extension of the standard Wess-Zumino model (3.2.11). Using the technique of Section 5.8, the above action can be easily reduced to components. We leave this as an exercise for the reader. Let us comment, however, about the coupling of the scalar component fields C(X) = 0 1
C(X) =
dil
to the vierbein. Setting the gravitino and supergravity auxiliary fields B and Ad to zero and keeping only C and C among the component fields of 0 and 0,from equations (5.8.40,50) we deduce (6.3.8a)
s Copyright © 1998 IOP Publishing Ltd
(6.3.8b)
Dynamics in Supergravity
507
We see that the kinetic term in action (6.3.7)leads to Weyl invariant coupling of the scalar fields to gravity (see also subsection 1.7.7).This is not incidental, since the kinetic term is super Weyl invariant (see subsection 5.7.5).The mass term and the (-term in action (6.3.7) break this invariance. Decomposing C into real and imaginary parts, C =-
1
(9 + i9)
$
we observe from expressions (6.3.8) that Sd"i E-'{.@
+z1 (02+- @D2 2
-!Sd4x 2
-1
e-'
where we have taken 5 to be real. It is seen that in the particular cases 5 = & 1 one of the scalar fields does not couple to the scalar curvature (minimal coupling with gravity). Now, let us determine the supercurrent and the supertrace of the system with action (6.3.7). When (=O, their expressions are given by equations (5.7.40)and (5.7.41),respectively. Hence, it remains to vary the superfunctional d8z E - { (02+
m}
with respect to the supergravity prepotentials by the rule given in subsection 5.7.1. Namely, we are to calculate the variation
"6
6(v)--e-2iH"0, (5
to first order in the superfields Ha and 0,where b-' reads as in equation (5.6.57). After simple calculation, one obtains the supercurrent
(6.3.10) and the supertrace T = -'(@-44R) 4 Copyright © 1998 IOP Publishing Ltd
+3yr(O).
(6.3.11)
508
Ideas and Methods of Supersymmetry and Supergrauity
Remark. Analysing the above expressions, we note an important property of the supercurrent and supertrace in flat global superspace. In the flat-superspace limit, the action (6.3.7) reduces to the standard Wess-Zumino model action (3.2.1l), which is 5-independent, but the corresponding supercurrent and supertrace reduce to
--
and
1 - 2 1 T$i=-(D,@)D,@+-i@ a,,@+-iia,,(5@2-@2) 3 3 3
:-I
Fi)=--D2 @@+-502 +3YC(@) 4l - ( -
(6.3.12) (6.3.13)
respectively, where some (-dependence is present. For every 4, we can take T!$ and Fi)in the roles of supercurrent and supertrace of the Wess-Zumino model. Different choices correspond to different possible ways of extending them to curved superspace. On the other hand, all choices are physically equivalent due to the fact that if T i t and T'Sj satisfy the conservation law
2 T(5' D*Ti$)= --D 3 "
(6.3.14)
then so do the improved supercurrent and supertrace
for an arbitrary chiral scalar R. This is a fundamental arbitrariness in the choice of supercurrent and supertrace in flat global superspace. The action (6.3.7) leads to the following dynamical equations 1 -
--(2?2 -4R)G
4
1
+ m@ + 5R@+ -21 io2= 0 +
-- (99- 4R)@
4
(6.3.16)
+ ZRG + 1&2 =0 2
where we have taken m real. If ( = E. =0, this system is easily separated: 1 16
-(9' - 4 R ) ( g 2- 4R)Q = m2Q
(6.3.174
1 16
(6.3.17b)
-(
9 2
- 4 R ) ( P - 4R)G = m2G.
Owing to the chirality of 0,the fourth-order operation in the 1.h.s.of equation (6.3.17~1)can be reduced, with the help of the commutator algebra (5.5.6), Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
509
to second-order. Then, the equation (6.3.17b)takes the form
( +
+
1 - ---(9'R)+
+ 41
= 9"9%, -R2'
RR-m'
(6.3.18)
+ iGaB0+ -41 (9'R)9&.
Similarly, the equation in (5 takes the form (6.3.19)
The operator + ( -) transforms every covariantly chiral (antichiral) scalar superfield into a covariantly chiral (antichiral) one. For this reason + ( U -) may be called the 'chiral (antichiral) d'Alembertian'. The operator + is unique in the sense that any second-order differential operator constructed from the covariant derivatives, with the leading term 9EBaacting on the space of covariantly chiral scalar superfields turns out to have the form O++Y
5BiY =o
where Y is a covariantly chiral scalar superfield. 6.3.2. Vector multiplet models The massless vector multiplet model (3.4.9)is uniquely continued to a curved superspace when taking the superfield variable V ( z ) to be a real scalar superfield (with respect to the general coordinate transformation supergroup and the superlocal Lorentz group) and requiring the invariance of the curved-superspace action under the gauge transformations 1
V+ V'=V+-(A-A)
2
G'iA=O
(6.3.20)
with A(z) being an arbitrary covariantly chiral scalar. Namely, the gauge invariant action superfunctional is of the form
where W, =
-: 4
(G" -4R)g,V
aiW, =O.
(6.3.22)
As a result of the identity (5.5.12), the action is real. Its invariance under Copyright © 1998 IOP Publishing Ltd
5 10
Ideas and Methods of Supersymmetry und Supergracity
transformation (6.3.20) follows from the observation GhA = 0
=>
(9' -4R)9,A
=0
(6.3-23)
which is a direct consequence of the identity (5.5.9e). Therefore, W, and its conjugate W kare invariant superfield strengths. The action superfunctional proves to be invariant under the super Weyl transformations (5.5.13) if we take V to be super Weyl inert. This can be easily checked using the transformation laws for R (5.5.14) and for E - ' , (E-1)LeU+eE- Therefore, S does not depend on the chiral compensator, which means that the supertrace vanishes identically. To find the supercurrent, we must, following the prescription of Section 5.7, calculate the variation
'.
where
to first order in Ha.With the help of equations (5.6.40, 58-60), one can derive the identity (?-4R)i,=(@
-4R)i,-A,P(G'
- 4 R ) l g + ( 9 ' -4R)[A,Pi.P+2A.83.,] (6.3.24)
which is convenient when performing variations. It is also necessary to recall relation (5.6.63). After some calculation, one arrives at the supercurrent 1 Ta,=2W,Wk+-(@Wp)[9a, 2
$h]V.
(6.3.25)
Making use of the dynamical equation 9W, = 0, we can change the above expression to Tu,=2W,Wi.
(6.3.26)
As can be seen, the supercurrent is not gauge invariant off-shell. The point is that the gauge parameters in expression (6.3.20) are covariantly chiral, hence they depend on supergravity prepotentials. For completeness, let us also consider the massive vector multiplet model in a curved superspace with the action
W'W,+m2
Copyright © 1998 IOP Publishing Ltd
s
d'z E - ' V 2
(6.3.27)
Dynamics in Supergravity
511
One can readily verify that this system is characterized by the supercurrent
1 T,, = 2 WEE', +- (& Wp)[9*,a,] V 2
and the supertrace 1 T = --mZ(gZ-4R)V2. 4
(6.3.29)
On-shell, the supercurrent takes the form
6.3.3. Super Yang-Mills models Now we are going to obtain a curved-superspace extension of the super Yang-Mills dynamical systems (3.5.46). In a curved superspace, superfields Oiand i6iforming the matter multiplet should be treated as covariantly chiral and antichiral scalars, respectively. Superfields VI forming Yang-Mills superfield V = V'T'= Vf, where TI are Hermitian generators of the gauge group, should be treated as real scalars. The most natural generalization of the super Yang-Mills gauge transformations (3.5.25,26) reads @' = eiA@
A = AITI
= &-ix
A =A I T I ,2 V' = eiA e 2 V
g,AI = 0 9 , A I =0
e - iA
(6.3.31a) (6.3.31b) (6.3.31~)
with the parameters A' being covariantly chiral scalar superfields. Invariant under these transformations the action superfunctional, with the flatsuperspace limit (3.5.46), is of the form
s
S = d8z E-'i6i(eZv)ijOj+
(6.3.32) where 1 W, = - - (9' - 4 R)(e- '9,e 8
Copyright © 1998 IOP Publishing Ltd
')
GkW, = 0
(6.3.33)
5 12
Ideas and Methods of Supersymmetry and Supergravity
are gauge covariant superfield strengths possessing the transformation laws
w; =eiA wXe- iA
= eiA
Fie- i A.
(6.3.34)
Note that the last term in equation (6.3.32) turns out to be real modulo total derivatives, see below. In order to describe the dynamical properties of the system with action (6.3.32),it is convenient to introduce two sets of gauge covariant derivatives: (6.3.35a))
and (6.3.35 b)
~k-)=g~ DL-)=e2Vgie-2V
Di;)=- 1 @(-ID(,-) U > 2 1
2
with the transformation laws D;+)
=eiAID,af)e-iA
D:-) = eiAID,a-)e-iA
(6.3.36)
connected to each other in the manner
Dy
)
= e - 2 v ID,a- )e2 v.
(6.3.37)
The operators Dy)and ID,L-) are curved-superspace versions of the flat ones (3.6.40a) and (3.6.40b), respectively. They satisfy commutator algebras of the form [U&*),
Db*))= TABCD~*)+R,,+iFs)
are the Einstein supergravity supertorsion where TABCand RA, = $RABcdMcd and supercurvature, F s )= FgB’T’ is the Yang-Mills supercurvature. Explicitly, we have {ID,:’),
D r ) }= -4RM,p
{Db -(*) ,D - (b *) =4RR,b
[@*’, DS)] = -ie,&RD$i)+ G,~D~*))-i$iB,RA,B
+ is,,@’G,’R9s
+ 2i4b W r )
- 2 WgY6Mj6)
(6.3.38)
[E):*), ID,$’] = i s U p ( R r ) +G’&!DS,*))+i$pRMUp - isU&WG6b MYs- 2 Wfrg”‘ rigs)
+ 2icXPLT’F
where WL*) and Wi*)are expressed in terms of the superfield strengths (6.3.33) as follows (6.3.39)
Copyright © 1998 IOP Publishing Ltd
5 13
Dynamics in Supergravity Together with the Bianchi identities (5.5.8), we also have
(6.3.40)
To derive equations (6.3.38-40), one simply repeats the steps performed in subsection 3.6.4 when obtaining (3.6.45-48). Now, let us consider the dynamical equations of the system. Varying the gives the equations action (6.3.32) with respect to
ai
1 dPc(@) --(92-41?)(e2V@)i+v=0. 4 aQ,j
(6.3.41)
Next, varying the action with respect to V leads to (6.3.42~) or
depending on which variation, A ( + ] V or A ( - ) V (3.6.57), was used. Clearly, the equations (6.3.42~)and (6.3.42b) are equivalent. Remark. Reality of the last term in equation (6.3.32) follows from the fact that its arbitrary variation over V can be represented as
and this expression is real, owing to the Hermitian conjugation rules
(A(+)V)+= A ( - l V Let us set the chiral superpotential has the consequence
Since
Q,
(DN+)W1)+ = Q$-)@i,
g(@) to zero. Then, equation (6.3.41)
is gauge covariantly chiral, DclO=O, making use of the algebra
Copyright © 1998 IOP Publishing Ltd
5 14
Ideas and Methods of Supersymmetry and Supergravity
(6.3.38) reduces the above system to
-14 (a28) + RE}@=0.
1
-- (W(+)W,)
2
(6.3.43)
These equations represent the Yang-Mills covariantized f q m of equation (6.3.18). To calculate the supercurrent and supertrace of the system with action (6.3.32),it is helpful to operate with the identify (6.3.24).Tedious calculations lead to the supercurrent 1 2 Tzk=-- [9% aj1](6e2"O) , +i6e2"~,~*-i(~,j16)e2"a -- G,,@e2"@ 6 3
2wC
)mc)+41 ( D P ( +)wF))(e-2"[9,, B J e " -
I
+ 2 ( e - 2 Y ~ ~ e 2 " ) ( e - 2 " ~ u e 2(6.3.44) ")) and the supertrace 1 T= --(~2-4R)(&eZY@)+ 3230). 4
(6.3.45)
As may be seen, the supercurrent is not gauge invariant off-shell. However, using the dynamical equations (6.3.42), one can easily rewrite Tah in the gauge invariant form:
(6.3.46) Further, due to the dynamical equations (6.3.41), the on-shell supertrace reads
(6.3.47) Therefore, the supertrace vanishes if and only if 1
YC(*) = - %ijk*i@Ok* 3!
In this case the action (6.3.32) turns out to be super Weyl invariant. Copyright © 1998 IOP Publishing Ltd
(6.3.48)
Dynamics in Supergravity
5 15
6.3.4. Chiral spinor model In conclusion, we would like to consider the model of a covariantly chiral and its conjugate jih with the action spinor superfield I,, superfunctional
aj,x,=O,
(6.3.49) where
In the flat-superspace limit, the theory reduces to the massless chiral spinor model (3.7.10). Owing to the identity (5.5.12), the above action is invariant under the gauge transformations dx,=i(a2-4R)9,K
K=K
(6.3.50)
with K being an arbitrary real scalar superfield, hence [1 is a gauge invariant superfield strength. It is worth also pointing out that, by virtue of relation (5.5.1 l), 11 is characterized by (G2-4R)11=(92-4R)[L=0.
(6.3.51)
Each real scalar superfield under this constraint will be said to be covariantly linear. To obtain dynamical equations, one uses the variational rules (6.3.52)
Then one obtains (G2-4R)9,11=(92 -4E))a,11=0.
(6.3.53)
To calculate the supercurrent and supertrace of the theory, we must evaluate the variation
where the derivatives T A are given by equation (5.6.40), to first order in the superfields H" and 0. Note, X(v)z should be a covariantly chiral spinor = 0. In the linear approximation in Ha, superfield with respect to TA, Copyright © 1998 IOP Publishing Ltd
5 16
Icitws unci M e t h i i s qf Supersymmetry und Supergravity
we have h
a=
x? - A ,
-
/j
xp
= ;Ck - Afi/j;C,j- 2iHbgbxk
&v)k
with Ax,j being defined by equation (5.6.60).After some effort, one arrives at the supercurrent
+-23 L[9,,
2 T,, =- (!2h[L)9,IL 3
9JIL
j $ ( Q 2-4R)Q%L+c.c.
3
(6.3.54) and the supertrace T=
-! (G2- 4R)L2 - 3f(a2-4R)gJ. 8
4
(6.3.55)
On-shell, the supercurrent and the supertrace become gauge invariant: 2 2 2 Tz,=-(9fiL)9a[i+-L [ g a ,GJL-- GakL2 3 3 3
(6.3.56)
1 T = --(9?'-4R)L2 4
It is instructive to compare the expressions obtained here with the supercurrent and supertrace of the chiral scalar model with action S=-
'I
2
d8z
ah@=O
which is obtained from action (6.3.7) by setting choice in equations (6.3.10) and (6.3.11) gives 2 3
2 3
(6.3.57)
< = 1, $Pc(0)=O. Making this 2 3
Tsk=-(9kM)5?aM+- M[9m,~~]M--G,kM2
(6.3.58)
T = -L(Q2-4R)M2 4
where we have introduced the notation M =-
1
4
(@
+ G).
(6.3.59)
We observe that the expressions (6.3.56) and (6.3.58) have the same form. Remarkably, this is not a formal coincidence. We have seen that [L was submitted to the kinematical constraints (6.3.51) and the dynamical equations Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergruvity
5 17
(6.3.53). As for M, from its definition we directly deduce
( S 2 - 4 R ) 9 % M= ( S 2 - 4 R ) g + M
(6.3.60)
with the help of identities (5.5.9d,e). On the other hand, the dynamical equations of the theory (6.3.57) are
(g22-4R)M = ( 9 2 - 4 R ) M
=o
(6.3.61)
hence M is covariantly linear on-shell. As a result, L and M satisfy the same differential equations on-shell. Therefore, under a proper choice of initial conditions for the theories (6.3.49) and (6.3.57) (and gauge conditions for the former theory) their supercurrents (supertraces) coincide on-shell. This implies classical equivalence of the theories under consideration. The equivalence can be also seen with the help of the first-order model with action (6.3.62) which constitutes duality of the above theories. Indeed, varying S[V, 0,@] over V gives the equation -V=@+@ which can be used to eliminate V resulting in the action (6.3.57). On the other hand, varying S[ V,@, $1 with respect to @ and di gives
(a2- 4R)V= (g2-4R)V= 0 which has the following solution: 1
1/= -( g z x , + g aX .+)
2 -&
= 0.
$
Substituting this expression into S[V, @, @] and using the identity (a2-4R)V=0
g@=O
3
d8z E-'V@=O
(6.3.63)
one obtains the action (6.3.49). 6.4. (Conformal) Killing supervectors. Superconformal models
We now turn our attention to looking for superspace analogues of conformal Killing vector fields-the objects discussed in Section 1.7. We also consider superconformal dynamical systems and superconformal massless superfields. Copyright © 1998 IOP Publishing Ltd
5 18
Ideas and Methods of Supersymmetry and Supergrauity
6.4.1. (Conformal) Killing supervector fields
To begin with, let us introduce the notion of conformal Killing supervector fields. Consider a curved superspace described by covariant derivatives 9 A =E A
+
1
n A = E A M ( z ) d M $-
2
RAbc(z)M,,.
Let < = t A ( z ) E A be a real C-type supervector field and K”(z) be a real antisymmetric tensor superfield. t Aand Kabgenerate an infinitesimal general coordinate and superlocal Lorentz transformation acting on the covariant derivatives and tensor superfields as follows: hXgA=[X,gA]
(6.4.1)
6Xx=Lfx
where
+ R”M&fi
+
X = t A g ~KapM,p
with x being a tensor superfield (with suppressed indices). Using the (anti)commutation relations (5.5.6,7), it is easy to calculate explicitly the variations hXgA.In particular, in the undotted spinor case one obtains
(6.4.2)
\
L
L
/
Let us also recall how infinitesimal super Weyl transformations act on the covariant derivatives:
(6.4.3)
Here o is a covariantly chiral scalar superfield. Now we proceed by finding a set of transformation parameters { C A , Kab,o} such that their combined action on the covariant derivatives is trivial,
(6, Copyright © 1998 IOP Publishing Ltd
+ 6@,4
= 0.
Dynamics in Supergravity
5 19
In accordance with the relations (6.4.2)and (6.4.3),this requirement means the parameters t,, &, K e p , R&g,0 and 8 should be expressed via 5" in the manner:
(6.4.4~) (6.4.4b) 1
0 [ 4 ]=-
3
(6.4.4~)
(W5,+2Bar6-ii5"Ga).
In addition, the following consistency conditions (6.4.5a) (6.4.5b)
B"K[<]py=-dcl(kv) or53 -46,@5Y)R+ 1G , ( p t y ~ ~ ~+$- tz&9(BGyM ( 6 . 4 . 5 ~ ) 2 2 1
G50[5]= 0
(6.4.5d)
should hold. From relation (6.4.4a) it is easy to read off the closed equation on the vector superfield 4": 9&
+ 9p5,g = 0
* G&tpg+ G&&
= 0.
(6.4.6)
Then, making use of the covariant derivatives algebra (5.5.6),one immediately obtains the important consequences ( 9 2
+ 2R)ta= 0
+ 2R)5, =0
(a2
(6.4.7)
and
(6.4.8) Remarkably, the last relation coincides in form with equation (1.7.5) defining conformal Killing vector fields of a space-time. But the highly important ) satisfied identically fact for us is that the consistency conditions ( 6 . 4 . 5 ~ - dare under equation (6.4.6). For example, let us check the requirement (6.4.5d). We have
Copyright © 1998 IOP Publishing Ltd
520
Ideas and Methods of Supersymmetry and Supergravity 1 -
= - [9*, 9p,j]
2
rpp' + 1 9pa*a&pp +12 B5(5@Gp&
= -1 [9+, - g p , ~ ] 4' p p1 + 2 ~ ~ R s p a ) + i ~ * ( 5 8 8 G g p )
2
2
where we have used equations (6.4.6,7). Expanding the commutator in this expression by the rule (5.5.6), one obtains gaa[5] =O. Slightly more work is necessary to check the conditions (6.4.5b,c). As a result, we come to the conclusion that the only way to achieve the goal formulated above is to choose a solution of the equations (6.4.6) in the role of 4" and to take the other parameters as in (6.4.4). A real C-type supervector field (6.4.9) is said to be a 'conformal Killing supervector' (for the supergeometry determined by gA)of 4" satisfies the master equation (6.4.6). The basic property of conformal Killing supervectors is that they act on the covariant derivatives like superlocal Lorentz and super Weyl transformations: 'X[<]pA
=[ x [ { I ,
= - 'c7[
(6.4. 0 )
where
x[5]= t A Q A
+ K[4]"phii2p+ R[<]"fl&fi
(6.4.
with K [ 4 ] , p and a[(] being defined as in expressions (6.4.4). A conformal Killing supervector ( is said to be a 'Killing supervectoi if it satisfies the requirement: a[<] = 8 [ 5 ] = 0
e.
9"(+ ,i("G, = 0.
(6.4.12)
The basic property of Killing supervectors is that they define symmetry transformations of the curved superspace: transformations generated by X [ ( ] leave the covariant derivatives unchanged, b [ 5 ] 9 ' 4
= C3,"[51,
9'41= 0.
(6.4.13)
From the point of view of supersymmetric field theory in curved superspace, Killing supervectors are of great importance. Indeed, consider a theory of matter superfields x with an action superfunctional S [ x ; 91 invariant, as usual, under general coordinate and superlocal Lorentz transformations 6.4.1). Then equation (6.4.13) shows that S[x;9] is unchanged under the transformations 6%= Jf[51x
(6.4.14)
for every Killing supervector 5. So, Killing supervectors constitute symmetries Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
521
of any dynamical system. As for arbitrary conformal Killing supervectors, they generate symmetry transformations of super Weyl invariant models only. If S [ x ; B] is a scalar with respect to super Weyl transformations of the form (5.7.33),then it does not change under the transformations
~ x = . x C t l x - ~ c + , ~ ~ t l x - ~ ~ - , ~ ~ 4(6.4.15) 1~ conformal Killing supervector 5. This statement is a trivial
for every consequence of transformation (6.4.10). For a given supergeometry, the set of all conformal Killing supervectors and the set of all Killing supervectors form real super Lie algebras. The latter assertion is almost obvious. If t1 and t2 are two Killing supervectors, =O
[x[51,21,
then
[[.x[tll, .x[t211?
9Al
=[[.x[tll? gAl, x [ ~ 2 1 1 +[.x[
=O
therefore
C.x[tiI, 3fC521I = x C 5 3 I for some Killing supervector
t3.One can readily see that 53=ct17
t21
which completes the proof. The fact that the set of conformal Killing supervectors forms a super Lie algebra follows from the theorem below. Before formulating the theorem, let us give one more definition. Two supergeometries determined by covariant derivatives BAand BAare are connected by some super Weyl said to be Weyl equivalent if g Aand transformation
aA
Theorem. Weyl equivalent supergeometries have the same conformal Killing supervectors. Proof. Let 5 be a conformal Killing supervector with respect to BA.5 is given in the form (6.4.9),where 5" obeys the equation (6.4.6).Decompose with respect to the supervierbein EA (aA = EA + f i A ) :
<
5 =t
A ~= rAEA , p=(~L)-'t"$ = L - ' Copyright © 1998 IOP Publishing Ltd
?A
=
(F,p,
(6.4.17)
522
Ideas and Methods of Supersymmetry and Supergravity
where we have used relations (6.4.16). It is not difficult to check that be represented as
and
p turns out to satisfy the equation i3&4 + spzab=0.
Therefore, 4 is a conformal Killing supervector with respect to also prove the important relations:
can
aA.One can
X [ ~ ] ~ ~ A ~ A + K [ ~ ] r B ~ , s + R [ ~ ] ' b(6.4.19a) BR,~=~[4]
d l = 4 4 1 - tA%o
(6.4.19b)
where
and G, is the supertorsion component corresponding to The relation (6.4.19a)shows that the operator X [ 5 ]is super Weyl invariant. The relation (6.4.19b) tells us that the property of a conformal Killing supervector 5 to be a Killing supervector is not super Weyl invariant.
6.4.2. The gravitational superfieid and conformal Killing supervectors Here we would like to look at conformal Killing supervectors from the point of view of their action on the supergravity prepotentials. In the gravitational superfield gauge (5.4.38), Einstein supergravity covariant derivatives Q A are experessed in terms of the superfields X"', cp and Cp, with cp being the chiral compensator, G,cp=O. Residual gauge transformations, generated by parameters jv"'(x,0) and j!(x, e), act on X"', BAand cp in accordance with the laws (5.4.41), (5.4.49)and (5.5.5), respectively. Let us choose (if possible) a set of superfields {i"'(x, e), LP(x, e)} such that
6X"=O
(6.4.20)
and only cp may change. Owing to (5.4.41), A"' and AP should satisfy the equations
+
nm(x iut: Copyright © 1998 IOP Publishing Ltd
e)-
P ( x - iz 8)= 2i52"'
(6.4.21)
Dynamics in Supergravity
523
where
E: = K'dM
1 1 K m=- % m ( x + i Ze)+- xm(x-i% 8) 2 2
(6.4.22)
Under these conditions, the covariant derivatives vary by the rule (6.4.3)with o = 6 In cp. On the other hand, their change is given by equation (5.4.49). Therefore the supervector field (6.4.22)is a conformal Killing supervector! Remarkably, the reverse assertion is also true. Namely, it can be shown, using the formulae (5.4.43-47) and (5.5.4) expressing gAvia X", cp and @, that every conformal Killing supervector has the form (6.4.22). Therefore, the basic property of conformal Killing supervectors is that they leave &m unchanged. Given a Killing supervector 5, it leaves gAunchanged. In accordance with ( 5 . 5 9 , this means that the parameters i m ( x ,e) and P ( x , e) should satisfy the equation
5 In cp = - -1 e, (amim- a,;.,)
(6.4.23a)
3
in addition to equations (6.4.21).Since cp(z) = cp((x+ iX: e)= ewp(x, e), this equation can be rewritten, with the help of conditions (6.4.22),in the following useful form:
(iemam + Pd,) In cp(x, e)= --31
-8 p ) .
(6.4.23b)
6.4.3. (Conformal) Killing supervectors in $at global superspace We are going to find (conformal) Killing supervectors of flat global superspace. In flat global superspace, a conformal Killing supervector E: has the form
where
4" satisfy the master equation Dxtpj+Dpt,b=O
*
Dh
(6.4.25)
The relations (6.4.7)and (6.4.8)take the form (6.4.26)
aatb
Copyright © 1998 IOP Publishing Ltd
+
1 ?ab(actc)*
(6.4.27)
524
Ideas and Methods of Supersymmetry and Supergravity
<"
From relations (6.4.26) we see that is a linear vector superfield, and 4" = - (i/8)Dfip@is a chiral spinor superfield,
D&" = 0
(6.4.28)
Equation (6.4.27) has been investigated in Section 1.7. In particular, it was shown that every solution 5" of this equation must be at most quadratic in space-time coordinates,
dadbdc(d(x,0, 8)= 0.
(6.4.29)
Owing to this observation, the general solution of the master equation (6.4.25) can be found quite easily. The result reads <"(.x, 8, 8)= b"
+ AX" + K a b X b +fax2 - 2xa(x,f) + 2i(00aC- EfJ"8)
+ 2i(8200aij- O2qa"8)- Q'BZfn
(6.4.30~)
or, in spinor notation, t1k(x, 8, 8)= b,?
+ AX,& + KXpxp&+ If k f i x , ~+ xzfixpfpB
- 48,(iCi
+ t+xgh)-48,(ic,
- x,pP)
+ 2e,B,n
+ 2ie,Bp(~,b -fp(&xpfi1) + 2 i ~ , e P ( ~+f(,Bxp,b) ,~ -4i0,B2ij5,-4i8,8'q,-
e282fEjl.
(6.4.30b)
Here A, R, b", f " and Kab= - Kb" are constant real c-number parameters, E, and q1 are constant a-number parameters. It is instructive to compare expression (6.4.30~)with the expression (1-7.12) determining conformal Killing vectors of Minkowski space. Fom (6.4.30b) one immediately obtains
5
1
-.
= - D*<,&= ei2&(x,
" 8
e)
x = eaaeaa
(6.4.31)
with &(x, e) being defined by equation (2.9.16) (recall that the parameters (2.9.16) generate infinitesimal superconformal transformations (2.9.17)). Moreover, it is a simple exercise to check that
< = SAD,= (5"+ i
= (eixia(x, e) e -i"~(x,
2
~ ) a+, (eixix(x, e))a, + (e
( x , G>)P
(6.4.32)
with J.LI(x,e) defined as in equation (2.9.16).Therefore, infinitesimal coordinate Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
525
transformations
+
(6.4.33)
z A+ z ' = ~ z A gBDBzA
generated by conformal Killing supervectors are exactly the superconformal transformations (2.9.17). Now, let us find Killing supervectors of flat global superspace. In accordance with equation (6.4.12), Killing supervectors are conformal Killing supervectors that obey the constraint
D"D'g,, = 0
o
D'D'4,k
= 0.
(6.4.34)
Those superfields (6.4.30b) which respect this constraint are given by
+
t2&(x,e, 8,= b,& K,BX~'
+ zQx,~ -4i(0,~&+ B~E,)+2 i ( ~ , @ ~ &+ &@K,,J. g (6.4.35)
Clearly, the transformations induced by Killing supervectors are exactly the super Poincari transformations. 6.4.4. Superconformal models As is well known, the conformal group is the space-time symmetry group of massless relativistic field theories (see Section 1.7). Similarly, there exist a variety of massless supersymmetric field theories possessing an invariance under the superconformal group. Such theories are called 'superconformal'. More precisely, a supersymmetric dynamical system with an action superfunctional S [ x ] , where K are tensor superfield dynamical variables, is said to be superconformal if (1) the space of superfield histories V is endowed with an action of the superformal group
m):x(z)
-+
G%)x)(z) = x&)
v xEv
T(g,)T(g2) for every elements g 1 and g 2 from the superconformal group; and (2) the action superfunctional is invariant with respect to arbitrary superconformal transformations, n?192)=
v x E v. S[XJ = S[X] In all known superconformal models, we have linear superconformal representations, and they can be described as follows. Let g be a group element of the form i( - b'po--f(lva
-1
+ -21 KQbjllb+ Ad + f2a + E q +E(I + qs + qs)
(6.4.36)
and let ( = rADAbe the conformal Killing supervectors having in the Copyright © 1998 IOP Publishing Ltd
526
Ideas and Methods of Supersymmetry and Supergravity
expansion (6.4.30~) the same coefficient as in expression (6.4.36).Denote by (6.4.37) the restriction on R414(0a0') of the Berezinian of the superconformal transformation yo + g .y", 8" g 0" on C4I2(constructed in Section 2.9), and introduce the operator --f
<
associated with (in accordance with the general prescription (6.4.4,ll)). Now, for arbitrary numbers d ( + , and d,-,, on the linear space of tensor superfields we define the following superconformal representation: T(g):x(z)+xg(z) = ( ~ g ) - ( ' 1 3 1 d 1 + ) ( ~ g ) - ( ' / 3 ) d ~ - ~ e(6.4.39) -~[rl<(z). In the infinitesimal case, the above transformation reads (compare with equation (6.4.15))
ax = - a < 1 x + id, +,a1x+d, -,X<J)x
(6.4.40)
where
1 3
= - - ( a , n ~y,
e)- a , n ~Y , e))
(6.4.41) y=x+i@oB
with P ( y , 0) and P ( y , 0) defined as in relation (2.9.16).To derive equation (6.4.41),one can use the relation (6.4.32). When choosing in representation (6.4.39) the dilatations g = exp(iAd) and the axial rotations g = exp(iAa), one obtains x'(x, 0, Q)=e-(dI+t+d[-dAX(e-Ax, e-(lP)Ae, e-(lP)AP)
and
x'(x, 6, e) = e -Ci/3Xd1 I - 4 +
I)Q
x(x, e(i/2)R 0, e - W ) R
8)
(6.4.42) (6.4.43)
respectively. Hence (d, + ) + d, -)) is the dimension of x, and (d, + ) - d, - is proportional to the axial charge of x. d,,, and d ( - ) will be called the 'superconformal weights' of x. To find generators of the representation under consideration, we rewrite, Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
1
D=-{(xa( + ) + XI2 the axial generator
))aa+ 8'8,
+ 8'ah}+ i(d, + d, - )) +)
1 1 A =- (6'8, - Pa,)- - (d,+, - d, -)) 2 3
527
(6.4.44a)
(6.4.44b)
the special conformal generators
and the S-supersymmetry generators
It is not difficult to check that the generators satisfy the (amti)commutation relations (2.9.21). Now, consider two examples of superconformal models. Our first example is the super Yang-Mills theory described by the action superfunctional S[@,
a, V ]=
s
d8z Gi(e2")ijW'+
+ C.C. (6.4.45)
DhQi=O Copyright © 1998 IOP Publishing Ltd
1 Wa= --D2(e-2"D,e2"). 8
528
Ideas and Methods of Supersymmetry and Supergravity
The action is invariant under the superconformal transformations
6 0 i = -
(6.4.46)
<
with = tADAbeing an arbitrary conformal Killing supervector. This follows from the fact that the curved-superspace analogue (6.3.32) (with the same chiral superpotential) of the above action turns out to be invariant under the super Weyl transformations (6.4.3) supplemented by
doi=-oat
61'=0.
The second example is the chiral spinor model with the action superfunctional (6.4.47)
G=-(D"z c( + D&(b -) 2 I
D&=O
k being a constant. Here x1 is a chiral spinor superfield. The action is invariant under the superconformal transformations
Jx, = - SADA X 1 - ~CSlc("p +j5 0CtIx1
(6.4.48)
for an arbitrary conformal Killing supervector g. The statement follows from the fact that SEX,j ] can be continued to a curved superspace by the rule
'S
S [ x , j ; 9 ] = --
k2
f
d8zE-1GlnG
G = I ( g u x1 + g &X-&)
2
(6.4.49)
%a=O
with 2, being a covariantly chiral spinor superfield. The superfunctional obtained proves to be invariant under the super Weyl transformations (6.4.3) supplemented by 3 6xu= -- ax,. 2
(6.4.50)
6.4.5. On-shell massless conformal superfields In Section 2.7, we studied on-shell massless superfields. Recall that a tensor superfield of Lorentz type (A/2, B / 2 ) (totally symmetric in its undotted indices and, independently, in its dotted indices) G 1 , , , , l A b , , , . kisL B said ( ~ )to be on-shell massless if it satisfies the system of supersymmetric massless equations D*G= j
j 2 =O ~
d,pDYG = d,pDtG = 0 Copyright © 1998 IOP Publishing Ltd
(6.4.51~) (6.4.51b)
529
Dynamics in Supergravity
where superfield indices are suppressed, and the supplementary conditions a “ ’ ” G a ,...z,h,.
..*E
(6.4.52a)
=0
(6.4.52b)
~YhlGal...aAhl.,.hs=O
of which the former (latter) must be imposed when A # O (BZO). Now, we would like to obtain an answer to the following problem. Suppose, G transforms according to some representation of the superconformal group, 6G = - a
t l G + jd, + pC51+ d, -,5.Ctl}G
(6.4.53)
for any conformal Killing supervector 5. The question is: in what cases is this transformation law consistent with the equations (6.4.51,52)?In other words, under which conditions does 6G remain on-shell massless? Let us remind ourselves that the similar problem for on-shell massless fields in Minkowski space was solved in Section 1.8. To begin with, consider the case A#O. Under the special conformal transformations, G changes according to the rule f
i ..h , =- f P b V p b G a l . ,
2
.@Ah].
.aAkl
,,
.h B
with Vldrwritten as in expression (6.4.44~). We demand the requirement (6.4.54) i F t 6 f G a ,...a,&. h, = 0. With the help of condition (6.4.52a),it is not difficult to verify that ,,
_.
B
+ ‘bDYGpa2...aAhI,,
.hE
-
...
G ~ ~ z . . . n n b bk...b# l ()
k=l
‘6t)&k
I
- 88O,Da1GaIa2.,.a,& I ...hB
*
(6.4.55)
Let us point out that the second equation (6.4.51) and the supplementary lead to the consequence condition (6.4.52~) DE’G a l a 2 . . . a A h l . . . h E0=
(6.4.56)
(for more details see Section 2.7). Hence the last term in relation (6.4.55) vanishes. To cancel the other terms in relation (6.4.55), one must demand the system of conditions: (1) d, +
+ d,
1
-)
= -jA
+1
(2) B=O (3) D,G*,. . . a , = O Copyright © 1998 IOP Publishing Ltd
(6.4.57)
530
Ideas and Methods of Supersymmetry and Supergravity
So, our superfield must be chiral with undotted indices only. Finally, let us recall that 451 is chiral. Then the variation (6.4.53) proves to be chiral, Dk6G =0, when
-,
(6.4.58)
d, = 0.
-,
As a result, the superconformal weights d ( + ,and d, of G are fixed completely. Further examination of the compatibility of the superconformal law (6.4.53)
with the massless equations (6.4.51,52) can be shown to give no additional restrictions on G. The case B # 0 is treated similarly. Finally, let us examine the case A = B = O . We begin by observing that equations (6.4.51) lead to the on-shell equation G = 0. Then, imposing the requirement
(6.4.59)
n6,G=O
proves to be equivalent to d ( + ) + d ( - ) = 1.
(6.4.60)
D26,G = D26,G = 0
(6.4.61)
Next, imposing the conditions leads to the equations (1 -d,+))D,G=O
(6.4.62)
(1 -d,-))D,G=O.
The system of equations (6.4.60,62) has the solutions: d,+,=l
d,-,=O
DkG=O
(6.4.64~)
d,+,=O
d,-,= 1
D,G=O.
(6.4.636)
or
From the above considerations we see that only undotted (or scalar) chiral and dotted (or scalar) antichiral on-shell massless superfields constitute representations of the superconformal group. Such superfields are said to be ‘on-shell massless conformal superfields’. In conclusion, let us summarize all the conditions determining on-shell massless conformal superfields: The scalar case ( A = B = 0) DkG=O d(+)= 1 Copyright © 1998 IOP Publishing Ltd
D2G=0 d,-,=O
(6.4.64~)
Dynamics in Supergravity
53 1
or
D,G=O
D2G=0
d,+,=O
d , - , = 1.
(6.4.64b)
The undotted case ( A # 0, B = 0) D,G,I...m(A=O
DZ1G,,,, ..,=O
d,+,=A/2+1=~(G)+l
(6.4.64~)
d,-,=O.
The dotted case ( A = 0, B # 0)
D,Gj,,,i,=O d,+,=O
DiX'Cklb2,,,ks=0
(6.4.64d))
d,-,=B/2+1= --K(G)+1/2.
Here x(G) or x(G) denotes the superhelicity of the superfield under consideration.
6.5. Conformally flat superspaces, anti-de Sitter superspace In Section 5.1, we introduced the notions of flat superspace and conformally flat superspace. Recall that a superspace is said to be flat if there exists a coordinate system in which the gravitational superfield and the chiral compensator have the flat form 2
m =8,meoa?j
cp = 1
(6.5.1)
a superspace is said to be conformally flat if in some coordinate system the gravitational superfield has the flat form 2
m
= bameaa?j
(6.5.2)
with cp being unrestricted. Now, our goal is to reformulate these definitions in terms of the supertorsion tensors R, G, and WEBY, that is, in a covariant fashion. We also consider andi-de Sitter superspace-the simplest solution of the Einstein supergravity equations with cosmological term (6.1.22). 6.5. I . Flat superspace To begin with, let us obtain necessary and sufficient conditions for a superspace to be flat. In spite of the simplicity of the problem, it deserves some discussion. In the case of a supergeometry determined (in the gravitational superfield gauge) by the prepotentials (6.5.l), the corresponding supertorsion tensors R, Ga and WxPyare zero, R = G, = W,pt = 0. Copyright © 1998 IOP Publishing Ltd
(6.5.3)
532
Ideas and Methods of Supersymmetry and Supergravity
So, every flat superspace should be characterized by the conditions (6.5.3), they are necessary conditions. Remarkably, they also turn out to be sufficient. Let us argue this statement. We choose the gravitational superfield gauge (5.4.38)supplemented by the Wess-Zumino gauge (5.8.1) and analyse the consequences of equations (6.5.3). First, from relations (5.8.17, 19) we see that the supergravity auxiliary fields vanish,
B = A" = 0.
(6.5.4)
Secondly, from relations (5.8.21,22)we deduce 2obed
=
(6.5.5)
where g O b c d is the curvature tensor associated with the space-time covariant derivatives V, defined by equations (5.8.16,23-25). Finally, owing to relation (5.8.30),the gravitino field strengths (5.8.27) vanish,
(6.5.6) VQbY = Tab= 0. The system (6.5.4)-(6.5.6) represents the total set of component requirements which equations (6.5.3) lead to. Recalling relations (5.8.20) and (5.8.27), we can rewrite expressions (6.5.5) and (6.5.6) as follows:
As may be seen, space-time may possess non-trivial torsion even if all the superfield strengths characterizing the supergeometry are zero. However, the torsion turns out to be gauge dependent and, under the conditions (6.5.4-6), it can be switched off using the local supersymmetry invariance (recall that in the Wess-Zumino gauge, the transformations (5.8.43) are also at our disposal). Indeed, the relations (6.5.7) and (6.5.8) show that y a y = Vat,
(6.5.9)
for some undotted spinor field &(x). On the other hand, under equation (6.5.4) the gravitino transformation law (5.1.45) takes the form 6Y
= - 2V,E,.
Therefore, the local supersymmetry invariance can be used to impose the gauge (6.5.10) Ya,=Y0'I=O * 2Ta;=0. Note that the system of relations (6.5.4)-(6.5.6) is locally supersymmetric invariant. Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
533
In the gauge (6.5.10), the only non-zero field in the expressions (5.8.1) is the vierbein, and the covariant derivatives V, satisfy the algebra P a ,
V b l =O-
Hence, the space-time general coordinate and local Lorentz invariances can be used to take the gauge
eam= 6,".
(6.5.11)
This completes the proof. In summary, a supergeometry (under the Einstein supergravity constraints) is flat if and only if the corresponding supertorsion tensors R, G , and WolSv vanish. 6.5.2. Conformally $at superspace Consider a supergeometry admitting the gauge fixing (6.5.2). Evidently, the corresponding covariant derivatives 9, are connected to flat global derivatives D, by means of some super Weyl transformation:
ga= FD, -2(DPF)M,p
gk =FD, -2(DBF)a4
1 - i 9,&= FFazj!+ - F(D,F)D, + - F(D&F)D,-iF(D,DBF)a,B 2 2
(6.5.12) -iF('(DkDPF)MaP
where F=q1i2@-l
Dkq=O.
Expressions for the supertorsion tensors can be read off from relations (5.5.14): R=
_ _1 q - 2 D 2 @
(6.5.13)
4
q)Dhln@+ia,,ln(@/cp) and
(6.5.14)
Wasr= 0.
We see that the super Weyl tensor of any conformally flat superspace is zero, Now, we are going to argue that the requirement (6.5.14) turns out to be the sufficient condition for conformal flatness, for a wide class of supergeometries. Let 9Abe a set of covariant derivatives which satisfy equation (6.5.14). If the corresponding chiral scalar R Z O , we change QA by some super Weyl transformed derivatives 9;, according to rule (5.5.13), such that R'=O. To achieve this, we must choose, due to relation (5.5.14), a covariantly chiral scalar superfield 0 satisfying the equation 1
-- (9z -4R)e"=0 4
Copyright © 1998 IOP Publishing Ltd
gkO=
0.
(6.5.15)
534
Ideas und Methods of Supersymmetry and Supergravity
As a particular solution, one can take
eo= 1 +4
+
-!(g2K)+R R 4
)-'(a2
-4R)R
(6.5.16)
where + is the chiral d'Alembertian (6.3.18). In order that the expression (6.5.16)is well defined, all supergeometry superfields should obey appropriate boundary conditions (for instance, be rapidly vanishing at infinity). Note that WiP,=O, due to relation (5.5.14). Suppose that G i = O also. Then, the supergeometry determined by 9; is flat, and, as a result, the supergeometry determined by g A is conformally flat. Thus, it remains to show that the relations R = W,p,=O
(6.5.17)
G , = 0.
(6.5.18)
imply Our proof will not pretend to strictness, we simply sketch the basic steps. Let gAbe a set of covariant derivatives characterized by conditions (6.5.17). We choose the Lorentz gauge (5.4.8) resulting in
9@ = E , + QaBgMPY
a, = & + nijfHfi9.
(6.5.19)
Such a gauge fixing leaves the possibility of making some residual Lorentz transformations of the form
(6.5.20) Next, since {ga,9p) = 0, the superconnection Rap, is trivial and we can write
. changing e' to There is some inherent arbitrariness in choosing U X pIndeed, e"'=e'eh
h=hbMpi
&hp,=O
(6.5.22)
leaves BAunchanged. The gauge freedom (6.5.20,22)can be used to impose the gauge E2Up)= t 2 U p y= 0.
(6.5.23)
Further, due to relation (6.5.17),the commutator [g,,Qpj] does not contain Mp)-terms (see expressions (5.5.6)). In the gauge (6.5.19), this requirement is equivalent to a2napt= 0. (6.5.24) Under reasonable boundary conditions, the system of equations (6.5.23)and Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
535
(6.5.24) has only trivial solutions, so
zz,,
As a result, the commutator leads to
[a,,
9,0]
= 0.
does not involve Mbp-terms, which
*
agG,=O
Ga=O.
In summary, it is the super Weyl tensor that measures whether or not a superspace is conformally flat. Superspace is conformally flat if and only if the corresponding super Weyl tensor is identically zero. 6.5.3. Physical sense of conformal jiatness We have seen that every curved superspace admits the choice of the Wess-Zumino gauge (5.8.1). What are the peculiar features of conformally flat superspaces? In the Wess-Zumino gauge, the residual gauge freedom consists of the space-time general coordinate, local Lorentz and local supersymmetry transformations. When our superspace is conformally flat, the above gauge freedom can be used to impose the gauge
earn = e1148,m Ym,= eam(cr"&
A" = g""i3,zz
(6.5.25)
+ O ( Y ,T)
for fields $(x) and O(x). Equivalently, the fields eam,Ym,and A" propagate only those modes that are generated by the Weyl transformations (5.1.28), the local S-supersymmetry transformations (5.1.30) and the local chiral transformations (5.1.29), respectively. In this and only this case, we are in a position to move from the Wess-Zumino gauge (5.8.1) to the gauge (6.5.2), applying a proper Weyl +local S-supersymmetry + local chiral transformation. As a result of equations (5.1.39,40), the chiral compensator, transformed from the gauge (5.8.1) to the gauge (6.5.2), is of the form
e)
(p3(x, = e- 3/4(ei'(x)
+ eaijc,(x)+ P e - 1/4(x)~(x)).
(6.5.26)
Here 0= fi and 4, are fields uniquely connected to the similar ones in gauge (6.5.25). Given a conformally flat superspace admitting the gauge choice
eam= e1/48am y m c , = ~ m 0= (6.5.27) the corresponding supergeometry can be described by the superfields
Copyright © 1998 IOP Publishing Ltd
536
Ideas und Methods of Supersymmetry and Supergravity
6.5.4. Anti-de Sifter superspace We are going to construct a supergeometry determined by W,a7= G, = 0
(6.5.29)
R = p = const # O or, equivalently, possessing the covariant derivatives algebra
a,}= - 2i9ui
{gz,
[Qzh,
9 p / d = - 2FP(%/IMzp
+ &,pA,/J).
I t will be called the 'anti-de Sitter supergeometry'. Clearly, anti-de Sitter supergeometry is conformally flat. One can verify that the supergeometry under consideration admits the gauge fixing (6.5.27). Therefore, it is within our power to take the prepotentials in the form (6.5.28). In accordance with the relations (6.5.13) and (6.5.29), flat (anti)chiral superfields @(z)and cp(z), D,(P =0, should satisfy the equations
(6.5.3 1a) 1
- (D, In cp)DjlIn (p + i2,, ln(@/cp)= 0. 2
(6.5.3113)
Note that the equations (6.5.31~)coincide in form with the dynamical equations (3.2.9) of the Wess-Zumino model with chjral superpotentials $pc(@)= -(p/3)(D3. A particular solution of the above problem is
(6.5.32) where x 2= qmnxmx". From solution (6.5.32) we may read off the space-time vierbein
(6.5.33) and the space-time metric 1
ds2= (1
-+')
qmndx"' dx".
(6.5.3 4)
This is the metric of the well-known space from general relativity, the anti-de Sitter space. As a manifold, anti-de Sitter space represents locally the domain Copyright © 1998 IOP Publishing Ltd
531
Dynamics in Supergravity
of R4 contained inside the hyperboloid X’ = 4/ F p .
(6.5.35)
The Lorentz covariant derivatives associated with the vierbein (6.5.33)read (6.5.36)
where xa &xm, and satisfy the algebra
Before going further, we would like to make one comment. In general relativity, there are known to be two (maximally symmetric) space-times of constant non-zero curvature, that is, of the form (6.5.38) The choice W>O corresponds to de Sitter space, and the choice W < O corresponds to anti-de Sitter space. Clearly, the set of requirements (6.5.29) represents the supersymmetric analogue of that in (6.5.38). So, every supergeometry of the type (6.5.30) can be called a ‘supergeometry of constant supertorsion’ (recall that supertorsion determines supercurvature). Our consideration shows that, for any complex p, the supergeometry described leads to a space-time geometry of constant negative curvature. Therefore, it is anti-de Sitter geometry which admits supersymmetrization. The de Sitter geometry cannot be extended to a supergeometry of constant supertorsion. Now, we formulate what will be understood by ‘anti-de Sitter superspace’. As a supermanifold, it is locally identified with the domain of R4I4such that the bodies of c-number variables x m lie inside the hyperboloid (6.5.35). The relevant supergeometry is determined by the prepotentials (6.5.28), with q ( x , e) defined as in expression (6.5.32). 6.5.5. Killing supervectors of anti-de Sitter superspace To develop supersymmetric field theory in anti-de Sitter superspace, it is necessary to determine all possible Killing supervectors. Recall that for every curved superspace, the corresponding Killing supervectors (if they exist) generate symmetry transformations, that is, ones leaving the covariant derivative unchanged. The simplest way to find the Killing supervectors of anti-de Sitter superspace is to use the observation that Weyl equivalent supergeometries possess the same set of conformal Killing supervectors (see subsection 6.4.1). Copyright © 1998 IOP Publishing Ltd
538
Ideas and Methods of Supersymmetry and Supergravity
Evidently, anti-de Sitter and flat supergeometries are Weyl equivalent. Above we have found all the conformal Killing supervectors of flat global superspace, hence, of anti-de Sitter superspace too. They are given by the expression (6.4.32), with P(x,6) and E."(x, 6) written as in expression (2.9.16). Killing supervectors of anti-de Sitter superspace are those conformal Killing supervectors which satisfy the equation (6.4.23),with q(x, 6) being the relevant chiral compensator (6.5.32).Solving equation (6.4.23)restricts the parameters in expression (2.9.16) as follows: 1 2
ij. = --PE&.
"
(6.5.39)
Notice that no restrictions on the Lorentz parameters K,p appear, since the chiral compensator (6.5.32) is Lorentz invariant. Inserting the above expressions into relations (2.9.16) gives
(6.5.40)
The Killing supervector associated with these superparameters can be read off from expression (6.4.27). Given a Killing supervector < = < A D Aof anti-de Sitter superspace, it generates the operator X[5],defined by expression(6.4.38),which commutes with the anti-de Sitter covariant derivatives (6.5.12,32). X [ < ] can be represented in the form
(6.5.41)
where Pa&,Jap, Jib, Q, and Qiare the super Poincare generators (2.4.37), and Vak, Sa and Si are the same as in expressions (6.4.37) but with d , + ) = d , - ) = 0. The operators just introduced satisfy, due to relation (2.9.21), Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergraviiy
539
the following (anti)commutation relations {Q,, Qdr} =2Pak {Qw
Qp>= - 4iPJap
CQ,,
Pp#1= - E@Q#
CP16r, Pp#l= - 2iPP(%aJ,p
(6.5.42)
+ E&#)
[J,p, QJ = i ~ ~ ( , Q p ) CJ,p, Pr91= iEr(,Pp)p
JysI= i"(nJlp)s + i"(ccJp)y. The remaining (anti)commutators vanish or can be found by Hermitian conjugation. The relations (6.5.42) define the superalgebra known as 'super de Sitter algebra'.
CJ,,
6.6. Non-minimal supergravity 6.6.1. Preliminary discussion This section is devoted to consideration of a family of supergravity-matter dynamical systems classically equivalent to the pure supergravity theory with the action (6.1.3). The existence of such systems implies that there are several off-shell supergravity formulations realized in terms of different sets of superfields, but possessing the same dynamics on-shell. One naturally comes to the results given below, when trying to solve the following dynamical problem. Suppose, we have a super Weyl invariant theory of supergravity coupled to a (constrained) scalar superfield Y. As usual, by super Weyl invariance we understand that the total supergravity-matter action superfunctional S[Y, gA]is unchanged under transformations of the form cp + cp'= e'cp
tp+yI=e-di+io-d(-t5
~,D=O
(6.6.1)
where cp is the supergravity chiral compensator, d ( + ) and d ( - ) are fixed numbers and D is an arbitrary covariantly chiral scalar superfield. Suppose also that our system is dynamically equivalent to the pure supergravity theory. By equivalence we understand that one can choose a super Weyl gauge in which the dynamical equations of the system under consideration (6.6.2) imply the Einstein supergravity equations R=Ga=O
(6.6.3)
and vice versa (in equation (6.6.2) T and Tu are the supertrace and supercurrent, respectively). As a result, S[Y, gA]can be taken to describe the supergravity dynamics, instead of the original supergravity action. Copyright © 1998 IOP Publishing Ltd
540
Ideas and Methods of Supersymmetry and Supergravity
Off-mass-shell, the super Weyl invariance (6.6.1)can be used to choose the gauge
(6.6.4)
q=l.
Then, due to relation ( 5 . 5 4 , every K - and A-transformation should be supplemented by a special super Weyl one to preserve the gauge, and the transformation law of Y takes the form
(6.6.5) instead of the original law &'= K M d M Y ' .
(6.6.6)
Therefore, the superfield Y loses its scalar nature and turns into a density. In the case of non-chiral Y , we then obtain a supergravity formulation with non-chiral compensator. The above dynamical problem is trivially solved when setting Y to be covariantly chiral. One introduces the action
(6.6.7) obtained from the supergravity action (6.1.3) by the replacement cp
+
(6.6.8)
cp@.
Evidently, S [ @ , g A ]possesses the super Weyl invariance q+e"q
@+e-a@
Bka=O.
(6.6.9)
The corresponding matter and supergravity dynamical equations are
(6.6.10) where we have used equation (6.3.10).Now, imposing the super Weyl gauge @= 1
(6.6.11)
reduces the action (6.6.7) to the supergravity one (6.1.3), and the dynamical equations (6.6.10) to the supergravity ones (6.6.3). Remark. The theory considered represents the supersymmetric analogue of the Weyl invariant version of the Einstein gravity described by the action
(6.6.12)
Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
541
where C is a scalar field. This action is obtained from the gravity action (6.6.1) by the replacement (6.6.13)
eam+ eOmC-
so it is explicitly Weyl invariant. In the remainder of the section we shall analyse another, non-chiral, choice with Y being a 'complex covariantly linear scalar superfield' defined by
Y#Y
(G2-4R)Y =O
((g2-4R)Y #O).
(6.6.14)
Wishing to work with unconstrained superfields only, one must represent, due to relations (5.5.10), the superfields Y and in the manner
y=
q=
(6.6.15)
with xa being a spinor superfield. Then, any theory of the superfields and Y can be equivalently treated as a gauge theory of the superfields and & possessing the gauge invariance 6xa = @Asp
AIp = ADz.
(6.6.16)
In such an approach, Y and Y appear as gauge invariant superfield strengths. One more useful representation for Y follows from the relations (5.4.2) and (5.4.23).The requirement (6.6.14) means that (6.6.17) It is seen that every complex covariantly linear superfield Y is determined by a flat complex linear superfield 9. 6.6.2. Complex linear compensator We begin by examining the question: what are admissible superconformal weights d ( + , and d,-,_for which the transformation law (6.6.1) is consistent with the constraint (g2 -4R)Y = O? Under the super Weyl transformations (5.5.13), the operator (g2 -4R) changes as
(a2-4R) + e - 2a(9-4R)e5+ R&B-dependent terms. -2
(6.6.18)
Therefore, the most general transformation law for Y is Y -P Y' = exp[ - d(+,a- L?]Y
(6.6.19)
with d ( + ,being arbitrary. The same result also follows from relation (6.6.17). Indeed, the relation (6.6.17) shows that Y depends explicitly on @ as Y a cp-' (recall, F a @ l i 2 c p - ' ) and can be taken to depend on cp as Y a c p - d l + l by means of the redefinition y + c p - d [ + ) ~ . Remark. It is not difficult to check that the spinor superfield &, which Copyright © 1998 IOP Publishing Ltd
542
Ideas and Methods of Supersymmetry and Supergravity
arose in expression (6.6.15), should transform according to the rule ( 1 -d,+,)o--o 32
-1
jidr
(6.6.20)
to produce the law (6.6.19). Further, we restrict ourselves to the consideration of cases with real d,+, # 1. The choice d ( + ) = 1 seems to be very special, since here the representation defined by the law (6.6.19)becomes reducible, and the constraint Y = Y can be imposed. The case d ( + )= 1 will be studied in the next section. By tradition, we will write d ( + ) in the form d(+)=-
1 -3n 3n-t 1
n# - 1/3, 0.
(6.6.21)
The action superfunctional invariant under the super Weyl transformations q+q'=e"cp
3n-1 Y + Y = e x p [3n+lo-0]Y
Bdra=O (6.6.22)
reads (6.6.23a) Note that the superfield Y(, 5 # 1, is not linear and so the choice of the super Lagrangian in the form (YY)( seems to be natural. With the help of the relations (5.4.19) and (5.5.4), the action can be rewritten in terms of the supergravity prepotentials and the superfield y (6.6.17): S,,,=-
'S
nK2
t
d8z E"[($q~)~(l .e~)(l.e~)]("1)/2(py)'3"1)~2, (6.6.23b)
The constant ( l / n ) has been introduced into the action to normalize EI" (&"x(l +e28L9Y= 1 +ne28L9). The action (6.6.23) was studied by W. Siege1 and S. J. Gates. Below we shall show that it describes supergravity. This supergravity formulation is known as 'non-minimal supergravity'. Remark. Let us set n= - 1/3 in equation (6.6.23b). Then, the superfields y and 7 drop out of the action and we recover the supergravity action (6.1.4). So, the choice n= - 1/3 is said to correspond to the minimal supergravity described in detail above. Owing to the super Weyl invariance of S("),we are in a position to impose the super Weyl gauge (6.6.4). After doing this, the supergravity gauge group acts on according to the law (6.6.5), where d,-,= 1 and d ( + , is given in equation (6.6.21). Now, let us determine the transformation law of the Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
543
superfield j j (6.6.17).F changes according to the rule
where we have used equation (5.4.14) and the fact that in the gauge (6.6.4) every A-transformation should be supplemented by the su r Weyl - 8,P). Next, the operator e changes transformation with o = as in relation (5.4.14). As a result, one arrives at
r
+
Sf = (Amam A’8,)f
- 8‘(A,9)
n+l +(8,A”3n+ 1
8,A’)f.
(6.6.24)
It is worth recalling that here the parameters A” and P are8-independent while R, are arbitrary. On these grounds, the variation Sjj turns out to be linear,
Ff=O * F2Sf=O. A complex linear superfield 9, Ff= 0, transforming by the law (6.6.24)under
the supergravity gauge group, is said to be a ‘non-minimal supergravity compensator’ or, simply a ‘complex linear compensator’. The arbitrariness in the choice of Ab can be used to eliminate the compensator, that is to impose the gauge j j = 1.
(6.6.25)
Then, the parameters of residual gauge transformations should obey the constraint (3n + l)FA, = (n + 1)(a,,,Am- a,A”).
(6.6.26)
6.6.3. Non-minimal supergeometry Up to a factor, the action (6.6.23)is obtained from the minimal supergravity action (6.1.3) by changing the chiral compensator cp to the super Weyl invariant combination
1
( p [ y 3 n - l T 3 n t l ( 3 n - 1)/12n
(6.6.27)
This observation can be used for constructing super Weyl invariant and generally covariant derivatives. Namely, let us introduce the set of operators D, =(Do,D, Da) as
=E& - 2(@0)&#&~
Dt,
1
D, = -- (zap{ ID, ID,}.
4
Owing to the scalar nature of Y , the operators D, change covariantly under Copyright © 1998 IOP Publishing Ltd
544
Ideas and Methods of Supersymmetry and Supergrauity
the supergravity gauge group:
Since D,In(UJ4U2)#0, D, are obtained from the minimal covariant derivatives 9, (possessing the algebra (5.5.6)) by applying a special generalized super Weyl transformation (compare with equation (5.5.13)). It is a simple exercise to verify that the derivative D, are invariant under the minimal super Weyl transformations (6.6.22). That is why one can impose the gauge cp = 1 and this does not spoil the transformation properties (6.6.29). Covariant derivatives D, obey (anti)commutation relations of the form (5.3.53), and the corresponding supertorsion tensors will be denoted as T,, R, Go and WXp2.With the help of straightforward but tedious calculations, one can check that they are expressed in terms of the supertorsions R , G , and WXP), associated with the minimal covariant derivatives gA,as follows:
T,= D,T
I-= i n ( u 4 ~ 2 ) (6.6.30)
Finally, the linearity condition (6.6.14) proves to be equivalent to the following relation (6.6.3 1 ) which is known as the 'non-minimal supergravity constraint'. The covariant derivatives algebra (5.3.53) under the constraint (6.6.31) defines the 'non-minimal supergeometry'. Remark. Representing D, in the form D, = [ E A M a M
+'
2
Y / A ~ ~ M ~ ~
(6.6.32)
and defining the Berezinian of the non-minimal supervierbein E = Ber( ), E ,
one readily obtains [E=E(OU)Z=E(~Y)-'3"+1)'2. Copyright © 1998 IOP Publishing Ltd
(6.6.33)
Dynamics in Supergravity
545
Therefore, the action (6.6.23) can be rewritten as
'S
S[,,,=nti'
(6.6.34)
d8z E-'.
In such a form, S(n) looks completely similar to the minimal supergravity action (6.1.3). 6.6.4. Dynamics in non-minimal supergravity We are going to show that the supergravity-matter theory with the action (6.6.23) is dynamically equivalent to the Einstein supergravity. Dynamical equations corresponding to the action S(n) are
U,= U,=o -
(6.6.35)
Ga=O.
(6.6.36)
and Note that the equations (6.6.35) lead to (6.6.37)
R=R=O
owing to the non-minimal constraint (6.6.31). Let us comment on the derivation of equations (6.6.35,36). First, it is necessary to vary S[,,) with respect to Y and T. Any variation 8" satisfies the constraint -4R)dY = 0, but it can be represented, due to (6.6.15), in the form 8Y =g&a, with 82' being arbitrary. Then, simple calculations give equation (6.6.35). Secondly, it is necessary to find the supercurrent and to set it to zero. To determine the supercurrent, one must calculate the variation
(a'
2, =e - iHYBo Xa
2a=e Xi where and are defined by equations (5.6.40,57-59) with o=O=O. Then, one obtains (6.6.36). Consider the consequences of the dynamical equations. From relations (6.6.30) and (6.6.35) we deduce -iHnBa-
vk
z-', v,
U = exp[
*
p
-: p]
g,p=O
=. Y=exp
3n+ 1
for some covariantly chiral scalar superfield p. Hence, due to the super Weyl invariance (6.6.22), the super Weyl gauge (6.6.38) is admissible on-shell. Under this gauge fixing, the equations (6.6.36) and (6.6.37) reduce to the Einstein supergravity equations (6.6.3), as is seen from Y=l
Copyright © 1998 IOP Publishing Ltd
546
Ideas and Methods of Supersymmetry and Supergravity
relations (6.6.30). Thus, the action S(,,) describes the Einstein supergravity, for any n # -3,O. The S(,,)is known as the 'non-minimal supergravity action'. The equivalence of the two theories with classical actions S,, (6.1.3)and S(,,), respectively, can also be seen as follows. Consider the auxiliary supergravity-matter model with the action
where V is an unconstrained complex scalar superfield and 0 is a covariantly chiral scalar superfield, BkQ= 0. The action is invariant under the super Weyl transformations q-+egcp
[::r: 1
V+exp - 0 - 5
V
(6.6.40)
.9ka=0.
Varying S[V, 0,gA]with respect to 0 and i6 gives (g2-4R)V=(g2-4R)P=0
*
V=Y
with Y a complex covariantly linear scalar superfield. Then we have
and the above superfunctional reduces to the non-minimal supergravity action. On the other hand, imposing the dynamical equations for V and P makes it possible to eliminate these superfields. As a result S[V, 0,gA] reduces to the action (6.6.7), which represents the super Weyl invariant version of the minimal supergravity action. Therefore, the minimal and non-minimal supergravities are dually equivalent. 6.6.5. Prepotentials and j e l d content in non-minimal supergravily Now it is time to discuss the results obtained. We commence by recalling that imposing the conformal supergravity constraints (5.3.15) leads to the covariant derivatives algebra (5.3.53) and expresses all the supergeometry objects in terms of the unconstrained superfields Nap, W Mand F (with Nup, Wp and Wb purely gauge degrees of freedom). The minimal supergeometry formulation is characterized by one more constraint (5.5.1) manifesting the fact that F becomes a function of W M ,, ' W 4 and G, where 8 is a &independent superfield, Jb+ =O. The superfields NuF, W M and ijj are the prepotentials of minimal supergravity. The non-minimal supergeometry formulation is characterized by the set of constraints (5.3.15) and (6.6.31), Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
547
where n # -3, 0. The last constraints leads to the fact that F becomes a function of W M ,W M , 4 and i, where 9 is a superfield under the restriction F.j =O. The superfields N a p , W Mand .j are the prepotentials of non-minimal = 0 is stronger than 3fi8bf = 0. supergravity. Clearly, the requirement As a consequence, .j describes more field degrees of freedom than 4, and the non-minimal algebra (5.3.53)looks much more complicated than the minimal algebra (5.5.6).That is why we use the term 'non-minimal'. With respect t@ the supergravity gauge group, the prepotentials W M ,4 and .j are characterized by the transformation laws (5.4.14),(5.5.5) and(6.6.24), respectively. In both minimal and non-minimal supergravities, one can gauge away the compensators resulting in 3 = 1 and 9= 1 . Then we end up with residual gauge transformations constrained by equation (5.1.32) in the minimal case and by equation (6.6.26)in the non-minimal case. Such a gauge fixing does not seem to be the most useful. The point is that equation (6.6.26) restricts the parameters i,and , this makes it impossible to gauge away all Wp and W,. The most useful gauge choice is the following. In both minimal and non-minimal supergravities, it is in our power to choose the gravitational superfield gauge (5.4.38) supplemented by the Wess-Zumino gauge (5.1.17). Residual gauge invariance is described by the superparameters (5.4.42), (5.1.18). Then, the local scale +y,-transformations and the local Ssupersymmetry transformations can be used to bring the minimal and non-minimal compensators into the form: (6.6.41~) ?(x,
e, 8)= I + eapl(x)+ s2B(X) + e d v , ( X ) + B ~ B ~ F ~ ( ( (X6)..6 . m )
Here B and U'' are complex scalar and vector fields, respectively, pa and F E are spinor fields. The real vector field Am from(5.1.17)andthe fields presented ) (6.6.41b) do not propagate on-shell. So, the in expressions ( 6 . 6 . 4 1 ~and minimal and non-minimal supergravity formulations have the same dynamical fields, the graviton and the gravitino, but different sets of auxiliary fields. 6.6.6. Geometrical approach to non-minimal supergravity Non-minimal supergravity has been introduced above in the dynamical approach as the supergravity-matter system classically equivalent to Einstein supergravity. Remarkably, it possesses a purely geometric realization analogous to that developed for minimal supergravity in Section 5.1. Here we describe the approach suggested by E. Sokatchev. Consider complex superspace C4I4parametrized by four complex c-number coordinates ym and four complex a-number coordinates: undotted P and dotted p,, pb #&,,(e")*= ~ ~ , fCoordinates ?. will be unified in
iM= (y", P,pic) and Copyright © 1998 IOP Publishing Ltd
gM = ( y",
W)
(6.6.42)
548
Ideas and Methods of Supersymmetry and Supergravity
looking on "5 as the coordinates of some complex superspace C4I2.We take the point of view that C4I4is the trivial fibre bundle over base C4I2with fibre Coi2. Then, natural coordinate transformations on C4I4 are 'triangular' transformations of the general structure:
Y = m y , e)
with
e/ji=fP(Y,
e)
p;,=ffi(Y,
e, D)
(6.6.43)
The infinitesimal form for such transformations is Y'm
= y m - E,"( y,
e)
t P = P - j,p( y, e)
(6.6.44)
p'.P = p P. - ~ P. (Y ,e,p) with A", i P and Afi being arbitrary superfunctions of the proper arguments. The supergroup of coordinate transformations (6.6.43)has a natural family of subgroups, each of which is singled out by a restriction of the form
[ (~ Ber
) ~ " =+[Ber( 1
$)y"
(6.6.45)
for some real number n. In the infinitesimal case, this restriction coincides with equation (6.6.26). Further, let us define an embedding of real superspace Iw4l4, parametrized by coordinates x", QP and B,, into our superspace C4I4in the manner:
Here 2"and A? are superfields on R4I4 which determine the embedding. Obviously, every coordinate transformation (6.6.44) induces, first, the coordinate transformation on IW4l4: 1 1 x" = xm--Am(x+ixe)-- Xm(x 8) 2 2 (6.6.47) O'P = Ob- AP(x + e)
ix
-ix
F, = Bfi- I,(x -i x 8) and secondly, the change of superfields Zmand 2fi: i
ix 0)--2i Xm(x - i x 8) (6.6.48a) 2'L(x'. 6 , Q )= Sfi(x, B,8)+ &(x - ix 8)-iii(x+ i3y: B,8+2) (6.6.48b) Here (xk iX) and (8+2)mean (x"& i 2 " ) and (gi,+ 2c), respectively. *"(XI,
@, B') = Pm(x,0, Q) + -2 JWm(x+
Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
549
Based on the above consideration, one can make the following proposal. For a given real n f O , the supergravity gauge group is the supergroup of coordinate transformations (6.6.43)constrained by (6.6.45). A curved real superspace is described by a 'generalized gravitational superfield' .N"(z)= 4 M ( z ) = ( P " ( Z ) , . H y z ) , 3 q Z ) ) .
(6.6.49)
The supergravity gauge group acts on P M ( z )according to the law (6.6.48). Let us argue the given proposal. To start with, consider the case n = -+. Then, the left-hand side in restriction (6.6.45) is identically unity, and the superparameters 2, in expressions (6.6.44) and (6.6.48b) are completely arbitrary. On these grounds, we are able to impose the gauge 2YM= (P 0,O) 1
After this, we arrive at the Einstein supergravity formulation described in subsection 5.1.3. Next, consider the case n f -3,O. We intend to show that there is a special gauge in non-minimal supergravity under which the situation described above is realized explicitly. In non-minimal supergravity, the prepotentials are N $ , W Mand 9. The invariance under the superlocal Lorentz group can be used to set N a p= dEp.The invariance under the general coordinate transformation supergroup can be used to gauge away Re W" (leaving some part of Re W", if we wish) and Wp. As a result, the following gauge
(6.6.50) is admissible. Evidently, this gauge fixing is weaker than the gravitational superfield gauge (5.4.38).So, the choice (6.6.50)will be called the 'generalized gravitational superfield gauge'. Under equation (6.6.50),every %-transformation should be supplemented by some K-transformation to preserve the gauge. Similarly to the derivation of equations (5.4.40-42), one now finds dew= KMaMeW-ewdMd, where
e), ~ q xe), , n,(x,
A~=(P(X,
1 K" = - Am(x 2
0, 8))
+ irt: e)+-21 xm(x - i.X: 8)
K!-'= A p ( x + iut: e) and the transformation laws
Copyright © 1998 IOP Publishing Ltd
R, = Ib(x- ix 8)
(6.6.51)
550
Ideas and Methods of Supersymmetry and Supergravity
It is seen that the relations (6.6.52)coincide with those in (6.6.48).Finally, imposing the gauge .j = 1 restricts superparameters AM in accordance with equation (6.6.26). 6.6.7. Linearized non-minimal supergravity In conclusion, let us describe a linearized version of non-minimal supergravity. Recall that the same construction for minimal supergravity was described in Section 6.2. Similarly to the minimal case, it is now convenient to use the prepotential parametrization described in subsection 5.6.1 and the chiral representation defined by equations (5.6.15,16). Upon imposing the A-supergroup gauge (5.6.18)and the super Weyl gauge rp = 1, the non-minimal supergravity action (6.6.236) turns out to be a superfunctional of the prepotentials H"= R", y and 7, with y being a flat linear complex superfield, D 2 y = 0 (D'y #O). The action reads
Here the objects H and ?!, are exactly the same as in equation (6.2.2).The action is invariant under the gauge transformations ae - 2 i H - - 2 i H -e - Z i H A (6.6.54) n+l &, - (A"i3, + A"D& + D'(A'y) + -(a,Aa - D , A U ~ ? . 3n+ 1 The explicit form of the gauge parameters A = A A D Awas given in equation (6.2.3). The set of superfields (H"=0, y = 1 ) constitutes a stationary point of (6.6.53) corresponding to flat global superspace. To linearize at this stationary point, we must expand the action up to third order in the superfields Ha, r and f, r being defined by y = 1+ r D2r= o ( D 2 r# 0). (6.6.55) Since the relevant computational work has been done in subsection 6.2.1, here we reproduce only the final result:
9n2-1 +-( 3 n4n+ 1 ) * rr+8n
Copyright © 1998 IOP Publishing Ltd
(6.6.56)
Dynamics in Supergravity
55 1
Note that we have made the rescaling Ha+ rcHa,r + Kr. The action obtained is invariant under the gauge transformations 6H,i = DkL, - D,L,
(6.6.57)
6r= 1 n + l D2DzL,+ D'D2Lj, 43n+l 4
1
L, being arbitrary. The equation (6.6.57) presents the linearized version of equation (6.6.54). One can readily see that the theories (6.2.5) and (6.6.56) are equivalent. It is sufficient to construct an auxiliary model realizing the dual transform for these theories. In the role of such a model, one can choose HaiDPD2D,H,,
n+l +(dol&Haa)' 8n
Dk]Hai)2- ( n + 1 X 3 n f 1 ) 4n
+-([DE, n+l 32
3n+ 1 (3n+ H~~(D,D&-D~D,P)+4 4n
--
1 +-9n2( v2+ P2) - 3( + 58) 8n
i HC?&a,&(V-7 )
-
W
(6.6.58)
v(T
V being an arbitrary complex scalar superfield. As a result, the theories (6.2.5) and (6.6.56) describe two on-shell massless super Poincare states of superhelicities (- 2) and 3.
6.7. New minimal supergravity
In the previous section, we have carefully avoided dealing with the value n=O. This case, similar to n= -f turns out to be very special, for at least three reasons. First, the Y-transformation law (6.6.22) is consistent with the constraint Y = Y if and only if n = 0. Secondly, the action superfunctional (6.6.23) becomes ill-defined for n = 0. Thirdly, the prescription for the construction of super Weyl invariant and generally covariant derivatives, given in subsection 6.6.3, does not work here. Now, we get down to n=O. 6.7.1. Real linear compensator The chief hero of our subsequent considerations will be a real covariantly linear scalar superfield [L:
(a2-4R)L Copyright © 1998 IOP Publishing Ltd
=(a2 -4R))[1= 0
L= C
(6.7.1)
552
Ideus und Methods of Supersymmetry and Supergravity
There exists a covariantly chiral spinor superfield qccsuch that 1 2
a.=-(Pq
%
a& =0.
+GhV+)
(6.7.2)
Clearly, qx is defined modulo redefinitions of the form: q,-q;=qa
+i(G2-44R)gzK
K=lf
(6.7.3)
with K being an arbitrary real scalar superfield. So, every dynamical theory of ii can be equivalently treated as a gauge theory of the superfields qz and ijh with the gauge invariance (6.7.3). Owing to the identities (5.4.23) and (5.4.2), the conditions (6.7.1) lead to
Therefore,
[I is
determined by some superfield
9 satisfying two equations:
? = ( F ) * = ~ - W [ F ~e EJ F ?I* ~
?,j=o
(6.7.5)
Note that in the case of a complex covariantly linear scalar superfield Y, the superfield .j defined by (6.6.17) and its conjugate 9 are independent. In accordance with equation (5.5.4),the reality condition (6.7.5)can be rewritten as t
63y-e e-
-
e~(iO~P)]
-w
(6.7.6)
[(l*eGl where cp=eii'+ and @ = e w e . Then, making use of the identities (1.1.13,14) leads to the very elegant formula:
Suppose, the supergravity chiral compensator cp acquires some displacement. To preserve the linearity condition (6.7.1), ii should also acquire some change. Remarkably, the transformation law of [I is uniquely fixed. Due to relation (6.6.18), one finds cp + cp'=e"cp
iL + IL'=e-"-"ll
$*o = 0.
(6.7.8~)
Remark. To produce the relations (6.7.8a), the chiral superfield qa (6.7.2) should be transformed according to the law q,+q;=e-3"/2
Vcc.
(6.7.8b)
An action superfunctional possessing invariance under the super Weyl transformations (6.7.8) can be obtained from the non-minimal supergravity action (6.6.23), if one suitably defines the limit n+O. Namely, let us set Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
553
Y='P=[i in S(n)and take the limit n-0:
S(n)=-
's
dEzE-'ll+-
nti2
'S
ti2
d8zE-'QIn 11.
The first term here vanishes identically, since
s
's
dEZE-'U.= -- d8~-(92-4R)11=0 4 ER1 -
(6.7.9)
Thus we obtain the action (6.7.10) It is invariant under the super Weyl transformations (6.7.8),since
( E - '[i)'= E -
'[i
and, therefore,
'S
dS= --
IC2
d8zE-'L{a+8)
The superfunctional (6.7.10), known as 'the action of new minimal supergravity', was invented by P.S. Howe, K.S. Stelle and P. Townsend and independently by S . J. Gates, M. RoEek and W. Siege1. Owing to the super Weyl invariance (6.7.8), one can impose the gauge cp = 1. Then, the transformation laws of R and fj under the supergravity gauge group take the form 1 1 d [ i = K M ~ , [ i + - ( e ~ ( d , A m - d , ~ ~ ) ) [ I + - ( e W ( d , ~ m - ~ ~ ~ ~ )(6.7.11) )[I and
3
3
+
dp = 3% = d,(A"?)
- d,(P.i)
- P(%$fj)
(6.7.12)
respectively. The derivation of equation (6.7.12) is the same as of equation (6.6.24) (formally, equation (6.7.12) is obtained from equation (6.6.24) by setting n =O). The superfield 9 constrained by equations (6.7.5) and transforming by the law (6.7.12) is said to be 'real linear compensator'. Due to arbitrariness of the superparameters Ab in equation (6.7.12), it is possible to choose the gauge .j= 1.
(6.7.13)
In this gauge, the residual gauge transformations are generated by Copyright © 1998 IOP Publishing Ltd
554
Ideas and Methods of Supersymmetry and Supergravity
superparameters iM constrained as +
(1 *A)= 0.
(6.7.14)
Next, from equation (6.7.6) we deduce t
c
(l.ew)=(l.ew).
(6.7.15)
Therefore, imposing the gauge (6.7.13) restricts the prepotentials W M according to rule (6.7.15). 6.7.2. Dynamics in new minimal supergravity Our next goal is to show that the supergravity-matter system with action (6.7.10) is dynamically equivalent to Einstein supergravity. Using relations (6.7.2), we represent S in terms of the (antikhiral superfields ij, and vc(,
'S
S=d8z E-'(ac(v,+B,4")1n(aBv~+~gijg). 21c2
This model, considered as the dynamical system of the supergravity prepotentials and the superfields qc(and qk, is characterized by the dynamical equations (6.7.16) with T and Ted,being the supertrace and supercurrent, respectively. In accordance with the variational rules for (anti)chiral spinor superfields (6.3.52), the equations of motion for vu and ijk are (B2-4R)geIn L=(g2-4R)B,ln L=o.
(6.7.17)
Here we have used equation (6.7.9) and the fact that variations 61, and dij, must be covariantly chiral and antichiral, respectively, hence 6L = &W6qc( + &3ijk) is covariantly linear. The supercurrent of the theory under consideration can be found in the same way as was done in subsection 6.3.3. for the theory (6.3.49). The result reads I C T,, ~
= L(2Ga,
+ (aZIn R)B, In 11 - [ac(, a,]
In L}
3 +-qa(92-4R)B,1n L-2ij,(532-4R)9a1n 2 2
L
(6.7.18)
or, after taking into account equations (6.7.17), as K ~ T , , = L { ~ G , , + ( ~L)a,ln ~ I ~ L-[g,,@]
In L}.
(6.7.19)
The general solution of equations (6.7.17) is k=e-P-P Copyright © 1998 IOP Publishing Ltd
gkp=O
(6.7.20)
Dynamics in Supergravity
555
for some covariantly chiral scalar superfield p. Then, the super Weyl invariance (6.7.8) can be used to gauge away p. In other words, the gauge choice [I= 1
(6.7.21)
is admissible on-shell. After this, the linearity condition (6.7.1) means R =0, and the dynamical equation T,,=O implies GaSr=O.As a result, we obtain the Einstein supergravity equations. In summary, we have shown that the action (6.7.10) leads to the same dynamics as the supergravity action (6.1.3) So, the action (6.7.10) can be taken in the role of the Einstein supergravity action. This supergravity formulation is known as new minimal supergravity. The term 'new minimal' is used due to the fact that the chiral superfield (the compensator of the minimal formulation is chiral) and the real linear superfield (the compensator of new minimal formulation is real linear) carry equal numbers of bosonic and fermionic field degrees of freedom, 4 real bosonic ones + 4 real fermionic ones. There is one more way to establish the equivalence of the supergravity theories (6.1.3) and (6.7.10). Consider the auxiliary model with the action superfunctional S[V,
's
[I, g A ] = -
K2
d8zE-1(V[I-eV)
(6.7.22)
where V is an arbitrary real scalar superfield and [I is a real covariantly linear scalar superfield. The action turns out to be invariant under the super Weyl transformations
It may be seen that V can be interpreted as a gauge superfield or the super Weyl group. Varying the action with respect to I/ gives the equation I/= In II, which can be used to eliminate K Then one obtains the new minimal supergravity action. On the other hand, by varying S [ V , [I; g A ]with respect to the (anti)chiral superfields ij6 and qa, from which [I is composed, we obtain the equations
(g2- 4R)9, V =(g2- 4R)Bi V =0. which have solutions K=ln@+ln$
&@=O
for some covariantly chiral scalar superfield. After doing this, our action reduces to the action (6.6.7) representing the super Weyl invariant form of the Einstein supergravity action (6.1.3). As a result, the minimal and new minimal supergravity formulations turn out dually equivalent. Copyright © 1998 IOP Publishing Ltd
556
Ideas and Methods of Supersymmetry and Supergrauity
6.7.3. Gauge fixing and field content in new minimal supergravity Upon fixing the superlocal Lorentz invariance (by setting N z p= daP) and the super Weyl invariance (by setting cp = l), new minimal supergravity is described by the prepotentials W Mand f , with W Mbeing arbitrary and 9 subject to constraint (6.7.5).These superfields possess the transformation laws (5.4.14) and (6.7.12), respectively, under the K - and A-supergroups. Above we have pointed out that the A-invariance can be partly fixed by imposing the condition (6.7.13),which requires the prepotentials W Mto be constrained as in (6.7.15). Then, one ends up with residual Ebtransformations under equation (6.7.14) and with arbitrary K-transformations. The latter can be used to take the generalized gravitational superfield gauge (6.6.50). In this gauge, we have
(6.7.24) where E, and 8 b are defined by equation (5.4.43).So, the requirement (6.7.15) takes the form
+
+
det(b," i&$") det(b,' E,#') = 1. det(b," - iamXn) det(Sfi;+ l ? b 2 b )
(6.7.25)
Remark. To derive equation (6.7.24), one can consider the change of variables
i
+
x;"L)= xm i X m
Z (L)-e MWzM -
-
O&)=Op
8,)
= 8b
+ 2b
a--z aTM
q
6,"+iia,&'" ia,Xn i@&'"
0
8mA?i,
6,'' 0
6 b GO+ P 2 +I (6.7.26)
and use the relations (1.11.17) and (1.10.64). In subsection 6.6.6, we saw that minimal (n= -3) and non-minimal (n # -4,O) supergravities can be treated as gauge theories of an (unconstrained) generalized gravitational superfield ZM= 2Mwith the gauge group being the supergroup of coordinate transformations ( 6 . 6 4 4 5 ) .It follows from the above consideration that in the case n = 0 such a treatment leads to a theory differing from new minimal supergravity. New minimal supergravity is to be understood as follows. It is the gauge theory of a generalized gravitational superfield &'OM = 2Msatisfying the constraint (6.7.25).The gauge group is the supergroup of coordinate transformations (6.6.44)under the unimodularity condition
(6.7.27) which coincides with constraint (6.6.45)for n = 0. The gauge group acts on Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
557
Z Maccording to law (6.6.48). Note that requirement (6.7.14) gives the infinitesimal form of requirement (6.7.27). Instead of imposing the gauge 9 = 1, let us take another, more convenient, course. The K - and A-transformations can be used to choose the gravitational superfield gauge (5.4.38). In this approach, the gravitational superfield &'M=i%?Mis unconstrained, but the compensator .j is governed by the equations
?i
=
o
c
c
p = (?ewe- w,
(6.7.28)
where W = WMamis some functional of X M(see equations (5.4.38)). In the gravitational superfield gauge, the residual gauge invariance is described by the superparameters (5.4.42). It is in our power to choose the Wess-Zumino (5.1.17) for our exploration. Since now X3=0,we have
W M= MM+ -1 Z"C?,YP 2
and the requirements for where X = &"a,. jqx,
9 take the form b9=0 ? = 9 + 2 i ( j 2 ) - - 2 ( 9 2 2 ) It follows from here that 9 has the structure:
e, 8,= c(x) + ezq,(x) + ~&cp&(x) + eoagu,(x) + e2gjIp(x)
(6.7.29) (6.7.30)
with C(x) being real. Further, the transformations surviving in the Wess-Zumino gauge are described by the superparameters (5.1.18). In accordance with the law (6.7.12), the Wey! and local S-supersymmetry transformations (corresponding to the parameters o and qa in (5.1.18)) can be used to gauge away the components C and qEto give ?(x,
e, 8,= e - '(x)( I + e&h,(x) + e28J*(X)).
(6.7.31)
Here the factor e - l is introduced to make the fields U, and %" scalars with respect to the space-time general coordinate transformations (setting 1"= b"(x) and ip=Ab = 0 in equation (6.7.12) gives 89 = dm(bmf)).Now, the reality condition (6.7.29) means
(6.7.32~)
and = - 2ea,(e-
Yma),
(6.7.32b)
The system (6.7.32~) has the solution U, = w,
Copyright © 1998 IOP Publishing Ltd
+ 2A, -ied,(e-
le,*)
(6.7.33)
558
Ideas and Methods of Supersymmetry and Supergravity
where (6.7.34) We see that the only independent component of 9 is an antisymmetric tensor field Bk,possessing the gauge invariance (6.7.35) There is no mystery in the appearance of B,, and corresponding gauge invariance. One can easily recognize Bk,as a component field of the (anti)chiral superfields & and rla from which II(and 9) is composed. Next, the invariance (6.7.35) represents a fragment of invariance (6.7.3). The most important feature of new minimal supergravity is that its component gauge group includes not only the space-time general coordinate, local Lorentz and local supersymmetry transformations (as in the minimal and non-minimal versions), but the local chiral ones also. It is worth reminding ourselves that the Weyl and local S-supersymmetry transformations have been used to achieve the gauge (6.7.31). If one chooses in the law (6.7.12) j."
=0
1
j."= -
2
ne"
which corresponds to some y,-transformation, then the result will be
60, = en ,
= - id,(nTmi)
hence
These are exactly the y,-transformation laws (5.1.29). Owing to the equivalence of minimal and new minimal supergravities, the fields A, and B,, do not propagate on-shell. They are the auxiliary fields of new minimal supergravity. At the same time, A , and B,, are gauge fields for the local chiral invariance and the invariance (6.7.35), respectively. One can check that A, and B,, enters the action (6.7.10) as
s
d4x e - l ( C 1 w m w , + C,w"A,)
(6.7.36)
C , and C , being non-zero constants. Here w m is defined in equation (6.7.34). The non-dynamical nature of A, and B,, is now explicit. Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
559
6.7.4. Linearized new minimal supergravity A linearized version of the new minimal supergravity can be obtained by means of direct examination of its action superfunctional (6.7.10)or by finding a dual transform for the linearized minimal supergravity action (6.2.5). We choose the second course. Let us consider a globally supersymmetric theory which is described by unconstrained real vector and scalar superfields Ha and U as well as by a chiral scalar superfield 6, D60=0, and its conjugate 3. The action superfunctional is given by 1 H"'D'!?D2DpH,, -- (d,iH"')z 4
+-161 ([D,, D&Y')2
> :I +-U2
(6.7.37)
and turns out to be invariant under the gauge transformations 6H,k
= D'L, - D,L,
1 6U =- (D'D2L,+ D6D2L6) 4
(6.7.38)
1 6~ = - - D ~ D ~ L , 12
L, being arbitrary. Note that the variation 6U is linear, D26U=0. The equation of motion for U makes it possible to express U via the other superfields. Then, the action (6.7.37) is reduced to the linearized supergravity action (6.2.5).On the other hand, the equation of motion for o means nothing more than the linearity of U ,
U=U=U
(6.7.39)
D2U=0.
Then, o and 3 drop out of the action and it takes the form: $2' NEW MIN.SG =
s I:,
d8z -- Ha6D'!?D2D,&'
1 4
--
1 1 +([D,, D,]Hak)2+ - U [D,, D6]Ha' + 16 2 2
This superfunctional represents the linearized version of the new minimal supergravity action. It is obviously invariant under the transformations
6H,'
= D'L,
- D,&
1 6U =- (DaD2L,+D,D2L"I) 4
where L, is an arbitrary spinor superfield. Copyright © 1998 IOP Publishing Ltd
(6.7.41)
560
Ideas and Methods of Supersymmetry and Supergrauity
Using the invariance (6.7.41), one can set the gauge U =o.
(6.7.42)
Parameters of the residual gauge transformations are constrained by DWL,
+ DbDzLb=0
which has the solution L, = iD,K
+ iDzlzi
K = R.
As a result, in the gauge (6.7.42) we end up with an invariance of the form
6 H , = o',K
K=R
+ i(A, - A,)
(6.7.43)
DfiA,=O.
Remarkably, one can look upon this law as a simultaneous generalization of ordinary Y ang-Mills and super Yang-Mills linearized transformations.
6.8. Matter coupling in non-minimal and new minimal supergravities
Owing to the existence of several supergravity formulations discussed in detail in Sections 6.6 and 6.7, it seems reasonable to investigate peculiarities of matter coupling in the non-minimal and new minimal supergravities. Strictly speaking, this problem is not a fundamental one. By their very origin, each of the n# -,: 0 and n=O versions represents a theory of (minimal) supergravity coupled to some special matter. On these grounds, every dynamical system describing an interaction, for example, of non-minimal supergravity with some matter superfields { V'} can be equivalently reformulated as a theory of minimal supergravity coupled to the extended matter { V I ,Y, T},where Y is a complex covariantly linear scalar superfield. In this sense, minimal supergravity is unique. Nevertheless, sometimes use of the new minimal formulation (or, less often, the non-minimal one) may turn out to be preferable. 6.8.I . Non-minimal chiral compensator We are going to introduce a chiral superfield, constructed from the non-minimal prepotentials W Mand y, playing the same role in non-minimal supergravity as the chiral compensator cp in minimal supergravity. Our consideration will be based on the non-minimal covariant derivatives (6.6.28) and the corresponding supertorsion tensors (6.6.30,3 1). Let us introduce the following complex scalar superfield:
E=- n+
3n+ 1
(4[W)-'fk8k=-
Copyright © 1998 IOP Publishing Ltd
3n+ 1 +(4R)-'D,fk n+l
n# -1
(6.8.1)
Dynamics in Supergravity
561
Its basic properties are:
(6.8.2~) (6.8.2b) (6.8.2~) In deriving equations (6.8.2),we have used the identities
{Di,D& =4RR&#
D,R=O
Tu = D,T -
-.
and the fact that for n= - 1 we have R=O and DhT1 is chiral. In what follows, we choose n # - 1. The case n = - 1 can be treated similarly, with evident modifications.
Remark. C contains the chiral superfield R in the denominator. It is not difficult to see that the leading term in the power series expansion in 0 for R is proportional to the auxiliary field B (see equation (6.6.41b)modulo bi-linear fermionic combinations. So, z turns out to be well-defined only for some special non-minimal supergeometries. In particular, 2 becomes singular in the flat superspace limit. Since Ti= DkB,from equation (6.8.2b) we deduce that -
n = In
~3
D&A= $&A =o
(6.8.3)
for some covariantly chiral scalar superfield A. One can check that the transformation (6.6.22) changes A according to the rule
A +A'= e-"A
(6.8.4)
coinciding with the transformation of @ in equation (6.6.9). Further, the non-minimal derivatives (6.6.28) admit the form
Here $A =(Bo,$a, &)are the minimal covariant derivatives obtained from BAby replacing the compensator cp by the super Weyl invariant combination cpA. (In other words, if one indicates the explicit dependence of aAon the , then B A = a A [ W Mq,A ] . Similarly, the prepotentials as gA= B A [ W Mcp], non-minimal supergravity action (6.6.34)can be rewritten as
E-'
= E-'&A.
(6.8.6)
It follows from equations (6.8.5) and (6.8.6) that the minimal compensator cp Copyright © 1998 IOP Publishing Ltd
562
Ideas and Methods of Supersymmetry and Supergravity
enters into all non-minimal quantities only in the super Weyl invariant combination cpA. The chiral superfield A transforms as a scalar under the supergravity gauge group as long as cp is not eliminated. However, in the super Weyl gauge cp = 1 every E.;transformation must be supplemented by the super Weyl one with cs= -+ew(c',i." - 2$p), and the transformation law for A takes the form (6.8.7) So, non-minimal supergravity possesses its own chiral compensator. But it is necessary to remember that A, similarly to 2 from which A is constructed, turns out to be well defined only for some special non-minimal supergeometries. A is known as the 'non-minimal chiral compensator'. With the help of A, we are in a position to introduce chiral superpotentials into non-minimal supergravity. Given a covariantly chiral scalar superfield 49,= e w g , aFg= 0, we define the integral of 49, over superspace in the three equivalent forms:
I=
s
d6z(@)3g
= s d s z [E-
(6.6.84
e'& (6.8.8b)
= --Id% n+l
2n
[E-l z. R
(6.8.84
Let us comment on the derivation of expressions (6.8%) and (6.8.8~).Using the chiral integration rule (5.5.21), we can write
Choosing here U =exp(iC-$Z) and noting that (6.8.9)
one obtains equation (6.8.8b). Next, the relation (6.8.2~)can be used to write
n+l 8n
eE(D-4R)(e-'z).
Then making use of the rule for integration by parts Sd"z E-'V(D-4R)U= leads to equation (6.8.8~). Copyright © 1998 IOP Publishing Ltd
s
dsz E-'UeE(D-4R)(e-'V)
(6.8.10)
Dynamics in Supergravity
563
6.8.2. Matter dynamical systems in a non-minimal supergravity background Now it is easy to to generalize the results of subsection 5.5.7, where we have discussed general properties of matter systems coupled to a minimal supergravity background, to the non-minimal case. Generalization consists merely in choosing the structure
(6.8.11) in the role of action superfunctional describing the dynamics of matter superfields x in some non-minimal supergravity background, instead of (5.5.24). In addition, it seems reasonable to admit a non-polynomial dependence on the supertorsions U, and Vifor 2 and Yc. All models of matter superfields coupled to non-minimal supergravity can be naturally divided onto two classes: 'effectively minimal' and 'essentially non-minimal'. A matter model is said to be effectively minimal (essentially non-minimal) if after expressing the non-minimal covariant derivatives D, via the induced minimal ones & A (6.8.5) the action S [ x ; D,] turns out to be independent (explicitly dependent) on the superfields C and 2. Examples of effectively minimal models are the chiral scalar superfield model with the action
(6.8.12) and the vector multiplet model with the action
w' w, I/=
(6.8.13)
E
Making use of the representation (6.8.5) reduces the former to the model (5.7.37), and the latter to the model (6.3.21). As an example of essentially non-minimal systems, one can consider the chiral scalar superfield model with the action
(6.8.14)
<
where 23. Many essentially non-minimal theories are potentially pathalogical due to non-analyticity in the supergravity auxiliary fields. Copyright © 1998 IOP Publishing Ltd
564
I d e a and Methods of Supersymmetry and Supergravity
6.8.3. New minimal supergravity and supersymmetric a-models In conclusion, we would like to elucidate one of the most important applications of the new minimal formulation-describing locally supersymmetric nonlinear a-models. Supersymmetric a-models have been studied in Section 3.3. The underlying a-model symmetry principle is invariance under the Kahler transformations (3.3.11). The a-model action superfunctional (3.3.10) possesses this invariance, since the operation of integration over flat global superspace is characterized by the property
DiA=O
*
I
dezA=O.
When trying to extend the a-model to a curved superspace, we must treat the a-model variables mi as covariantly chiral scalar superfields, = 0, as well as covariantize the integration measure. However, the naive covariantization (6.8.15)
j d 8 r E-' K(@, @) proves to be inconsistent with the Kahler invariance (3.3.1 l), since
D&A=O =- j d 8 1 6 - ' A = j d 6 z ~ ' e - " ( R A ~ ~ * as long as R # 0. On the other hand, the relations (6.7.1) imply
DkA=O
s
3
As a result, the action
d8z E-'[LA=
s
d'z E-'[LA=O.
{d8z E - 'L K(@, @)
(6.8.16)
is explicitly invariant under the Kahler transformations (3.3.11). Remarkably, it is also invariant under the super Weyl transformations (6.7.8). We see that the Kahler invariance appears to be consistent with the new minimal supergravity formulation. By adding the pure supergravity term (6.7.10) to expression (6.8.16), one obtains the action
]
(6.8.17)
defining the new minimal supergravity-o-model dynamical system. It is interesting to notice that the action (6.8.17) can be obtained from the Copyright © 1998 IOP Publishing Ltd
565
Dynamics in Supergravity auxiliary action SCO', 11,
1!
which generalizes action (6.7.22), by eliminating the Lagrange multiplet V. The action introduced proves to be invariant under the super Weyl transformations (6.7.23) and the Kahler transformations
a)+ A(@)+ A(6) V + V + Iff (A(@)+ A@)). 3
K ( 0 , G ) --t K(@,
(6.8.19)
If one now varies action (6.8.18)with respect to [I(obtaining ( g- 4R)@V= 0) and then imposes the super Weyl gauge V=O, the action reduces to
'J
S[Oi,9J= --
KZ
(Y
d * z E - l exp - - K ( @ , @ )
- --?Jds;E-l
)
+ J d s i E - 1 K ( @ i , @ ) + 0 ( ~ 2 ) , (6.8.20)
K2
Here the first term represents the Einstein supergravity action and the second term describes the coupling of the o-model superfields to minimal supergravity. In the flat superspace limit and K+O, S[@', gA]takes the flat form (3.3.10). In the super Weyl gauge V=O we have adopted, every Kahler transformation (6.8.19) must be accompanied by a special super Weyl transformation with B = ( ~ ' / 3 ) @ ( @ ) A(@)).O n these grounds, the a-model action possesses the following super Weyl-Kahler invariance:
+
a)+ K(@,a)+ A(@)+ A(5).
K(@,
(6.8.21)
Clearly, supergravity-matter models (6.8.17) and (6.8.20) are equivalent.
6.9. Free massless higher superspin theories As has been shown, linearized Einstein supergravity describes free massless particles of spin 3 and 2. So, Einstein supergravity can be treated as a supersymmetric field theory of interacting massless spin-? and spin-2 particles. It is quite natural to ask: Do supersymmetric field theories exist which consistently describe an interaction of massless particles having higher spins s > 2?. A consistent system of equations for interacting massless fields of all Copyright © 1998 IOP Publishing Ltd
566
Ideas and Methods of Supersymmetry and Supergravity
spins, including the gravitational field, was constructed by M. Vasiliev]. It is not clear, however, whether there exists an action functional leading to such equations of motion. Description of Vasiliev's construction is out of the scope of this book. The chief goal of the present section is to describe free supersymmetric field theories of massless higher spin particles. These models may happen to be linearized versions of some fundamental higher spin theories extending, in a nontrivial way, Einstein supergravity. We begin with two definitions. A free relativistic field theory is said to describe a massless particle of spin s if the Poincare representation which acts in the physical dynamical subspace @dphys), is given as a direct sum of two irreducible massless representations carrying the helicies (- s) and s. By (Pkphys) we mean the dynamical subspace factorized over all the purely gauge histories (massless theories turn out to be gauge when s 2 1). It is clear that Obphys) may be parametrized by (functionally independent) gauge invariant field strengths which remain non-vanishing on-shell. A free supersymmetric (super)field theory is said to describe a massless multiplet (superparticle) of superspin Y if the super Poincare representation acting on the physical dynamical subspace is equivalent to a direct sum of two irreducible massless representations carrying the superhelicities ( - Y and Y. A massless multiplet of superspin Y describes four massless particle Y and + ( Y + $ ) or, equivalently, two massless states of the helicities particles having spins Y and ( Y -ti). This is why the massless superspin-Y multiplet is often denoted as (Y, Y 4). Previously, we have discussed a lot of supersymmetric models realizing multiplets with lower superspins Y =0,3, and 3.In particular, the cases Y = O and Y =3 were realized by the models (3.2.5)and (3.4.9),respectively. Three equivalent formulations were found, given by equations (6.2.5), (6.6.56) and (6.7.40), for a superspin-2 multiplet. Below we shall present the superfield realizations for massless multiplets of higher superspins Y = 2, 2, 3, . . ., as well as for the so-called gravitino multiplet ( Y = 1).
-4)
+
6.9.1. Free massless theories of higher integer spins In this subsection we discuss free field theories describing massless particles of higher integer spins s=3, 4, 5, . . .. These theories were constructed by C . Fronsdal? Let us fix some integer s > 2 and introduce into consideration a set of two real bosonic tensor fields over Minkowski space of the form cp' = V x .,,1,4,. . ..h,(X)> hx,...x s-*h,. .U,- Ax)>. (6.9.1) ,
These fields are assumed to be totally symmetric in their undotted indices and, independently, in their dotted indices.
Remark. Throughout the section, excepting one specially mentioned case,
'
*
M . A. Vasiliev Phys. Lett. 243B 378, 1990; Class. Quantum. Grav. 8 1387, 1991. C. FronsdaJ Phys. Rev. 18D 3624, 1978.
Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
567
all (super)fields which appear will be totally symmetric in their undotted indices and, independently, in the dotted ones. To simplify expressions involving numerous indices, we will often use compact notation of the type F ~ ( A ) ~E(FBa ),...a,hl
...drs
- F ( a ,. . . a A ~ .. i . ,h,)
indicating in brackets the number of indices. We will also use the following summation convention F ~ ( A ) ~ ( B ) G ~E ( ~ ai ) ~... ( Z~ A)
~ I . , . ~
Ga,...X A h 1 ...hs'
Remark. Spinor indices of the fields (6.9.1) can be converted into vector ones according to the rule h a l . , , a ~ = ( - l / 2 ~ ~ ~ ' ~ ~l ~~ 1( a~1 ~ ~ ~ a l , . , a ~ ~ l . . . ~ ~
and similarly for ha(,-2)b(s-2). The resultant real tensor fields ha,...a, and hal,,,as-2are obviously symmetric and traceless. They can be unified in the symmetric tensor field @u l . . . u s = ~ a l . . , a s + ~(a,a2ha3...a~
which has vanishing second trace, @bcbcal...a,-4 = 0.
The fields (6.9.1) play the role of dynamical variables in the theory with action
(6.9.2) The action remains invariant under the following gauge transformations
where the real tensor parameters ( a ( s - l)b(s- I ) are completely arbitrary functions over space-time. Note that the operations of symmetrization over undotted and dotted indices are always done independently. We are going to show that the theory (6.9.2) describes a massless particle of spin s. It is worth beginning with the following simple table. Copyright © 1998 IOP Publishing Ltd
568
Ideas and Methods of Supersymmetrjq and Supergravity Table 6.9.1.Number of fields and gauge parameters. Fields
Gauge parameters
Next, let us prove an important theorem. Theorem I. On-shell, among the components of ha(s)k(s)and h,(,-2)+-2) there are only two independent components modulo gauge transformations. Proof. Suppose that h,(s)i(s)and h,(,-2)6(s-22)satisfy the dynamical equations determined by the action (6.9.2).Using the gauge freedom, one is in a position to impose the gauge condition hl(S
(6.9.4)
- 2)h(s-2) = 0
which constrains, due to transformations (6.9.3),the residual gauge invariance via (6.9.5)
aPBrpl(s-2)Bb(s-2)=0.
implies Note also that the dynamical equation for h,(s-2)h(s-22) (6.9.6)
dpr’dYi.hp,,(, - 2)jth(s- 2) = 0.
If equation (6.9.5) is satisfied, the divergence 8%p,(s- l)fik(sthe manner
1)
transforms in
6(dPBhpa(s-1)Bjr(s-l))X n L [ s - I ) & ( s - l )
and, owing to relation (6.9.6), can be gauged away (6.9.7)
d@hpcc(s- l ) & - l ) = O .
Under the gauge conditions (6.9.4) and (6.9.7), ha(s)h(s) satisfies the equation (6.9.8)
0hor(s)h(s) = 0
and the residual gauge invariance is constrained by 0L [ s -
l)&(s- I ) =
0
(6.9.9)
in addition to equation (6.9.5). For further analysis, it is convenient to transform to momentum space both the fields and residual gauge parameters. In accordance with equation Copyright © 1998 IOP Publishing Ltd
569
Dynamics in Supergrauity
(6.9.8), we have
Ps=(Po>
8)
Po=lbl
and similarly for the residual gauge parameters. It is also useful to choose a standard reference system in which p " = ( E , 0, 0, E )
==
(6.9.10)
where p 1i = p22 = 2E. Now, the relation (6.9.7) means (6.9.11)
h21(s-I,i&(s-I)(P)=0.
Analogously, equation (6.9.5) leads to r 2 1 ( s - 2 ) 2 & ( S - z ) ( P ) = 0.
Making the gauge transformation generated by such a parameter, one has ~~la(s-l)ik(s-l)~P~~P~~r1(s-l,a(s-l)~P~.
Therefore, we can impose the gauge condition (6.9.12)
h l 1 ( s - l)i &(s- 1)( P ) = 0
which turns out to completely fix the gauge freedom. The relations (6.9.11) and (6.9.12) tell us that the only non-zero components of ha(s)k(s)(p) are the following: h
mu 5
h
(PI,
(P).
(6.9.13)
+2+i
s
This completes the proof. The next important observation is the fact that the theory under consideration possesses the following gauge invariant field strengths:
c1
I... a z , = a h P 1
* '
~s8.~1~~,...,*~)al...8. (6.9.14)
On-shell, these field strengths do not vanish and satisfy the equations 8PQc/71(,s-~)=0
a~~c~a(Zs-l)=O.(6.9.15)
The last statement can be immediately proved by choosing the gauge (6.9.4,7). It follows from equation (6.9.15) that C1(2s)and Ck(2s)are on-shell massless fields (see Section 1.8)and their helicities are s and (-s), respectively. Copyright © 1998 IOP Publishing Ltd
570
Ideas and Methods of Supersymmetry and Supergraoity
Now, let us recall that each on-shell massless field has only one independent component. If one fixes the reference system in the manner (6.9.10),then the and Cb(Zs) are the following: relevant components Ca(2s) In the guage (6.9.4, 7, 12), these components are uniquely connected with the fields (6.9.13) according to the rule: Cl . . . l ( p ) ~ ( ~ ~ h l . . . l ~ . . . ~ ( p ) Cl...~(p)~(~Yh2...2~...i(p).
Therefore, Ca(zs)and Ci(2s)are the only (functionally independent) field strengths surviving on-shell. As a result, the theory (6.9.2)describes a massless particle of spin s. 6.9.2. Free massless theories of higher half-integer spins In this subsection we discuss free field theories of massless particles having half-integer spins (s + i),where s = 2, 3, 4, . . ., These theories were constructed by J. Fang and C . Fronsdal'. Let us fix some integer s 2 2 and introduce into consideration a set of fermionic tensor fields over Minkowski space with the structure
v i= { Y a ( s + I)&),
V a ( s - I)&),
%s-
%(s)h(s+
l)i(S-2)(X)
1)(X), Y ? ( s ) h ( s - 1 ) ( 4 , %(s-*)h(s-
1)(4).
(6.9.16)
Remark. The fields can be unified in a four-component Majorana and in spin-tensor field as follows. First, let us combine the field
-
~ ~ ~ l . . . ~ : l i i . . . b l - ~ ~ ~ ( a i . . . a r ) h i . . . h r = ~ p a l . . €. a~,(haii.y. a. ~* ,. + ,.a~)h,...i~
which is not completely symmetric in its undotted indices. Secondly, introduce the two-component spin-tensors Aal,, ,a,@= ( y a i ...a, - 2p = (
1/ 2 ~ ( 8 a , Y ' .~ '* (cajbsasRBai.. asii...it, ,
- 1P)S- 2 ( c a , Y i a i. . .(8aT- 2 P - 2 a 5 - 2 \ f l ~ a 1...a, *al.. . .a, . -2 -
and then construct the four-component Majorana spin-tensors
Both spin-tensors are symmetric and traceless, with respect to the vector satisfies the relation indices, and Yal.,.a,_2
*J. Fang and C. Fronsdal, Phys. Reu.D 18 3630, 1978. Copyright © 1998 IOP Publishing Ltd
571
Dynamics in Supergravity
Finally, we construct the spin-tensor @a l . , . a , = A a l . . . a ,
+V ( a l a 2 ~ a ~ . . . a , ~ )
which is symmetric with respect to the vector indices and satisfies the equation h c
? 'r'
Ydobcda l...as ,=O.
The fields (6.9.16) play the role of dynamical variables in the theory with action
s(s+
= i( -
i)'
Sd'x { qa(s)fih(s) 8'B Y Bz(s)h(s)
+S ( q z ( S ) b (s 1 ) aaPy + +
Pz(s)Bh(s - I )
+a PB.ipipa(s-
I)h(s))
(2s + 1)y'"'s - 1 )Bk(s - 1) aa#qp a ( s - l ) k ( s - l ) (s + 1) ( q M S - 2)h(s- 1'3 Bcy i Ba(s-Z)@(s8 . $ p a ( s- 1,BkLCs - 2 ) yz(s- I ) h ( s - 2 ) )
+
1)
BB
- q a ( s- 2
) ) -~2) a Sfi y p a ( s - 2 ) h ( s - 2 ) ) *
(6.9.17)
The action turns out to be invariant under the gauge transformations a y z ,...q , , k
I...
b,=d(,,(h15z* ...z,,1)62...bs)
1 ~~al...a,-lhl...~c=- ~p(~l~~zl...a~_I~z...~,)
ss1
(6.9.18)
with arbitrary tensor parameters
Gauge parameters
Now we prove an important assertion useful for our analysis.
Theorem 2. On-shell, among the components of Ya(,+ I)&(~), I ) & ( ~ and ) there is only one independent component modulo gauge transformations. '€'a(s-
Copyright © 1998 IOP Publishing Ltd
572
Ideas and Methods of Supersymmetry and Supergravity
Proof. Let Y'sc(s+I)h(s), Y,(s-lI)b(s) and Y'sc(s-l)h(s-2) be solutions of the dynamical equations determined by the action (6.9.17). Using the gauge freedom, we can annihilate 1)cl(s-2), that is, impose the gauge condition Y,(s-
l)h(s-2)'0.
(6.9.19)
Then, the residual gauge freedom is constrained, due to transformations (6.9.18),by a%,(s- l)&S-2) = 0. (6.9.20) dynamical equation, the field Next, owing to the T'sc(s-2z)cl(ssatisfies the equation (6.9.2 1) d@By3u(S- 2)B&(((S- I ) = 0. The relations (6.9.20,21) and the YE(s- transformation law (6.9.18) show that can be completely gauged away I)h(s)= 0.
%(s-
(6.9.22)
This leads to additional restrictions on the gauge parameters of the form aP&?a(s-I)cl(s-
hence
l)=o
0. In the gauge (6.9.19,22), a B t ~ g , ( s ) h ( s= ) 0.
(6.9.23) satisfies the (6.9.24)
Similarly to the bosonic case, we can now make the transform to momentum space. Choosing our reference system in the manner (6.9.10), one immediately finds that equation (6.9.24) means ~ 2 4 s , h ( s ) ( P= ) 0.
(6.9.25)
Further, the residual gauge invariance described by equations (6.9.20,23) can be used to impose the condition YJ1a(s)l b(S - I)( P)= 0
(6.9.26)
which completely fixes the gauge freedom. Now, the expressions (6.9.25) and (6.9.26) imply that the only non-vanishing component of Yd,+ l)b(s)(p ) is
. .(PI.
Y
(6.9.27)
1...12...2
+-"--.--.-s+l s
Thus, the theorem has been proved.
Another crucial observation is the fact that the theory under consideration has the following gauge invariant field strengths: (6.9.28) Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
573
On-shell, they turn out to satisfy the equations
a w g u ( 2 s )= 0
dY43&(2r)= 0.
(6.9.29)
Therefore, 1) carry the helicities (s+i) and -(s +#, I ) and respectively. It is easy to see that, in the gauge (6.9.19, 22, 26), the only non-zero component of Cu(2s+ is expressed via the field component (6.9.27) and vice versa. As a result, Ca(2s+1) and Cdr(zs+l) prove to be the only independent field strengths surviving on-shell. We see that the theory (6.9.17) describes a massless particle of spin (s + i). 6.9.3. Free massless theories of higher half-integer superspins In this subsection we are going to discuss free theories describing massless multiplets of half-integer superspins Y = (s i),where s = 2, 3, 4, . . .. These theories were constructed by S . Kuzenko, V. Postnikov and A. Sibiryakov. Let us fix some integer s 2 2 and consider two sets of bosonic superfields over flat global superspace:
+
-
(6.9.30)
~ ~ f = { ~ a l . . . a ~ ~ l . ~. .u ~l ~ . .( .zu) ~ , - l ~ l . .ra,...u~-,&,...~~-,(z)} . ~ ~ - l ~ ~ ~ ~
and
us^, ...&~z), ~ u...us, ..&,- !(z),~ a...u,-,aI , . . . . E., - I(z)} (6.9.31) /lvf= which will play the role of dynamical variables in two different, but equivalent, supersymmetric theories. In both cases, Ha(s)dr(s) is an unconstrained real scalar superfield which transforms in the representation (s/2, s/2) of the Lorentz group. The complex superfields rU(+ and Gu(stransform in the same representation ((s- 1)/2, (s- 1)/2)) of the Lorentz group. However, they are subject to the following, essentially different, constraints:
D k ,...U,- l&4&l. .&,I
2
=0
(6.9.32) (6.9.33)
Obviously, the equations (6.9.32)and (6.9.33) imply that superfields,
r and G are linear
DZ~u(s-l)h(s-l)=O
(6.9.34)
1) = 0.
(6.9.35)
DzGu(,-I ) q s -
It seems reasonable to call a ‘transversally linear superfield’. Analogously, GU(+ is said to be a ‘longitudinally linear superfield’. It is worth pointing out that the general solution of equation (6.9.32)reads (6.9.36) where bu(s-i)dr(~)is an unconstrained fermionic superfield of the Lorentz type ((s- 1)/2, s/2). Such a superfield is defined modulo transformations of the form (6.9.37) Copyright © 1998 IOP Publishing Ltd
574
Ideas and Methods of Supersymmetry and Supergravity
with arbitrary bosonic tensor parameters Aa(ssolution of equation (6.9.33) is
).
Further, the general (6.9.38)
Here [z(s - I )h(s- 2) is unconstrained fermionic tensor superfield defined modulo shifts of the form (6.9.39) with arbitrary bosonic tensor parameters K ~ ( l)dr(s-3). ~ We see that the superfield rztsconstrained by equation (6.9.32) can be treated as a gauge invariant superfield strength of the superpotentials t5r(sl)h(s), with respect to the gauge invariance (6.9.37) (of infinite range of reducibility). subject to equation (6.9.33) can be Similarly, the superfield Gatstreated as a gauge invariant superfield strength of the superpotentials [ z ( s - l ) h ( s - 2 ) , with respect to the gauge invariance (6.9.39) (of finite range of reducibility). TI(s- l ) h ( s - 1 ) Let us define gauge transformations of the superfields Hz(s)h(s), and GI(s- I ) h ( s - according to the law: -
-
~ H z l , . . z ~ ~ l .(.h.lh~ z~l ,=~ D . e ~ h 2 . , . h ~ ) - D ( z l ~ zI... 2 .h,. . I ~ ) h
(6.9.40) (6.9.41)
+ i b - l ) @ ~ D ( h l L p z l , . , z,,C ~i , . . . $ _ , ) ~ .
(6.9.42)
Here Lz(s)b(s-is an arbitrary fermionic tensor superfield of the Lorentz type (s/2,(s - 1)/2). There exists a quadratic superfunctional of the superfields Hz(s)h(s), which is invariant under the transformations ~ ) h ( 1) ~ -and (6.9.40) and (6.9.41):
It turns out that this action superfunctional describes a massless multiplet of superspin (s i-f). Explicitly, the gauge freedom (6.9.40,41) makes it possible Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
575
to choose the following Wess-Zumino gauge: ~ z l . . . z T j r l . , , & T BBBBh( = Bal...
lsXB&I . . .
u,~)+~~~~(Bal...a,)&l~..&~-~Z~BQa,...ad8jr,...a,)
+ ~ZQ%,...a.&.&s r a 1 , , . z( I
*
I... *,.I-
- eieayBa,
Eh,,
(6.9.44) ...a,+ ,jrI.. .&,_
+4aI%2...a,7.1)jr
I... & , _ ]
I
+ @ q ( ~ a ..., as- I)&,...ir, -
I
+BB~”zl...z,_ l(Bcrl...cc,-I)
+e2Bzl...a,_l&~...b,~l + e B ~ B U ( ~ a l . . . z , _ , ~ ~ j r l , . . . h , _ , )
+ @4a1&z2...aT- lHBjrl ...&,. + 8 2 @ P a ,...a*-l(flcl ...a~-l)l 1)
all the component fields being totally symmetric in their undotted indices and independently in the dotted indices. Here the bosonic fields ha(,+ &($- I ) ~ c ( ~I -) and 4+)t+) are real, the fields &1) and Ua(s)k(sj are Be+ I ) & ( ~ -I), complex. One can readily see that the bosonic fields Uz(s)jr(s) and FK(s-*)i(s)as well as the fermionic fields La(s- l)jr(s), I ) & ( ~ )are auxiliary. Performing the integration over 8, 8 in equation (6.9.43) and eliminating all the auxiliary fields, we obtain a special theory of bosonic fields ha(s+l)&(s+l)
ha(s-l)l(s-I)
and fermionic fields
K(s+
I)&($
Y z ( s - I)jr(s)
T a ( s - I)&(s-2)
+ C.C.
Then upon trivial rescalings, the bosonic part of the action proves to coincide with the massless action S(s+I ) (see equation (6.9.2)),while the fermionic part coincides with the action S(s+ I/z) (6.9.17).Therefore, the dynamical system describes a massless multiplet of superspin (s +$). Now, we define an action superfunctional of the superfields (6.9.31)which is invariant under the gauge transformations (6.9.40) and (6.9.42):
Copyright © 1998 IOP Publishing Ltd
576
Ideas and Methods of Supersymmetry and Supergravity
This model also describes a massless multiplet of superspin (s + 1). Indeed, the theories (6.9.43) and (6.9.45)are connected by duality transformation via the auxiliary model with the action
(6.9.46) 1)6(sbeing an unconstrained complex tensor superfield of the Lorentz type ((s- 1)/2,(s - 1)/2). Thus, the theories (6.9.43)and (6.9.45)are equivalent. The two superfield formulations for the massless superspin-(s + 1)multiplet, which are determined by the actions (6.9.43) and (6.9.45), can be called transversal and longitudinal, respectively. There is a purely superfield way to prove that both the theories (6.9.43) and (6.9.45) describe a massless multiplet of superspin (s + f). Namely, they possess the following gauge invariant superfield strengths:
Wm(2s+and Wk(zS+ being chiral and antichiral, respectively. On-shell, they satisfy the equations DpWpa(2s)= 0
DfiWp42s)= 0
(6.9.48)
which can be readily checked, for example, by applying the gauge conditions
DQ,a(s- I)?&) = Tcc(s- l)b(s)= 0 which the theory (6.9.43) admits on-shell. Therefore, Wcc(2s+ 1) and i7k((zS+ 1) carry the superhelicities (s +f)and -(s + l),respectively. It is easy to see that Wa(2,+ and lT'&(zs+ 1) are the only independent superfield strengths surviving on-shell. In conclusion, let us turn our attention to the case s = 1, which was excluded above and, in fact, should correspond to linearized supergravity. This choice is seen to be exceptional for both formulations developed. Concerning the transversal formulation, the point is that the constraint (6.9.32) cannot be applied when s = 1. However, one can still impose constraint (6.9.34), a consequence of equation (6.9.32). Clearly, this is the most natural reduction of the transversal formulation to the case s = 1. If one now sets s = 1 and Copyright © 1998 IOP Publishing Ltd
577
Dynamics in Supergravity
r,
takes to be a complex linear superfield, D2r=O and r # then the action (6.9.43)proves to coincide (after the redefinition r -+ - r)with the linearized action of non-minimal n= - 1 supergravity (6.6.56). As for the longitudinal formulation, upon the choice s= 1, and only in this case, the equation (6.9.33) implies chirality, D,G = 0; the right-hand side in the transformation law (6.9.42)also becomes chiral. The corresponding action (6.9.45) coincides (up to a trivial rescaling) with the linearized action of minimal supergravity (6.2.5). It is quite possible that the new minimal formulation is a specific feature of supergravity (superspin $) and cannot surve as a basis for extensions to higher superspins. 6.9.4. Free massless theories of higher integer superspins We now present free superfield theories describing massless multiplets of higher integer superspins Y = s , where s=2, 3, 4, . . .. These theories were constructed by S. Kuzenko and A. Sibiryakov. Let us fix some integer s > 2 and consider two sets of bosonic superfields over flat global superspace:
J=
{ H a 1 . . . a,,_Ib I . . .h,.
J4, r a ,...a,h
h S @ h r aI...a,i I... h,(Z))
I...
and iIJ=
-
(6.9.49) (6.9-50)
{ ~ a l . . . u ~ ~ l i l1. (. ~ . i~ ) ~,a~~ . . . a ~ i t ~~ .a ,.. .. .~a ~ ~j l )~ .~. . k ~ ( ~ ) ) .
In both cases, the tensor superfield Ha(s- 1)6(s- 1) is real and unconstrained. The complex tensor superfields ra(s)h(s) and Gz(s)h(s) are transversally linear and longitudinally linear, respectively, (6.9.51)
D ~ ~ , , . . . , ~ B h , . . . ,= . ~0 -,
q,, Gz,...
?$t(l.., is* 1)
(6.9.52)
= 0.
We define gauge transformations of the superfields Hz(sand Gaz(s)dr(s) according to the law: d ~ a , , . . a $ -1 ., ..h&.,
.I
11, ra(s)h(s)
(6.953)
. .
= D ~ ~ ~ x l . , . a sI . ... la,.! u -D~~a,...a~-,Bhl...jr.~l
(6.9.54) 1-
~Ga,...a& .,,&? =
-- D,h1D2L, 4
I...
&9,1,..iJ~
(6.9.55 )
1) is an arbitrary bosonic tensor superfield of the Lorentz type Here La(s)h(s( S P , (s - 1)/2)* An action superfunctional for the superfields (6.9.49), which is invariant
Copyright © 1998 IOP Publishing Ltd
518
Ideas and Methods of Supersymmetry and Supergravity
with respect to the transformations (6.9.53) and (6.9.54), reads
We are going to show that the dynamical system determined by S&,s+l/2) describes a massless multiplet of superspin s. It turns out that the gauge freedom (6.9.53,54) makes it possible to choose the following Wess-Zumino gauge: ~ a l . , . z , ~ l i l I.=, 8. (~z 1, 4~& l h a 2...a,. I)i2...br-l)
+ 82&,%l . . . a , v _ ... I&*
U,$- I )
-Q 9 a 1 % . . z s -
(6.9.57)
+ 8 2 ~ A z I . . . * , - I... , &&,-I
- ,ieo@&
rzl..,z,il..,&~ -
[
..,&<- I
,)&I
+
+
~ a l . . . a s k l . . . & s ~ ~ ~ ( p z l . . , z s ) & l , 8, .( & a 1s~ a * , . . z s ) h 1 . , . h s
+ ~ ~ ~ z I . . . i , ( B & l . +. . &~ sz ~) a l . . . a s & l . . . &+s ~ B ~ ~ ( p a l . . . a , ~ ~ & l . . , & , ) + @8(aI ~ a ~ . . . a ~ ..X ~ t +t , 8 2 @ ~ a l...aS(4h,...&,)I* ,
Here the bosonic fields l)dr(s-ll)are real and the fields Ba(s)&(s) and are complex. It follows from considerations of dimension and Ua(s+ that the bosonic fields 1 ~ I), ~Ba(s)&(s), U a ( s + ~ ) i (I),s +F,(,-l)h(s+I ) and the fermionic fields ~ar(s)dr(s+,),”Par(.~)dr(,~+i)are auxiliary. Integrating over 8, in equation (9.6.56) and eliminating all the auxiliary fields, one arrives at a special theory of bosonic fields ha(s)&[s)
hz(s - 2)&(s- 2 )
and fermionic fields Y a ( s + l)&(S)
Ya(s-
l)&(s)
Y a ( s - l)h(s-2)
+ C.C.
It proves that, up to a simple rescaling of fields, the bosonic part of the action coincides with the action (6.9.2), while the fermionic part coincides with (6.9.17). As a result, our dynamical system describes a massless multiplet of superspin s. Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
579
An action superfunctional for the superfields (6.9.50), which is invariant under arbitrary transformations (6.9.53) and (6.9.55),reads
(6.9.58) This model also describes a massless multiplet of superspin s. The point is that the theories (6.9.56)and (6.9.58)are equivalent, since they are connected by duality transformation via the auxiliary action S [ H , r, v]=
(-:)'
[d";
{i
Ha(S-l)*(S-l)DPD2D 3!j H a(s- l)&(s- 1)
+ ----.(sHa(s1
I)h(s-
1)DPDBT/
p,(S-
ss 1 + (s + 1) p i S ) k ( s )I/a(s)h(s)+
l)Ba(s- I )
ya(s)sc(s)I/a(s)h(s)
+2 y ( M s )
+c 4
va(sMs) (6.9.59)
Va(S)k(S) being an unconstrained complex tensor superfield of the Lorentz type (SP, SP).
Let us present one more proof of the fact that the theory (6.9.58) describes a massless multiplet of superspin s. The chiral tensor superfield is WaI... . , , =2 - ~ ( a l ~ i . . . ~ , ~ _ l ~ s - ~ D ~ ~ D , ~ , _. l. d, a, ~. Ga, *- , . . . a * . , B l . . . B ~ ~)Bl,,.B~+a(,,a1. (6.9.60)
and its conjugate are gauge invariant superfield strengths (in fact, the only independent ones on-shell). When restricted to the dynamical subspace, they satisfy the equations DBWpa(2s- 1) = 0
DB Ff'~h(2~- I
= 0.
Therefore,Wa(zs)and @dr(z,.)carrythe superhelicities s and - (s+ i), respectively. Remark. If one expresses Ga(s)h(s) via an unconstrained fermionic tensor superfield according to the rule Ga(s)hl...hq = D ( h l c a ~ 8 ) h 2 . . . h J
then the expression (6.9.60) can be rewritten in the manner
Copyright © 1998 IOP Publishing Ltd
580
Ideas and Methods of Supersymmetry and Supergrauity
6.9.5. Massless graoitino multiplet In the case s = 1, it turns out that both the formulations (6.9.56)and (6.9.58) describe a massless multiplet of superspin Y = 1 (gravitino multiplet). To argue this assertion, it is sufficient to consider the transversal model in which the dynamical variables are
where H is a real scalar superfield, while rzbis subject to the constraint
Daras= 0.
(6.9.62)
The corresponding action (6.9.56)is invariant under the gauge transformations
6H = DEL, + DJ?
(6.9.63)
L, being arbitrary. This gauge freedom makes it possible to choose the following Wess-Zumino gauge (compare with (6.9.57)) H = 0282A
(6.9.64)
ha&being real. All component fields, except ha&,ylafib,\Yak),Y kand Tu,are auxiliary. If one inserts the above expansions into equation (6.9.56),integrates S $ , 3 , 2 ) . over 0, IT and eliminates the auxiliary fields, then the resultant action will coincide with a sum of two action functionals describing massless particles of spin 1 and $. Let us consider the s = 1 longitudinal formulation (6.9.58) developed originally by a number of authors3. It can be described by a set of unconstrained superiields of the form
{H,Ye,Yk}
H=H.
(6.9.65)
Here Y, is connected to the longitudinally linear superfield Gahwhich arose in equation (6.9.58)in the manner G,& = DkY,.
(6.9.66)
'E.S. Fradkin and M.A. Vasiliev, Nuovo Cimento Lett. 25 79, 1979; B. de Wit and J.W. van Holten, Nucl. Phys. B 155 530, 1979; S.J.Gates and W. Siegel, Nucl. Phys. B 164 484, 1980. Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
581
Using the vector multiplet strength
w,=- -41D-~ D , H one can rewrite Sji,,3,2)as follows: s11,3/2)=9(1,3/2)--
's
4
d6z W'W,-
where
J
(6.9.67)
d82 (W"Y,+ E'&"'')
(6.9.68)
As a result of conditions (6.9.53,55) and (6.9.66),the action (6.9.68)is invariant
under the transformations
SH = D"L, + D$ 1
+2
6Y, = Au - D,D@Lp
(6.9.70)
D'A,
=0
L, being arbitrary. As may be seen, it is in our power to impose the gauge fixing (6.9.71)
H=O. In this gauge, the action (6.9.68) coincides with (6.9.70) is reduced to
g(1,3,2), while the invariance
6Y, = A,
+ au~DZDkK
DkA,=O
K=R.
(6.9.72)
We conclude that the massless gravitino multiplet admits the formulation in terms of unconstrained spinor superfields Y, and T&,whose dynamics is determined by the action 9(1,3,2), invariant with respect to the transformations (6.9.72). There exists another formulation for the gravitino multiplet, in terms of unconstrained spinor superfields Y, and Yk,suggested by V. Ogievetsky and E. Sokatchev4. It can be obtained using the following scheme. The longitudinal formulation for the gravitino multiplet proves to be equivalent to that described by the action S[Y, H,01= S ; , , ~ / 2 , - I S d 8 ~ ( ~ 0 + 0 D , Y , + ~ k ~(6.9.73) ~~
4V.I. Ogievetsky and E. Sokatchev, J . Phys. A: Math. Gen. 10 2021, 1977. Copyright © 1998 IOP Publishing Ltd
582
Ideas and Methods of Supersymmetry and Supergraoity
where (D is a chiral scalar superfield. This action is derived from S/l,3,2)by the replacement Y, +Y,
1 +D, -0, 16 D 2 -
The action S[Y, H , (D] is invariant under the gauge transformations 1 2
S0=--D20
dH=U+O 1
w a= A, + 2 D,U
(6.9.74)
D&A% = 0.
Here U is a complex scalar superfield. The above invariance can easily be deduced from expressions (6.9.70) by making the replacement L, -P L, D,T and identifying U = D'L, + D2T, Next, setting the gauge (6.9.71) reduces the action (6.9.73) to
+
and the invariance (6.9.74) to the following: dYX=A,
+ iD,K
K=R
80 = iD2K D&A1=O.
(6.9.76)
The model (6.9.75) turns out to be equivalent to the dynamical system with action superfunctional
's
S[Y, 0,V ] = S ( 1 , 3 , 2 ) + - d8Z(DaY,+DbYh)2 4
+
s
+
+ +
d8z { V 2 V ( 6 0 Day, + D i p ) }
(6.9.77)
V being a real scalar superfield. The 'final action remains invariant under the transformations (6.9.76) supplemented by
6v=
1 ( D ~ A+, DhAa). 2
--
(6.9.78)
On using the equations of motion for @ and 0,the action (6.9.77)is reduced to
(6.9.79) Copyright © 1998 IOP Publishing Ltd
Dynamics in Supergravity
583
where 9 is a real scalar superfield subject to the linear constraint, that is,
p=g
(6.9.80)
D29=0.
1possesses an invariance of the form Obviously, S[Y,9 dYz+ A'
+ iD,K
1 2
6 9 = --(DEI\,
+ DjlAb)
(6.9.81)
with A, and K defined as in relations (6.9.76). This gauge freedom admits the gauge choice 9= 0, which reduces S[Y, 9 1to the superfunctional (6.9.82) which is invariant under the transformations dY,=iD2D,K, +iD,K,
Ki=R,.
(6.9.83)
Our considerations show that the dynamical system with action (6.9.82) describes the massless gravitino multiplet. This is the model proposed by V. Ogievetsky and E. Sokatchev.
Copyright © 1998 IOP Publishing Ltd
7 Effective Action in Curved Superspace Time present and time past Are both perhaps present in time future And time future contained in time past If all time is eternally present All time is unredeemable T.S. Eliot: 'Four Quartets' Burnt Norton
7.1. The Schwinger-De Witt technique
The Schwinger-De Witt technique (or the proper-time technique) is a powerful method for studying the effective action in theories with external fields. This technique was first introduced by J. Schwinger and later formulated in a general-covariant manner by B.S. De Witt. The SchwingerDe Witt technique is known to be well adapted to calculations of divergences and anomalies in the framework of the background field method. In the present chapter we describe a superspace extension of the Schwinger-De Witt technique and explore it for investigating the structure of effective action superfunctional, including the analysis of divergences, anomalies and so on, in simple locally supersymmetric theories in a curved superspace. To make the account complete and independent, we find it pertinent to give here a brief review of the standard Schwinger-De Witt technique. 7.1.1. When the proper-time technique can be applied Let us consider a non-gauge theory of bosonic and fermionic fields pi p'(x) which live on a curved background space-time to be described by a vierbein 5 84 Copyright © 1998 IOP Publishing Ltd
Eflective Action in Curved Superspace
585
e,"'(x).Let S [ q ] be the classical action of the theory. The one-loop correction to the effective action was shown in Chapter 4 to be formally given by i 2
I-("[@] x - sTr ln(;,S,I[@])
(7.1.1)
where the operator Srr= ;,S,j[@]determines the quadratic part of S [ q ] with respect to the mean field 3. The standard Schwinger-De Witt technique allows one to obtain information about the explicit structure of F1)[@] only when sDet(S") can be represented as a product of functional determinants from second-order differential operators of the general form H = V'V, - m2O + 2da(x)v,+ 2(x)
(7.1.2)
where V denote torsion-free covariant derivatives. Each operator H is assumed to act on a space of purely bosonic fields or purely fermionic fields, d"and 9 being matrix-valued functions over the space-time. The chief advantage of the Schwinger-De Witt technique is that it supplies us with a universal prescription to determine the short-distance behaviour of the Feynman propagator associated with any operator (7.1.2). It is worth recalling two simple theories meeting the above requirement. We first consider the model of a scalar field cp with the action
's
/I S[q] = -- d4xe-'{V"cpV,cp+m2cp2+gWcp2+-(p4} 2 6
(7.1.3)
where m, 5 and 1. are coupling constants and R is the scalar curvature of the background space-time. Here the one-loop quantum correction is 1
l-(')[@]= - Tr ln(V"V, - m2 2
- i@'). (7.1.4)
Another model describes a charged massive field YD,a four-component Dirac spinor, and its conjugate 6, coupled to external gravitational and electromagnetic fields:
s
~ [ ' Y D= ] - d4x e-'yD{iy"(v,-iqA,)+m)lD.
(7.1.5)
The quantum correction to the classical action is now given as
where
1 r =-iTrlnh('")= --TrlnH(") 2
h('")= iy"(V, - iqA,) +mO
iyaPa+ mi
~ ( " ' = h ' m ' h ' - m ) = ~ ' ~ , - (+;8), ~ 1' Copyright © 1998 IOP Publishing Ltd
+a
(7.1.6)
1
qF,b[y",
rb]
(7.1.7)
586
Ideas and Methods of Supersymmetry and Supergravity
P being the gauge-covariant derivatives. In equations (7.1.4) and (7.1.6) we
have taken into account that sTr coincides with (differs in sign from) ordinary functional trace denoted by Tr in the purely bosonic (purely fermionic) case. The second example considered illustrates the observation that any operator (7.1.2) can be brought to the form
H = W , - m * O +$x)
(7.1.8)
by redefining V, in the manner V , - t V , = V , + d , . In what follows, the generalized covariant derivatives will be assumed, for the sake of simplicity, to have no torsion, that is, to satisfy the commutation relations CVa, v b l
with some generalized curvature
(7.1.9a)
= &ab
gab,given by (7.1.9b)
g a b ( p i = &,bijv'.
In contrast to the definitions of Section 4.1, in curved space-time it is useful to define formal operations, like variational derivatives and sTr, with the help of the covariant delta-function d4(x, y) = e(x)d4(x- y) = a4(y, x) which is bi-scalar under general coordinate transformations. Thus we have *
s
s
'
sTr F = d4x e - '( - l)E'Fii(x,x) ZE d4x e- s t r F ( x , x). Here F denotes some operator, and its kernel F:(x, y ) is determined by
s
Fq'(x) = d4y e - ' ( y ) F i i(x, y) cpj(y) for any cpi with compact support. 7.I.2. Schwinger's kernel We introduce a Green's function G'Xx, y) of the operator (7.1.2)as a solution to the equation
H'j(X)Gjk(X, y ) = - dikd4(x,y)
(7.1.10)
To pick out a single solution of this equation, one must take into account suitable boundary conditions. We choose those boundary conditions which specify the Feynman propagator by giving to the operator H an infinitesimal imaginary part (H ie), E -, 0. Then in the limit of turning off ail the external fields, in particular when the background metric g,,(x) takes its flat value
+
+
Copyright © 1998 IOP Publishing Ltd
587
Eflective Action in Curved Superspace
vmn. :be Green’s function G reduces to the Feynman propagator in flat space-time. To solve equations (7.1.10) under the boundary conditions chosen, we use the integral representation G(x, y)=i
1;
ds
w,
YlS)
(7.1.1 1)
where U ( x ,yls) satisfies the Schrodinger-type equation (7.1.12) This representation is known as the Fock-Schwingerrepresentation,with parameter s the proper time and operator U(x,yls) the Schwinger’s kernel. It is worth noting that the &-prescription guarantees convergence of the integral in (7.1.11) at the upper limit. Substituting (7.1.11) into equation (7.1.10) and making use of equation (7.1.12), we obtain H(x)G(x,y)= - U ( X ,y10)=
-O
~ ( X , y).
Hence the Schwinger’s kernel must be subjected to the initial condition lim U ( x , yls) = 0 6(x, y).
(7.1.13)
s-?+o
Therefore, finding G is equivalent to solving equation (7.1.12) under the initial condition (7.1.13), but the latter problem turns out to be simpler in practice than a direct examination of the Green’s function. Before discussing the general structure of the Schwinger’s kernel, we first compute this object in the case of Minkowski space and for vanishing background fields, i.e. when the operator (7.1.8) takes the simplest form
H~= (rmna,,,a, - m2) 0. Here, use of the Fourier transform
reduces the evolution problem
(*is+ 1
)
- Ho U,(x, yls) =0
to the following one
Copyright © 1998 IOP Publishing Ltd
UO(X,y10) = 0 64(x - y )
(7.1.14)
588
Ideas and Methods of Supersymmetry and Supergravity
Then we obtain (7.1.15) Using the integral identity
r"
a>O
dl: eiar2 + ibr ,
J -iL
a
where a and b are real numbers, a#O, it is easy to calculate the integral (7.1.15) resulting in (7.1.16) We note that U , involves the Poincare-invariant two-point function (7.1.17) which is the half squared interval between two points x and y in Minkowski space and satisfies the identity dmcodmcTO = 20,.
(7.1.18)
Returning to a curved space-time, we will search, following De Witt', V ( x , yls) in the form
Here c(x, y) and R(x, y ) are bi-scalar functions to be specified below, F'Ax, yls) is supposed to be analytic in s in a neighbourhood of s = 0, F'Ax, yls)=
1 anij(x,yKis)".
(7.1.20)
n=O
Comparison with equation (7.1.16) shows that each two-point function a,(x, y), for n = 1, 2, . . ., is created by the background fields and disappears in the limit of turning off all these fields. By virtue of equation (7.1.20), one can see the ansatz (7.1.19) is consistent with equation (7.1.12) only if a(x,y) is subjected to the requirement VaaVaa = 2u.
(7.1.21)
'B.S. D e Witt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965). Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Superspace
589
This is the very equation which identically satisfies an important geometric object known as the 'geodesic distance', which is half of the squared distance along the geodesic line between two space-time points x and y. Next, the function o should possess the boundary behaviour n(x, x)=V,u(x,
(7.1.224
x')I,,,,=O
and det(d,&o(x, x'))l,=,,
(7.1.22b)
= -e-*(x)
in order to satisfy the initial condition (7.1.13)and to have the correct limit in flat space. Such boundary conditions prove to pick out a unique solution of equation (7.1.21).We note also that for near points the geodesic interval is 1 2
- gm,(x)(x - Y)YX - Y)"
+ O((X -YI3)
(7.1.23)
and, hence is consistent with equation (7.1.22).Therefore, we identify o with the geodesic interval. The introduction ofR(x, y) in expression (7.1.19) is a matter of convenience; the role of this object is to make maximally simple the equation for F(x, yls). It is useful to choose R in the form of the Van Vleck-Morette determinant R(x, x') = - det( - V,Vbo(x, x'))
(7.1.24)
v'(nv,o) = 4 0
(7.1.25)
Q(X, x) = 1.
(7.1.26)
obeying the equation and the boundary relation Then equation (7.1.12) tells us that aF i -+Vaa'OaF=iR-1'2~'0,(R112F)+i~F. as s
(7.1.27)
Using the decomposition (7.1.20), we obtain the system of equations VaaP,uo = o
( v ~ ~+ P n, + l)a, +
(7.1.28~)
+
= n- 1/2PP,(n1'2a,)2a,.
(7.1.28b)
The zeroth coefficient must be subjected to the boundary condition aoi,(x, x) = Sij
(7.1.29)
which guarantees, together with equations (7.1.22,23) and (7.1.26), fulfilment of the initial condition (7.1.13). The two-point functions a,, n=O, 1, 2, ..., under equations (7.1.28) and (7.1.29) are known in the literature as the Copyright © 1998 IOP Publishing Ltd
590
Ideas and Methods of Supersymmetry and Supergravity
De Witt coefficients. Equations (7.1.28) are recurrence relations allowing one to calculate the De Witt coefficients step by step. In accordance with equations (7.1.21) and (7.1.25), the bi-scalars CJ and $2 entering the Schwinger's kernel are completely determined by the pseudo-Riemannian manifold under consideration and do not depend on the background fields except the gravitational field. In other words, ,these objects are universal ones for field theories in a fixed curved space-time. A different situation will be shown to take place in curved superspace. Remark. In equations (7.1.21) and (7.1.25) we have taken into account the fact that the derivatives V, and 0,differ only in an internal connection which does not act on Lorentzian indices. 7.1.3. One-loop divergences of efective action We turn to the analysis of the first quantum correction to S [ q ] in a bosonic field theory
l-(')[@]
i 2
i 2
=- Tr In H = -- Tr In G.
(7.1.30)
This definition of I-(')[@] is rather formal. Rewritten with the aid of the proper-time representations for G, r(')[$] turns out to acquire a more transparent form. Let us consider an arbitrary variation 6H of the operator H. Then r(')[@] has the displacement i 6r(')[@] =-Tr(H-'6H) 2 in which we replace H-'= -G by its explicit expression (7.1.11)'
Equation (7.1.12) tells us that Tr(U(s)GH))=(is)-' Tr(SU(s)),hence
has the form Therefore, up to some additive constant, r(')[@]
F[@]= -;j:${d"xe-'(x)tr
U(x, xis)
(7.1.31)
where tr denotes the trace over matrix indices. As a result, the one-loop effective action is determined by the Schwinger's kernel at coincident points. We come to the conclusion that the Schwinger's kernel should be a central object in one-loop quantum field theory. Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Supe'rspace
591
Use of the explicit form (7.1.19) for U ( x ,yls) in equation (7.1.31) leads to
Now, equation (7.1.20) shows that the first three terms of the series (7.1.20) cause the above integral to be divergent at the lower limit. Hence, one must regularize the integral over proper time in practical calculations. Below we list several popular regularizations. In the framework of the dimensional regularization scheme, the Schwinger's kernel, corresponding to the four-dimensional theory, should be replaced by its analogue for a d-dimensional theory
Here w = (4 - d)/2, and the objects um,R, and F , correspond to d-dimensional space-time. The mass parameter p has been introduced into (7.1.33) to keep fixed dimension of U,. At o = O the kernel U, turns into the original one (7.1.19). The dimensionally continued effective action is then given by
Treating w as a complex variable, the integral will be convergent for Re w > 2 and can be readily computed in this region. Making use of the decomposition
dW)being the De Witt coefficients in d dimensions, and introducing the notation ta
J
Ai@ = ddx e;
tr aiU)(x,x)
An= Aio)
(7.1.36)
one finds in the massive case
where T(z) is the gamma-function. We separate the part of I-:)[$] to divergences when w -P 0
Copyright © 1998 IOP Publishing Ltd
leading
592
Ideas and Methods of Supersymmetry and Supergravity
In the limit w +0 this expression reduces to A,-m2Al + - A o m4 2
1
+finite terms.
(7.1.38)
Therefore, we must calculate the coefficients a,(z, z) and a,@, z) to determine the one-loop divergences of the theory. The relation (7.1.37) constitutes an asymptotic expansion of the effective action in inverse powers of m'. It is obvious that such an expansion is inadmissible in the massless case, where one must apply some other technique to sum an infinite number of terms coming from the Schwinger's kernel. In practice, one usually introduces an auxiliary mass M at intermediate stages of the calculation to improve the behaviour of the integral over proper time at the upper limit. This mass parameter should be eliminated only after completing all manipulations with ultravioletly regularlized expressions. Any polynomial-in-M term in the effective action can be freely removed. In particular, the one-loop divergence of r(')[@] in the massless theory is determined by the first term in the figure brackets of equation (7.1.38).The corresponding counterterm in the minimal substractions scheme reads
sg',;,, = - -
'
d4x e- tr a2(x, x),
(7.1.39)
To find the counterterms, one can use a scheme which is technically analogous to dimensional regularization but less sophisticated. Such a regularization consists in simple insertion of an additional factor (is)" into the integral over proper time in equation (7.1.32), without any continuation of the Schwinger's kernel in &dimensional space. Then we obtain
r$)C@I =-p20 2 1
:s
ds j d 4 x e-'(x) tr U ( x , XIS). is)' -
(7.1.40)
This prescription will be called the o-regularization. One more way to regularize the integral over proper time is as follows
j'-
32n2
m
-e d(is)
-im2s
jd'xe-'(x)trF(x,x,s)
(7.1.41)
-iL2
L being a real regularization parameter of inverse mass dimension, L + + 0 at the end of calculations. Here the integration over s is performed along the positive part of the real axis, except a small region near s=O where the integration contour goes to the domain Im s < O to end at s = -iL*. The point is that everyone computes, starting from Schwinger, proper-time Copyright © 1998 IOP Publishing Ltd
Effective Action in Curved Superspace
593
integrals by rotating the integration contour into the negative imaginary axis or, which is the same, by making the replacement s = -is with real non-negative z. Hence, the above prescription is equivalent to cutting off time-rotated integrals at the lower limit. For s z 0 the decomposition (7.1.20) is applicable, and one can readily calculate the divergent part of (7.1.41).One finds
A 2 - m 2 A , +-Ao m4 2
)}
(7.1.42)
where p is a parameter of mass dimension. Dropping here all m-dependent terms, we obtain the one-loop divergences for the massless theory. Then counterterms can be chosen in the manner -ln(L2p2)A2}.
(7.1.43)
7.1.4. Conformal anomaly The Schwinger-De Witt technique is an ideal tool for the study of breaking local symmetries by quantum corrections. In this subsection we demonstrate its power by giving explicit calculations of the so-called conformal (or Weyl) anomaly. As has been discussed in Section 1.7, certain field theories in curved space-time possess a local scale (or Weyl) invariance of the form
e: "(x) = eo(x)ea*(X)
cp"(x) = ept+'(x)cpi(X)
(7.1.44)
P(i)beingfixedconstants determined by the field types under consideration. Such a symmetry is inconsistent with the presence of massive parameters in the classical action, hence it can originate only in massless theories. The Weyl invariance appears to be potentially anomalous at the quantum level. The problem is that one inevitably introduces some massive parameter in the process of regularization. As a result, there is room for anomalies in any classically Weyl invariant theory. We will consider two models describing bosonic fields cpi(x)and fermionic fields Y'(x), respectively, on a curved gravitational background. Their action functionals read as follows: S B = i 1 d 4 xe-'qijcpi(Hcpy'
H = VaVaCopyright © 1998 IOP Publishing Ltd
(7.1.45)
594
Ideas and Methods of Supersymmetry and Supergravity
5 being a constant, and
s
sF=-
d4x e-'vIJq'(h'Y)J
(7.1.46)
h= iy"V,.
Here q i j and vrJ are constant Lorentz-invariant matrices. It is easy to see that SBmay be invariant under transformations of the form (7.1.44) only if p ( t )= 1, for each index i, and the operator H changes according to the rule
(7.1.47) Owing to the first and fourth relations (1.6.33), such a transformation law proves to be admissible only under the conditions that <=* and the field family { v i }is a set of scalars. Analogously, SF turns out to be Weyl invariant only under the conditions: (1) pr =3 for each I ; (2) the field family ('Y'} is a set of Majorana spinors. Then, the operator h is characterized by the transformation law
(7.1.48) The squared operator HF = hh, which is used to define the fermionic effective action, transforms in more complicate way than the bosonic operator H
Now, we are going to investigate the behaviour of renormalized effective action for a massless conformal scalar field under the Weyl transformations (the case of a massless spinor can be handled similarly). Naive effective action 1
r=-Trln H 2
1 H=VaVa--W 6
(7.1.50)
is formally invariant with respect to the redefinitions (7.1.47) provided Det is defined such that Det(a(x)G(x,y)) = 1 for any Q. However, here we operate with infinite expressions. To make our manipulations sensible, we introduce some regularization. Let us first consider the effective action in the w-regularization scheme:
(7.1.51) where the kernel U(s) corresponds to the conformally covariant operator H given in equation (7.1.50). In accordance with equation (7.1.47), an Copyright © 1998 IOP Publishing Ltd
Effective Action in Curved Superspace
595
infinitesimal Weyl transformation changes H according to the rule 6H = qBoH -pBHC hence 6 Tr U(S)= iS(q,-p,)
Tr(aHU(S))= ( q B -pB)S
a -Tr(oU(s)). as
(7.1.52)
This leads to (7.1.53) Since K, is proportional to w, only the divergent part of the proper-time integral, appearing in equation (7.1.53),contributes to K , in the limit w +O. This part can be found in complete analogy with the derivation of equation (7.1.38), that is, by introducing the factor exp(-iM2s), M being an infrared regulator, into the proper-time integral. Then, after separating the divergent part, M should be removed. As a result, one obtains
6r lo
=--
'S
(47q
d4x e - 0a2(x, x)
+ O(w).
(7.1.54)
Let us define the renormalized effective action rren =
lim
CO-0
(rlo+ scourit)
with ScOunt given as in equation (7.1.39). Equation (7.1.54) shows that not invariant under the Weyl transformations:
rren is (7.1.55)
This phenomenon is known as the Weyl anomaly. Equation (7.1.54) allows one to observe an interesting property of counterterms in a classically Weyl invariant theory. The point is that 6r, has no divergent part. Since To is of the form
-s count + r r e n + O(w)
lTlo=
we conclude that the functional ScOunt is Weyl invariant. Therefore, the combination e - '(x)u2(x,x) varies in total derivative under the Weyl transformations. It is instructive also to re-obtain the Weyl anomaly in the framework of the L-regularization scheme. Starting with the regularized effective action i rL=--J
2
Copyright © 1998 IOP Publishing Ltd
-iLZ
ds -Tr s
U(s)
(7.1.56)
596
Ideas and Methods of Supersymmetry and Supergravity
we consider its variation with respect to an infinitesimal Weyl transformation. Making use of equation (7.1.52) gives
s=
i 1 6rL = - - ( q B - p ~ ) ds - Tr(a U(s))= - ( q B - P B ) Tr(a U(S))l = - i ~ z . 2 -iL2 as 2 Here and below we do not at this stage set explicit values for q B and pB. For s z0 the decomposition (7.1.20) is powerful, and the above expression can be rewritten as follows: 6rL=
1 -y(qB-pB) 3271
s
d4~e-'o{L-4+L-2a,(x,x)+a2(x,x)}+O(L2) (7.1.57)
where we have used condition (7.1.29).Then, the finite functional (r,+ SCOUnt), for S,,,,, given as in equation (7.1.43), is W
L
+
SC,",,)
= --
+L-'[(qB-pB)oe-'al(X,
X)+d(e-'a,(x, x))]
I
-ln(L2p2)6(e-'u2(x, x)) +(qB-pB)af?- 'a2(x, x) + o ( L ~ ) . Since the variation must be finite as L+O, there appear the consistency conditions:
qB-PB=2
(7.1.58~)
6(e-'al(x, x))= -2oe-'a,(x, x)+ total deriv.
(7.1.58b)
S(e- la2(x,x)) = total deriv.
(7.1.584
In addition, the renormalized effective action rr,,=(I-, + Scount)lL+Osatisfies the same relation (7.1.55) which arose in the w-regularization scheme. Therefore, the Weyl anomaly is a physical effect independent of the regularization chosen (in contrast with counterterms). Let us discuss the relations (7.1.58). Equation (7.1.58~)arose as a requirement necessary for Weyl invariance of the classical action. We note further that equation (7.1.586) is satisfied identically if a,(x, x ) - 9 (see equation (1.6.33)). As will be shown below, a,(x, x)=O for a massless conformal scalar field, but this is not the case for a massless spinor field where a,(x, x) coincides with the scalar curvature. Finally, equation (7.1.584 we met when describing the Weyl anomaly in the framework of w-regularization. The correctness of equation (7.1.58~)can be easily checked on the basis of explicit results concerning the general structure of a,(x, x) and given in the next subsection. Copyright © 1998 IOP Publishing Ltd
Effective Action in Curved Superspace
597
In conclusion, let us return to the relation (7.1.55). We recall that any Weyl invariant field theory is characterized by the condition that its energymomentum tensor is traceless (see subsection 1.6.8).The breakdown of Weyl invariance by quantum corrections appears to be equivalent to the fact that the averaged energy-momentum tensor ( Tm”) X , , , / ~ g m , acquires a non-zero trace (T;), the latter coinciding with the expression in the left-hand side of equation (7.1.55). Therefore equation (7.1-55) shows that the trace of the averaged energy-momentum tensor is given by the a,-coefficient.
-
7.1.5. The coeflcients a,(x, x) and a,(x, x) We have already seen that one-loop counterterms and anomalies are determined by De Witt coefficients a , and a, at coincident points, thus it is necessary to be able to calculate these objects. In principle, the calculation of De Witt coefficients at coincident points is a simple, but tedious task. It consists of numerous differentiations of the master equations (7.1.21),(7.1.25) and (7.1.28) together with the use of the covariant derivatives algebra (7.1.9) and the boundary conditions (7.1.22), (7.1.26) and (7.1.29). To find a,(x,x) and a2(x,x), it is necessary first of all to calculate purely geometric quantities of the form V,, . ..Vakc(x,X ’ ) I ~ = ~ , and V,, . .. V,jR1/2(x,x’)I,=,. to finite orders k and I. Direct calculations lead to the relations:
1
V a V b e(x, x’) x = x’ = q a b
(7.1.59a)
v,vbvco(x, x’)Ix=x‘=o
(7.1.59b)
vavbvcvd
2 o(x, x’) 1 x = x’ = - wa(cd)b 3
v, 0 0c(x, x’)I x
=-
v,w
(7.1.59~) (7.1.59d)
and
V,R1’2(X, x’)I
= x, = 0
(7.1.60a)
1 gab 6
(7.1.60b)
1 vow 6
(7.1.60~)
VoVbR1’2(X, x’) I
=-
v, 0 R”z(x,x’)l x
=-
(7.1.60d) Copyright © 1998 IOP Publishing Ltd
598
Ideas and Methods of Supersymmetry and Supergravity
where =V"V,. For example, let us derive the important relation (7.1.59~). Acting with V b v , on both sides of equation (7.1.21),we then put x = x' in the resulting identity: v b V " o v , v , o v " o V b v , ~ , o= vbvco
+
and use equation (7.1.22~). This leads to FbnfdFdc
= Fbc
where Fhc=VbVcO(X,x ' ) ( ~ = ~Since , . Fb, is a non-singular matrix as a result of equation (7.1.22b),the final relation is equivalent to relation (7.1.59~). Other information which is necessary for computing a , ( x , x ) and a2(x,x), involves multiple derivatives of a, at coincident points:
-
V,ao(x, x ' ) l x = x ' = o
(7.1.61a)
1 V,Pbao(x, x')1 x = 2' = 2
(7.1.61 b)
aO(x, x')I
=
1 2
=- PbBab
(7.1.61c)
where ? i =pP,,. Let us describe the derivation of equation (7.1.61b).Acting with p b p c on the left-hand side of equation (7.1.28~)gives
+
+
VbVcVaopaao {VbVaopcpaaO+(b++c)} V a b ~ b P c P a a O = O . We put x = x ' and make use of equations (7.1.22a), (7.1.59~)and (7.1.59b) resulting in
(Vbpcao
+P c v b a O ) I
x
which can be rewritten as follows: 2PbVca0 I x = x ' = [ p b , pcIa0 I ~ = x ' = ~ b c ~ O ( ~ ,
Then, the boundary condition (7.1.29) leads to equation (7.1.61b).
Remark. In the above expressions, gRb =Se,, denotes the Riccci tensor, while iii?,b = -Bba denotes the generalized curvature tensor taking its values in some matrix algebra (see equation (7.1.9b)). Now, we are ready to calculate a , ( x , x ) and a 2 ( x , x ) . In accordance with equation (7.1.28b), the equation determining a,(x, x') reads VaoPaal +a,
=a- 1 ' 2 ~ P a ( n 1 ' * a 0 ) + 2 a 0 .
(7.1.62)
We set here x = x ' and make use of equations (7.1.22a), (7.1.26), (7.1.29), (7.1.60a,b)and (7.1.61b).This immediately gives 1 WO. 6
a,(x, x ) = 2 + -
Copyright © 1998 IOP Publishing Ltd
(7.1.63)
Effective Action in Curved Superspace
599
Next, the equation determining a, reads V"oV,a2+2a2 =SZ-1'2VVa(n1'2al)+ s a 1
and reduces, for x=x', to
'(-+ : 0 )(-9+:BO)+ ;B
a,(x, x) = - 2
2
-B
-
-
(x, x')a l(x, x') I
=
The quantity aal(x, x ' ) [ ~ =can ~ ,be easily computed with the help of equation (7.1.62) and the relations (7.1.59-61). One obtains as a result:
1 180
a,(X, X)=-(BabcdWabcd-~abWab+
1 12
oa)o+ - P b 4 a b
The relations (7.1.63,64)solve the problem of finding the De Witt coefficients a, and a2 at coincident points. For applications it is sometimes useful to re-express (gabcd)2 in equation (7.1.64) via the squared Weyl tensor. Making use of the identity
+ 31 W 2
CabcdCabcd= BabCdBabcd - 2Pb&?ab -
(7.1.65)
which follows from equation (1.6.17), we obtain
(7.1.66) As examples, let us determine the coefficients a, and a, associated with the conformally covariant operators
1 6
H,,=0 --w
1 H,,=hh= 0 --WO 4
for h given as in expression (7.1.46).The H,, and H,,are assumed to act on spaces of scalar and four-component spinor fields, respectively. In the former case we have 3= -:a and &ab = 0, hence
Copyright © 1998 IOP Publishing Ltd
600
Ideas and Methods of Supersymmetry and Supergravity
In the latter case we have 3=-$go and M c d = -'[ f , y d ] , hence
&?ab=@?abcdMCd,
1 tr u ~ P ~ ( x , x )---% = 3
where (7.1.68~)
(7.1.68b) The results of subsection 1.6.6 imply that the functionals
l
Ay)= and A(;p)
=
d4x e - ' u f ' ) ( x , x)
e - tr u(zp)(x,x)
are invariant under arbitrary Weyl transformations. In summary, we have described the standard Schwinger-De Witt technique established in its final form in the 1960s, and still remaining the most efficient method for studying functional determinants of operators of the form (7.1.2). For more complicated differential operators there exists a generalized Schwinger-De Witt technique developed later by A. Barvinsky and G. Vilkovisky'. The interested reader can find the necessary details in the original paper.
7.2. Proper-time representation for covariantly chiral scalar superpropagator We proceed to investigating the one-loop effective action in theories of matter superfields on a curved superspace. Among the specific features of quantum theory in curved superspace (or space-time) is the fact that the matter effective action begins to depend here not only on the matter mean (super)fields, but also on the supergravity prepotentials (or gravitational field). Analysis of the latter dependence will be our primary goal. Throughout this chapter, only the minimal formulation for Einstein supergravity will be used to describe the background supergeometry. In the present section, we often make use of the prepotentials parametrization adopted in Sections 5.4 and 5.5. As mentioned above, the standard Schwinger-De Witt technique is most suitable for studying functional determinants of second-order differential 'A.0. Barvinsky and G.A. Vilkovisky, Phys. Rep. 119 1, 1985. Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Superspace
601
operators with the universal second-order term VT,. Most differential operators which arise in superfield theories involve spinor partial derivatives along with ordinary ones. It turns out that the Schwinger-De Witt technique preserves its power for superspace second-order operators of the general form 9%BO+fyz)[Qa,
a,]+f1(z)@
+f2(Z)@
+ . ..
(7.2.1)
where the dots mean all terms are at most linear in covariant derivatives. A mixed term with spinor and vector derivatives can, in principle, be included into expression (7.2.1), but such structures do not appear in practice. When considering the one-loop quantum correction in a superfield theory, we will try to re-express it via determinants of operators having the form (7.2.1).
7.2.1. Basic chiral model Let us begin with the curved-superspace action S(y,[0]=
s
{is
d8z E-'$.@+
d8Z---Y0'+~.~. ER1
}
(7.2.2)
describing the interaction between dynamical (0) and external ( Y )covariantly chiral scalar superfields, = BiY = 0. A set of chiral models, which will be discussed below, can be read from S(v).For Y = m we obtain the massive chiral action S,,,[0]=
Setting \I, = 0 action
s
d*zE-'@@+
gives the massless super
s
(7.2.3) Weyl invariant model with the
S [ 0 ] = d8z E - ' @ @ .
(7.2.4)
For Y = R we obtain another interesting massless action S(,)[@]=-
's
2
d'z E-'(@+$.)',
(7.2.5)
In curved superspace, the variational derivative for a covariantly chiral scalar superfield is given by equation (5.5.28). Hence the Hessian of S,,,[0] reads
"
S-(Z, z')
Copyright © 1998 IOP Publishing Ltd
(7.2.6)
602
Ideas and Methods of Supersymmetry and Supergravity
where (7.2.7)
For Y =0 this operator reduces to
Let us note that H“y) turns into the operator (4.4.5) in the flat superspace limit. We denote both flat and curved operators by the same symbol, but only the latter will appear in the present chapter and hence no confusion should arise. The quantum correction TCv,to S(vy)is defined by the superfunctional integral eirc.v= )
s
9~~a eiSc~~[@l
(7.2.9)
and can be represented in the form 1
TCv, = - Tr In H(y).
(7.2.10)
2
Remark. Throughout this chapter, the operations sTr and sDet are denoted by Tr( -Tr) and Det(Det- ’), respectively, in the purely bosonic (fermionic) case. Remark. We define the superkernel of an operator
which acts on a space of covariantly chiral-antichiral scalar columns
with the help of the covariant chiral delta-function 1 -
6 + ( z ,z’)= -- (9’ -4R)ES8(z- z’)= rp - e e IVS4(x - x’)S ’( 8 - el) 4 and its conjugate 6 - ( z , z’) in accordance with the rule A - +(z,z’) A--(z,z’) Copyright © 1998 IOP Publishing Ltd
0
).
6-(z,z’)
(7.2.11)
Effective Action in Curved Superspace
603
This definition is convenient in practice since the components of the superkernel are bi-scalar and covariantly chiral or antichiral according to which label, + or -, is attached to the corresponding argument (for example, $&A+ +(z,z’)=B&A++(z,z’)=O).Then Tr A reads as follows: TrA=
s
s s
d6ze- (rp A++(z,z))+ d 6 2 e - W ( @ 3 A - - ( ~ , ~ ) ) M , 3
A , +(z,z 1
+
=s d 8 z
dsz -A - - (z,z ) ER1
r T r + A ++ + T r - A - - .
(7.2.12)
The action (7.2.2) is the minimal extension of the flat action (4.4.6) in a curved superspace. As we have seen, the flat action was used to define the one-loop quantum correction in the Wess-Zumino model (see equation (4.8.27)). Analogously, the one-loop correction F1)[@] in the generalized Wess-Zumino model (6.3.7) is connected with the superfunctional (7.2.10) by the rule (7.2.13) r(”C@I = r c Y ) I Y = m + < R + 10. This is why Tr, deserves consideration for arbitrary Y. When “#const the operator H“y) has such a structure that the Schwinger-De Witt technique proves to be inappropriate to handle directly the Feynman superpropagator of H(y).There are two approaches, however, to reduce the problem of computing Det H(y) to similar problems, but for operators which meet the chief requirement of the proper-time technique. One of these approaches can be applied in the case when the body of Y(z) is non-zero at each point z of superspace. Under this assumption, we have (7.2.14)
where ea=Y/m
m=const#O
(7.2.15)
and m is a real parameter and H’@) is just the operator H‘”) constructed on the base of the super Weyl transformed covariant derivatives (5.5.13) with Q given by conditions (7.2.15). Equation (7.2.14)leads to the formal relation Det H(y)= Det H(”’)
which shows that Tcy, can be read from r(,,,). If we explicitly indicate the on the supergravity prepotentials W M and q, dependence of r(,,,) r(m) = r,,,[W, q ] , then r(yy) is given by (7.2.16) Copyright © 1998 IOP Publishing Ltd
604
Ideas and Methods of Supersymmetry and Supergravity
Another approach to computing Tn,,which needs no assumption about Y , will be described in the following section. 7.2.2. Covariantly chiral Feynman superpropagator Let us consider the Feynman superpropagator
associated with the operator H('"),By definition, G(m)satisfies the equation
m
-$(9'-4R) m
G',"! G(+m)_ G(?+ G(?L)=
)(
-(: 60_)* (7.2.17)
From this equation we obtain
(7.2.18) where
The bi-scalar Green's function Gim)(z,z') is covariantly chiral in both arguments and because of equation (7.2.17) satisfies the equation where
(Ye, - m2)GLm)(z, z') = - 6 +(z,z') 1 16
% =- (9' -4R)(9' -4m.
(7.2.19) (7.2.20)
Similarly, the antichiral superpropagator GLm)(z,z') obeys the equation
( X - m ' ) G Y ( z , z')= -6-(z,z')
(7.2.21)
where 1
= -(9' -4R)@'
16
- 4R).
(7.2.22)
As pointed out in subsection 6.3.2, the fourth-order differential operator Zc (Sa) turns into a second-order operator when acting on the full space of covariantly chiral (antichiral) scalar superfields; that is 1 -
=o+--(a2R)+RW %'chiral scalar space 4 1
%lantichira, scalar space
Copyright © 1998 IOP Publishing Ltd
=
- - 3 g 2 ~ ) +RR. 4
605
Egective Action in Curved Superspace
Here + and - are the chiral and anticharal d'Alembertians, respectively, defined in equations (6.3.18) and (6.3.19). Both operators U + and 0-have the form (7.2.1). The effective action r(,,,) can be readily expressed in terms of the superpropagators GLm)and GL'").With the aid of the formal relation
Det H(m)= Det1/2(H("H(-") one obtains i qm) = - - Tr
4
1
+
In Gim)-- Tr - In G:") 4
(7.2.23)
for Tr+ defined as in equation (7.2.12). In accordance with equations (7.2.18) and (7.2.23), the Green's functions GLm)and GLm)are central objects in the quantum theory. Both terms in the right-hand side of equation (7.2.23) produce in fact the same contribution modulo boundary terms at infinity, because of the formal equality 6Tr - In GLm)= 6Tr - In GLm)
(7.2.24)
which holds for an arbitrary variation of the supergravity prepotentials. Equation (7.2.24) is a simple consequence of the operator relations
(7.2.25) for H + - and H -
+
given in identity (7.2.8)' together with the identity Tr+(A+-B-+)=Tr-(B-+A+-).
(7.2.26)
To prove relation (7.2.25), it is sufficient to take account of the composite nature of Scand X, % , = H + -H-
+
Sa = H - + H +-
(7.2.27)
%,H+_=H+-Sa.
(7.2.28)
which implies
H-+Zc=XaH-+ As a result, we can rewrite
r(m) in the form 1 r(m) = - - Tr + In GLm).
(7.2.29)
2
In perfect analogy to the massive case, the quantum correction massless theory (7.2.4) can be represented as i 1 I-= --Tr+ In G,--Tr4 4 Copyright © 1998 IOP Publishing Ltd
1
In G,= --Tr+ In G, 2
in the
(7.2.30)
6%
Ideas and Methods of Supersymmetry and Supergravity
where the chiral (G,) and antichiral (G,) Green’s functions constitute the Feynman superpropagator G associated with the operator (7.2.8) 0
”)
(7.2.31)
Ga
and satisfy the equations #cG,(~,z’) = - 6 + (z, z‘)
(7.2.32a)
%aGa(~, z’)= - 6 - ( 2 , ~ ’ ) .
(7.2.32b)
7.2.3. The chiral d’Alembertian Before proceeding further, we would like to discuss the chiral d’Alembertian c]+
1 1 =9a9a+-992 +iGa.9a+-(9ER)3E 4 4
(7.2.33)
entering the equation for GLm),and related geometric objects. As has been already mentioned in subsection 6.3.1, the chiral d’Alembertian is unique in the sense that any operator of the form
a’9*+fEn(z)9&
+
+Y(z)Qa f“(z)%
which maps the full space of covariantly chiral scalar superfields into itself, coincide with C] +. Closely related to + is an operation of multiplication on the chiral space. This operation is defined by associating with arbitrary chiral scalars and O2 another scalar superfield 1 0 1 * @2 = @2 * @ 1 3 -(g2 -4R)(9’@19@2) 16
= go@ 1ga@2
+-41
R9a@,9g@2
(7.2.34) which is covariantly chiral by construction. The connection between * reads
+
and
(7.2.35) for arbitrary covariantly chiral scalars Q1 and 02. the *-product is uniquely defined in the sense that any Similarly to o+, binary operation (@I, @2)=.9“@19aQpz
+f(z)9’@1ga@2 + f ” ” ( ~ ) ( 9 ~ @ l 9 ~ @+29 a Q 1 . 9 E @ 2 )
which maps arbitrary chiral scalars O1 and Q2 into a chiral scalar, must coincide with ml * 02.This observation has important consequences. First, chiral generalizations of the equations (7.1.21) and (7.1.25) for the geodesic Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Superspace
607
interval and the Van Vleck-Morette determinant, respectively,are of the form
* = iBaaiBau+ - R9'aiBEa = 2a
1 4
(7.2.36)
1 9a~9,,S2 - RiB'a9'52 = (4 - 0 + C T ) ~ .
(7.2.37)
U U
and
+4
Analogously, the equation for an antichiral extension of the geodesic interval should have the form
+1 RB&5@5= 25. 4
9a5iBa5
(7.2.38)
From equations (7.2.36)and (7.2.38)we see that there is no universal extension of the geodesic interval in curved superspace. Different Green's functions will involve different super-intervals. Chirality of the final expression (7.2.34) implies that there is a natural =, e" zM chiral supermetric. Let us introduce the complex coordinates .i$ most suitable for operating with covariantly chiral scalars @(z)= e @(x,0 )= O(y M ) , where yM = (xz,, e&,). Next we re-express the supervierbein in the coordinate system introduced, E R M a M = E f I M Because of the relations (5.4.2,3),in the gauge (5.4.6) we have
ac).
(7.2.39)
A=(a,cr)
M=(m,p)
Now equation (7.2.34) can be rewritten as follows
(7.2.40) where the supermetric is given by g M G ( p T ) M A q AB1 N
-
(7.2.41)
lsT being the supertranspose of the supermatrix I (see equation (1.9.59). The sypermetric is seen to be a function of the chiral variables y M only;thoughgm is built in terms of the reduced supervierbein I", with non-chiral components. We note, however, that the Berezinian of 7," is chiral. With the aid of Copyright © 1998 IOP Publishing Ltd
608
Ideas and Methods of Supersymmetry and Supergrauity
equations (1.1 1-17), (5.4.19),(5.5.3) and (7.2.39),one can readily prove 1 = Ber(lAM) - =( P - ~ .
(7.2.42)
Because of the relations (7.2.35)and (7.2.40),the chiral d'Alembertian can be written in the form (7.2.43) completely analogous to the standard representation
v"V, = ea,(e-
' #"a,)
for the scalar d'Alembertian. 7.2.4. Covariantly chiral Schwinger's superkernel The structure of the chiral superpropagator GL") can be described in the framework of the proper-time technique. As in the case of propagators in curved space-time, we introduce the integral representation
:j
z') = i
G',")(z,
ds e- i(m2-
is)s Uc(z, zlls)
(7.2.44)
where Uc(z,z'ls) is now required to satisfy the equation
dU i"=-~U as
c
c
(7.2.45)
=-
and the initial condition Uc(z,z ' / s + + 0) = 6 + (z, z').
(7.2.46)
Similarly to Gym),the superkernel Uc(z,z'/s) must be covariantly chiral and scalar in both the superspace arguments. Due to the explicit form of U+, it is possible to search for a solution of equation (7.2.45) by applying the ansatz i ia(z, z') CT,(z, z' I s)= -exp (4nis)'
(7.2.47)
where U and a's are chiral bi-scalars. To ensure the fufilment of equation (7.2.45), U must be a solution of equation (7.2.36), whereas the remaining bi-scalars must obey the system of equations 1
U
+
(n 1)a; +
* a; + - ( 2
+U-
1
+U
Copyright © 1998 IOP Publishing Ltd
4)a; = 0
* a; + + -2 ( U+ U- 4)af,+ 1 =%ai
(7.2.48a) (7.2.486)
Efective Action in Curved Superspace
609
the *-product being defined in identity (7.2.34).The initial condition (7.2.46) proves to be fulfilled under the following boundary conditions:
a(z, z) = 9’4a(z, z’)I
= z’
= 9&p’4a(z, z’)I
= z,
=0
(7.2.49~)
9&9*bb(Z, z’)Iz=z’=929aa(z,z’)/z=z’=o a:(z, z) = 9Aa:(z, Z ’ ) I ~ = = ~9z9aa;(z, , z’)I z = z I = 0
(7.2.49b)
and
Equations (7.2.49-50) completely specify all the objects entering the right-hand side in equation (7.2.47). Rather than demanding condition (7.2.50a), it is sufficient in fact to require det( -.9a9bc(z))/z=z,#O; then is a simple consequence of the master equation (7.2.36). equation (7.2.50~) There are simple observations which lead directly to the above boundary conditions even without careful analysis concerning their compatibility with condition (7.2.46). We first note that in the flat superspace limit Uc(z,z’Is) should take the form (4.4.21), hence the flat values of c and a: read
and are seen to be consistent with the flat version of equations (7.2.49,50). Another remark is that all tensor superfields g A I ..g. A p ( z ,z ’ ) / ~ = and ~, g A.,. . gAI.ai(z,z‘)I where k, n = 0, 1, 2, . . ., should have purely geometric origin by their very construction. What is more, they should be polynomials of the supertorsion tensors R, R, G,, Wepr and W’,hb and their covariant derivatives in order to make possible the existence of a flat limit. Then, the equalities (7.2.49) follow from considerations of dimension. Dimensions of the basic geometric structures are
[RI = [Gal = 1
[W.pr] = 3/2
[g,J 2 1/2
but all objects entering equalities (7.2.49) have lesser dimensions ([a] = - 2, [a;] = - 1) or their tensor form differs from that of the invariant tensors cap, E’,D and (cJak. Finally, the relations (7.2.50)are dictated by the flat limit. Equations (7.2.48) can be rewritten in a simpler form if one redefines a; in the manner
a;(z, z’) = a2’”(z,z’)i;(z, z‘)
n = 0, 1, 2, . . .
(7.2.51)
where the chiral bi-scalar 42 is subject to equation (7.2.37) and the boundary conditions
R(z, z ) = 1 Copyright © 1998 IOP Publishing Ltd
9$2(z, z’) 1
= z, = 0
(7.2.52)
6 10
Ideas and Methods of Supersymmetry and Supergravity
Then B's satisfy the equations
a*PF,=O
(7.2.53~)
(n+ l)Pf,+ 1 +a * P;+ 1 = a -1 / 2 2 ' ( a l / ' - 'a").
(7.2.536)
By analogy with the terminology of Section 7.1, it is natural to call af, or 5; the De Witt supercoefficients. 7.2.5. aT(z, z ) and a;(z, z )
From the viewpoint of renormalization, of exceptional interest are the coefficients ay and a; at coincident points. To calculate these coefficients, one follows the same scheme described in the previous section. Namely, it is necessary to make numerous covariant differentiations of the master equations (7.2.36) and (7.2.48), with the use of the algebra of covariant derivatives (5.5.6),and take account of the boundary conditions (7.2.49-50). The immediate consequences of equations (7.2.48-50) are: aC,(z, z) = - R
(7.2.54)
and 2a;(z,z)=
n+a'l(z,~~)l,=,,+1(~~-4R)RR. 4
(7.2.55 )
To find o+a;(z,z')I,=., it is necessary to obtain a number of auxiliary relations. Straightforward but tedious calculation leads to 9a9b9cEb(z,z ' ) I z = z ' = ~ 2 ~ a ~ az b' )(l z~ ,= ~ . = o 9a9a92a(z, z')Iz=,, = - 16R
9a9a9b9ba(z, z')I ,=, = 8RR + G"G,
1 -12 aZRZ -- 9 ' R 2
9a9a9xa;(z, z')lZ=,, =9a9b9caz(z,z')l,=,,
9'9,a;(z,
= -9,9'a;(z,
=O
Z ' ) I ~ = ~ , =2iG,
(7.2.56)
53"9,,ka2a;(z, z')IZ=,,= 1 2 R R + 2 G a G a - 9 2 R 929a9aa;(z,
Z ' ) I ~ = ~ =. 4 R R + 2GaG, -a2R
1
1 1 9%,@gbaz(z, .q1 ,= = - wepy wusr+ - P R ~ ~+ R~ , G ~ P G , . 2 8 4
With the help of these identities one readily obtains the final relation 1 1 1 a%, z ) = - ~ x ~ ~ W u g y + - ( ( a Z - 4 R ) G " G , - - ( 9 2 - 4 R ~ 9 2 - 4 ~ R . 96 12 48
(7.2.57) Copyright © 1998 IOP Publishing Ltd
Effective Action in Curved Superspace
61 1
At intermediate stages of computing as(z, z) there appear non-chiral expressions, but the final result is explicitly chiral. It is worth noting that the superfunctional
is invariant under arbitrary super Weyl transformations (5.5.13), in accordance with the results of subsection 5.6.5. 7.2.6. One-loop divergences Once the proper-time representation for GY) has been established, the effective action (7.2.29) can be handled by the rules given in the previous section. We rewrite in the form
(7.2.59) and then improve the behaviour of the proper-time integral at the lower limit by applying the prescription of o-regularization. Since a;@, z) = 0, the divergent part of the effective action reads r(m),div
' J
r
c - ~
fi
d8z -{a;(z, z) - m2aE(z,z)} R 32n2w
=-
1 j d 8 zE-' 32n2w -
{A
aC,(z,z)+ m2
1
,
(7.2.60)
Because of the explicit form of aC,(z,z), the integral (7.2.60) is real modulo total derivatives. With the aid of equation (7.2.60) it is easy to obtain divergences of the and r(m) are connected as in equation effective action (7.2.10). Since r(yr) (7.2.16). we have (7.2.61) where we have accounted for the fact that the functional (7.2.58) is super Weyl invariant. Now, equation (7.2.13) tells us that the one-loop divergences in the generalized Wess-Zumino model (6.3.7) are
(7.2.62)
Copyright © 1998 IOP Publishing Ltd
612
Ideas and Methods of Supersymmetry and Supergravity
with A and ( being chosen real for simplicity. To eliminate these divergences = -r(l)[~P],~~~ into the classical action one must introduce counterterm ScOun, (6.3.7). Let us discuss the final result. As is clear, in the flat limit only the first term in equation (7.2.62)survives. In a curved superspace with R #O integrals like 1d'z E - '@ do not vanish, and there arise linear-in-0 divergences. As a result, the Wess-Zumino model turns out to be unstable in curved superspace. Both the third and fourth terms in equation (7.2.62)show vacuum divergences which appear due to non-trivial background supergeometry. Functional
- 3y 2j ~d ~8 0z is to be understood as a one-loop counterterm to the classical supergravity action in a full supergravity-matter quantum theory. The explicit form of the second term in equation (7.2.63) implies that the supergravity action should include the standard Einstein term (6.1.3)along with higher-derivative R*-terms (6.1.19). Because of the first term in (7.2.63), the supergravity action should also involve pathalogical structures (6.1.31) as long as 5#0. 7.2.7. Switching on an external Yang-Mills superfield The above results admit a natural extension to the case of a system of covariantly chiral scalars @ = (Oi(z)> coupled to an external Yang-Mills gauge superfields V = V'(z)T', with T i the generators of a real representation of the gauge group. The quantum correction in the chiral model
s
+
S ( J 0 ; V ]= d8z E - ' G e'"@
can be represented in the form 1 qm,[ V ]= - - Tr
2
+
In @)
(7.2.65)
where the matrix superpropagator @",(z, z') is covariantly chiral in both arguments and subjected to the equation (
1 a+--(BZR)+RR-m2 4
)@")(z,z')=
(7.2.66)
- O&+(Z,Z').
Here [j+ is the Yang-Mills covariantized chiral d'Alembertian, +
+-1 RDZ+iG"D, +-1 (WR)D, - WTDa--1 (DaWa)
= DUDu
4
Copyright © 1998 IOP Publishing Ltd
4
2
(7.2.67)
Eflective Action in Curved Superspace
613
where D denote the gauge-covariant derivatives (6.3.35a), and W" is the corresponding superfield strength defined in equation (6.3.33).In comparison with equation (7.2.47),the Schwinger's superkernel for Gym)involves the same chiral super-interval c, but all coefficients 6; are now matrix-valued. The coefficients iiq(z,z) and ri;(z,z) prove to be given simply by combining equations (4.8.25)and (7.2.54,57): (7.2.68~)
ii;(z, z)= a',(z, z)O
&(z, z) = a;(z, z)O
+ W"W,.
(7.2.68b)
In conclusion, it is necessary to note that the coefficients a;(z, z) and a;(z, z ) were originally computed by I.N. McArthur with the aid of a normal coordinate frame in curved superspace'.
7.3. Proper-time representation for scalar superpropagators
In the present section we give an approach to the computation of the quantum correction defined by equation (7.2.10).This approach does not require us to make any assumption concerning Y and is based on a remarkable link, between superpropagators arising in special chiral models and in the massless vector multiplet model. For the scalar superpropagators of interest we obtain the relevant proper-time decompositions.
7.3.1. Quantization of the massless vector multiplet model The massless vector multiplet in a curved superspace is described by the action
's
S [ V ] = - - dsz E - ' V . 9 a ( g 2 - 4 R ) 9 z V 16
(7.3.1)
where V(z) is a real scalar superfield. The classical properties of this model have been discussed in subsection 6.3.2 (in comparison with (6.3.21),S [ V ] contains an additional factor 3 for later convenience). In particular, it was shown that S [ remains invariant under the following gauge transformations V-+V*=V+A+A
GhA=O
(7.3.2)
A being an arbitrary covariantly chiral scalar superfield. As is obvious, the gauge generators are Abelian and linearly independent, so the theory can be quantized with the help of the Faddeev-Popov prescription (see Section 4.5). 'I.N. McArthur, Class. Quantum Grav. 1 233; 245, 1984. Copyright © 1998 IOP Publishing Ltd
6 14
Ideas und Methods of Supersymmetry and Supergravity
We fix the invariance (7.3.2) by imposing the gauge conditions K[
V ]E -
1 (a2-4R)V+ a =0 4 1
- - (9
E[ V ]
4
(7 * 3.3)
- 4R) V + d =0
where a(z) is an external covariantly chiral scalar superfield. The functions K [ V ]and E[V] have been chosen to be covariantly chiral and antichiral, respectively, since they should belong to the same functional space where the gauge parameters A and T\ live. Because of the chiral variational rule (5.5.28), the Faddeev-Popov matrix reads
(7.3.4)
Here H(R)is defined by equation (7.2.7) for Y = R and is the Hessian of the chiral model (7.2.5). As a result, the in-out vacuum amplitude is given by eirv = j B V 6
+[-A 4 (aZ-4R)V+ a
1 [: 6-
- - (9' -4R)V
+d
1
Det(H@))cis["]. (7.3.5)
In accordance with the results of subsection 4.5.2, r, does not depend on the external superfields a and d. Therefore, one can freely integrate the right-hand side of equation (7.3.5) over a and ii with some weight p(a, a) such that
s
9&ii p(a, a) = 1.
It is useful to choose p(a, a) in the form Det1/2(H(v))e- iSlW[~l being defined by equation (7.2.2).This leads to the following the action representation e i r v = Det(H(R)) Detl/Z((H(Y))e-(i/2)5d8zE - 'vab'v (7.3.6) where 1
1
{(L32--4R),(@-4R)} U"( V ) = --gP(G2-4R)Bm+8 16 1-
--Y(@--4R). 4 Copyright © 1998 IOP Publishing Ltd
1
--Y(P-4R) 4 (7.3.7)
Effective Action in Curved Superspace
615
When acting on scalar superfields, nly)reads 1 4
1 4
1
n$y)=909a-Cz‘[9,, G&]--Y(a2-4R)--\Y(9’-4R) 1 - - - . 1 --(gi R)$Y --( 9 ‘ R ) 9 a - A 4 4 4
4
1 (a2R) --(9’R)+ 2RR 4
(7.3.8)
as a consequence of the covariant derivatives algebra (5.5.Q and is a curved-superspace extension of the operator A (4.8.30). Let GIy) be the Feynman superpropagator associated with and defined by
Ly)
by)Gly)(~,2’) = - 6’(2,
2’).
(7.3-9)
Then, equation (7.3.6) can be rewritten in the form
Remark. Given an operator A acting on some space of unconstrained scalar superfields, it is useful to define its superkernel to be bi-scalar:
A(z, z’) = A ~ * ( z2’) , = AEd’(z - z’). Then Tr A is given as
s
Tr A = d’z E - ‘A@, 2).
(7.3.1 1)
(7.3.12)
Equation (7.3.10) contains more interesting information than may appear at first sight. The point is that the right-hand side in equation (7.3.10) does not depend on ‘4, which leads to important consequences. Let us first choose Y =O. Then equation (7.3.10) takes the form i 1 l-, = --Tr In G, - - Tr In H - iTr In H(R).
2
2
(7.3.13)
where G , GIo)and H H(O).Choosing Y = R, we obtain 3i r, = --1 Tr In GLR)-Tr In H(R). 2
2
Equations (7.3.10, 13, 14) allow us to express
(7.3.14)
r, and Tr In H(”’)as follows:
3i 3i l-, = - - Tr In G, + iTr In GkR)-- Tr In H 2 2
(7.3.15)
Tr In H(y)= Tr In G , - Tr In GIy) + Tr In H.
(7.3.16)
In the previous section we saw that Tr In H can be represented according to Copyright © 1998 IOP Publishing Ltd
616
Ideas and Methods of Supersymmetry and Supergravity
the rule TrInH=-Tr+lnG,
G , being the massless chiral superpropagator defined by equation (7.2.32a).
As a result, we arrive at the final relations:
3i G, + iTr In G$) + - Tr + In G, 2
(7.3.17)
Tr In H(vy’= Tr In G, - Tr In GLv)- Tr + In G,.
(7.3.18)
l-,
3i 2
= - - Tr In
The latter is a by-product of our consideration, and it supplies us with a simple expression for the quantum correction (7.2.10): i
Tcv,= - - (Tr In GtY)- Tr In G,) 2
+
(7.3.19)
where r = r(o) is the quantum correction in the massless chiral model (7.2.4). The above relations make it possible to compute r, and Tcy,in the framework of the proper-time technique. For any the operator miy) has exactly the form (7.2.l), hence the corresponding Green’s functions GLy) possess a well-defined proper time decomposition. The proper-time representation for G, follows from equations (7.2.44,47) by setting m =O. 7.3.2. Connection between GLy) and GcY) Let us consider the Feynman superpropagator (7.3.20)
associated with H(v), (7.3.21)
It turns out that G(y)is expressed via the scalar superpropagator GLy), defined by equation (7.3.9),in the manner z‘) hh’Gty)(z,z’) hr;’G$y’Y’(z, hhGty)(z, z’) hhfGty)(z,z’)
(7.3.22)
where 1 -
h= - - ( g 2 - 4 R ) 4
L=
1 --(g2-4R). 4
This relation generalizes equation (4.8.31) to the case of curved superspace. It can be proved as follows. First, we act with H(R)from the left on both Copyright © 1998 IOP Publishing Ltd
617
Effective Action in Curved Superspace sides in equation (7.3.21) resulting in H(R)H('~"G"W(~,= ~ ~) HW)
Further, we apply H=H(O) from the left to both sides of the relation
which is a consequence of equation (7.3.9). The result can be presented in the form
where is the operator defined by equation (7.3.22). Here we have used the identities
hB'(B2- 4R)9&= L9'(G2 - 4 R ) 9 %= 0
L2=&
h2==Rh
which are correct when acting on scalars. Since h O) = H ( h i ; oii h i
)
both G'v) and @lu) satisfy the same equation. Using equation (7.3.22), it is not difficult to obtain an independent proof of the relation (7.3.19). 7.3.3. Scalar SchwingerS superkernel The proper-time representation for G$?(z, z') reads
GIy)(z,z') = i
1
30
ds Ucy)(z,z' Is)
(7.3.23)
Jo
where the bi-scalar superkernel U$y)(z, z') satisfies the evolution equation
+
(i
)ulq)(z,z')
:'~)(z, z')
and the initial condition
=0
u:yz,z' 1s -+ +0)= P ( Z , z').
(7.3.24)
(7.3.25)
We search for a solution of equation (7.3.24) in the form
(
U:?')(Z,z' 1 s) = 1 exp icy ':' (4x1s) Copyright © 1998 IOP Publishing Ltd
'"> f aLv)(z, z')(is)" n=O
(7.3.26)
618
Ideas and Methods of Supersymmetry and Supergravity
ulv)and a(y) being real bi-scalar superfields. It is easy to see that utv) should
obey the equation
whereas the recurrence relations for a(vy)are ( a ~ )abv) , ) + (A(v)u~v) - 4)abv)= 0
(7.3-28)
1 ( n + l)aLT),’Y?, +(abv), aiy)l)+-(A(v)0,(v)-4)aiT)1= nbv)aiv) 2
where we have introduced a bracket ( , ) for real scalar superfields,defined by
The operator ACT)reads 1 A(v) = $Bag -a G‘’[$i3a, 4
Gk]
1 - - - . 1 1 - 1 --“9’--Y.9’ -- ($B,R)9*-- ( W R ) g s .
4
4
4
4
(7.3.30)
Equation (7.3.27) shows once again that in superspace there is no universal generalization of the geodesic interval. The initial condition for Uty) implies that the system of equations (7.3.27,28) must be solved under the following boundary conditions: Up(z, z’)= 9 A u p ( 2 2’) , I = z* = 9u9Auv (W(z, z’)I = = 0 zl
gagb@?z, z’)1 z = 2’ = ‘lab
a, 9’CT:yz, z’)I
=
= ga92 U (W , (z, z’)I = = z1
[ga,a,]u:yz, zl)I
(7.3.31) = zl
=0
9’gZ~iv)(~, z ’ ) I ~ = ~ ~ = 16 and ab?’)(z, z)=g,,,abv)(z, z ’ ) ~ ~ = ~ , = ~ ~Z~’ )~~ ~ =a~ b. =~O ) ( z , gag,,gBabv)(z,z’)I
zI=
o
9a9,&ggaby)(z, zr)lz = z , = - 4&,p,g. Copyright © 1998 IOP Publishing Ltd
(7.3.32)
Effective Action in Curved Superspace
619
These requirements are grounded in the same considerations as were made in subsection 7.2.4. In the flat-superspace limit U, z Uto) takes the form (4.8.43). Using equations (7.3.27, 28, 31, 32), one readily obtains
aiY)(z,z ) = o
(7.3.33)
aLy)(z, z) = - G“G, + TY.
7.3.4. Divergences of effective action Now, it is a trivial task to obtain divergent parts of the superfunctionals (7.3.17)and (7.3.19) with the help of the general algorithm given in subsection 7.1.3. We rewrite rvin the form Tv=
-i2
$ (Tr(3Uv(s)-2U~R)(s))-3Tr+U,(s)}
(7.3.34)
and modify the integral over proper time by applying, for instance, the prescription for w-regularization. This leads to the following divergent part of r,:
3a&, z ) - 2aLR)(z,z ) - 3 2)) R Similarly, the divergent part of
(7.3.35)
rn) reads
+
aT)(z,z ) - a2(z, z ) ”> (7.3.36) R which coincides, as a result of equation (7.3.33), with our previous result (7.2.61). In conclusion let us point out that the superfunctional (7.3.35) is invariant under arbitrary super Weyl transformations (5.5.13). This is a consequence of super Weyl invariance of the corresponding classical theory (7.3.1).
7.4. Super Weyl anomaly In Chapter 6 we met a number of super Weyl invariant models in curved superspace. The classical action S[x; Kcp] in such a theory remains unchanged under transformations of the form (5.7.33), the parameter o being an arbitrary covariantly chiral scalar. This implies vanishing of the classical supertrace T, defined by equation (5.7.15), on the mass shell S S / S x = O . Since equation (5.7.33) presents a supersymmetric extension of ordinary Weyl transformations, the super Weyl invariance is potentially anomalous at the quantum level (see also subsection 7.1.4). Renormalized effective action r[i;Kcp] can be chosen to be invariant only under the supergravity gauge Copyright © 1998 IOP Publishing Ltd
620
Ideas and Methods of Supersymmetry and Supergravity
transformations, but not with respect to the super Weyl ones. Both symmetries cannot be preserved in general, which means that the effective supertrace
remains non-zero when imposing the effective equation of motion d r / d i =O. As a result, the condition of general covariance in quantum theory is given by the following equation for the effective supercurrent (Tah)t
which replaces the classical equation (5.7.35). In the case when i = O is a solution to the effective equations of motion, the occurrence of the super Weyl anomaly can be studied by analysing the vacuum part r of the effective action, which is related to the in-out amplitude (out I in) = exp(ir). The breakdown of super Weyl invariance is expressed by the fact that r possesses some dependence on the chiral compensator,
7.4.1. Super Weyl anomaly in a massless chiral scalar model We are going to calculate anomalous supertrace ( T ) in the simplest super Weyl invariant model with classical action
(7.4.1) 0 being a covariantly chiral scalar superfield. The operators H +- and H- + are given by equation (7.2.8). Since (D changes as cp' = e - W with respect to local rescaling of the chiral compensator, the super Weyl transformation laws for H + - and H - + read H'+- = e - Z u H +-e'
H'- + --e-"H-
Therefore, the transformation law for Xc= H + - H -
x:=e-zuxce"-*.
+eu. +
(7.4.2)
is given by (7.4.3)
In the framework of the L-regularization scheme, the regularized effective action is given by (7.4.4)
Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Superspace
621
where
Uc(z,z’Is)=exp(is~,)b+(z,z’) is the chiral Schwinger’s superkernel described in Section 7.2. Then, the renormalized effective action reads
(7.4.5) the counterterm being chosen as E-’
(7.4.6) Our goal is to find the variation of rren under an infinitesimal super Weyi transformation. Let us vary rL.In accordance with equation (7.4.3),we have
STr+U,(s)=-is Tr+(g%,U,(s))-isTr+(2,i7UC(s)).
(7.4.7)
The first term in this variation can be rewritten with the aid of equation (7.2.45),in the form -is Tr+(gZcUc(s))= -s-
d
as
Tr+(aU,(s)).
(7.4.8)
To transform the second term in equation (7.4.7),we make use of the operator identities
%,O
H + -H - +if = H + -5H - +
and H - +U,(s)=H-+exp(isH+-H-+)=exp(isH-+H+-)H-+=U,(s)H-+.
Taking into account equation (7.2.26)also, we can write
Tr + (2cifU,(s))= Tr + (H + - iiU,(s)H- + ) = Tr - (iiUa(s)Sa) which leads to the relation
a
isTr + (%,ifU,(s))= - s -Tr - (XJ,(s)). as
Equations (7.4.47-9) show that i
(7.4.9)
rL varies as follows: 1
6rL= --Tr + (aU,(- iL2))- - Tr +@U,(- iL2)). 2 2
(7.4.10).
The proper-time decomposition for U, was described in Section 7.2. As is easily seen, an analogous representation for the antichiral Schwinger’s Copyright © 1998 IOP Publishing Ltd
622
Ideas and Methods of Supersymmetry and Supergravity
superkernel Ua(z, z' I s)= exp(is%'Jb
-(z, z')
(7.4.11)
is of the form
i Ua(z,z' 1 s)= -exp (4nis)'
(7.4.12)
with the antichiral bi-scalars ii and I' being conjugates of those presented in equation (7.2.47).As a result, equation (7.4.10)can be rewritten in the form
K,=L {3271
L2 [d'z E - ' o
+ {d'z (7.4.13)
where we have used the fact that a;(z, z ) = - R. Now, let us turn to the counterterm S,,,,,. As was noted above, the second term in the right-hand side of equation (7.4.6) is invariant with respect to arbitrary super Weyl transformations. Hence only the first term contributes to the super Weyl variation of Scount: 6Sc,,,,,=-[d8zE-10+ L32n2
C.C.
(7.4.14)
From equations (7.4.13) and (7.4.14)we obtain
q,, = -{d'z 32n2
7
+
oa;(z, z) c.c.
(7.4.15)
Therefore, the anomalous supertrace in the theory under consideration reads 1 < T ) =a;@, z). 32n2
(7.4.16)
Remark. Another example of classically super Weyl invariant theories gives the vector multiplet model (7.3.1).Calculation of the anomalous supertrace in this theory requires more subtle consideration'. We give only the final result:
I
( T ) = - - 32n2 { 3 a ; ( ~ , z )4+ ~ ( ~ ~ - 4 R ) ( 3 az)-2alR)(z,z)) ~(z, . (7.4.17)
Here the coefficients a2(z,z) and a\")(z, z) are obtained from aiv)(z, z) (7.3.33) by setting P ' = 0 and Y = R, respectively.
iI.L. Buchbinder and S.M. Kuzenko, Nucl. Phys. B 214 653, 1986. Copyright © 1998 IOP Publishing Ltd
Effective Action in Curved Superspace
623
7.4.2. Anomalous efective action The relation defining anomalous supertrace
(7.4.18) can be understood as an equation for the renormalized effective action provided we have at our disposal an exact expression for ( T ) obtained by some indirect method. This is just the situation under consideration. Substituting the explicit values of a;(z, z),a2(z, z ) and a(zR)(z,z ) into equations (7.4.16)and (7.4.I7), the anomalous supertrace can be written in the following general form: c
-
( T ) = aW2 + bP +- (g2 -4R)g2R 16
(7.4.19)
where w2 = W&W
=BY
P = W2-A(g2-4R)(GaGa+2RR). 4
These are only the numerical coefficients a, b, c which reflect the specific features of the initial theory. The chiral superfield P is originated as a density for the topological invariant (5.6.61),
s
-1 s
d'2E-l { i W 2 + G a G a + 2 R R = d'z-P. ER1
(7.4.20)
Now, our goal is to obtain a solution of equation (7.4.18). As is clear, the renormalized effective action consists, in general, of two parts:
r = rA+ r, where
r A
generates the anomaly (7.4.19):
-Arc (T) Acp whereas T, is a super Weyl invariant functional. Only the anomalous part of r can be restored by solving the above equation. Let us proceed by solving the equation
-ArA-aW2+bP+C(a2-4R)92R. A'
16
(7.4.21)
An important role in our considerations will be played by the covariantly Copyright © 1998 IOP Publishing Ltd
624
Ideas and Methods of Supersymmetry and Supergravity
chiral scalar superfield R and its conguate 3 ! defined by 1 +jd8z' E-'(z')G+ -(z,z')
).
l+~dsz'E-'(z')G-+(z,z')
(7.4.22)
Here G is the Feynman superpropagator for the massless theory (7.4.1). The superfield R satisfies the massless equation of motion (GZ-4R)R=0
(7.4.23)
and proves to obey the transformation law Q = e-'R
(7.4.24)
with respect to the super Weyl transformations (5.5.13). As an instructive exercise, the reader can explicity verify equation (7.4.24). It is worth pointing out that the flat-superspace limit for R is unity, and its dependence on the supergravity prepotentials takes the form R = q-1 + O(W).
(7.4.25)
In accordance with definition (7.4.22), R is a non-local functional of W and q. However, in the case of a conformally flat superspace, R turns into the inverse of q in the gauge W=O. It is necessary to note that the chiral scalar R was introduced by Grisaru, Nielsen, Siegel and Zanon' and is a supersymmetric generalization of the conformal scalar field 1 -~- +W/6)-'W of Fradkin and Vilkovisky3. Using the definition of the variational derivative A/Aq (7.4.18) and the super Weyl transformation laws (5.5.14) and (7.4.24), it is extremely simple to obtain the following identities:
(7.4.26~) AV
I d 8 . E-'(GaGa+2RR) ln(nR)=1(g2-4R)(G"G,+2RB) 4 1 -
-- (9' -4R).9'(G,h~h
8
In
n- R g e In R) (7.4.26b)
*
M. T.Grisaru, N. K.Nielsen, W. Siegel and D.Zanon Nucl. Phys. B 247 157, 1984. E. S.Fradkin and G. A. Vilkovisky Phys. Lett. 73B 209, 1978.
Copyright © 1998 IOP Publishing Ltd
Eflective Action in Curved Superspace
625
1 4
= --(~z-4R)(-9z(G,h~hln~)+i9,k(~'(ln~).9aIn0)} (7.4.264
A s d s z E-'g"(ln Acp
0)9,(ln O)Gk(lnn)@(ln
= 2 ( a 2- 4R){ig,@'(ln
A40
n)
n>B'(ln 0)) - B'(R9, In Cl)} (7.4.26d)
s d 8 z E-'RR=-
1 (a2-4R)g2R. 16
(7.4.26e)
In deriving these relations we have also made use of the equation of motion (7.4.23). Let us consider the functional (7.4.27) From equation (7.4.26~)we obtain (7.4.28) Introduce one more functional
s
(7.4.29)
ATY)--(gZ-4R).9'R. c -Aq 16
(7.4.30)
rf'=c Equation (7.4.26e) leads to
d8z E-'RR.
It remains to look for a superfunctional generating the term bP in equation (7.4.19). With the aid of equations (7.4.2664) one finds that the functional
I
x a,(ln a)gE(ln0)
solves the problem,
(7.4.31)
(7.4.32)
Copyright © 1998 IOP Publishing Ltd
626
Ideas and Methods of Supersymmetry and Supergravity
Equations (7.4.27-32) lead to the anomalous effective action
(
- b Gak+ 1g'(1n 0)9'(ln
I
n) gk(lnO)ga(lna)+ c R R . (7.4.33) 2 8 One more interesting representation for the anomalous effective action, which is based on the use of a natural generalization for R, can be found in the original publication4. The non-local functional r A turns into a local one in the case of a conformally flat superspace, i.e. when WE?,= 0. Then, in the gauge W = 0 the covariant derivatives and the supertorsions R, G, are given by equations (6.5.12,13), whereas R coincides with c p - l . As a result, r A takes the form r A=
- Sd'z
{
)
1 cda(ln @)d,(lncp) + - (c- b)D'(ln @)D"(lncp) 4
[D&, D,](ln
@cp)--
1 D&(ln@)D,(ln cp) 2
where DA= (d,, D,, D ') are the flat global covariant derivatives. This functional can be considered as the classical action of some supersymmetric theory in flat superspace. 7.4.3. Solution of eflective equations of motion in conformallyflat superspace It is natural to understand the renormalized effective action T[xcp] as a quantum correction, due to vacuum effects, to the classical supergravity action SsGwhich can be taken in one of two possible forms (6.1.3) or (6.1.17). In general, the effective gravity equation of motion
(7.4.35) cannot be written explicitly since the calculation of is an extraordinary technical problem. However, the above results supply us with a remarkable possibility to obtain exact effective equations in the case of classically super Weyl invariant theories. In such a theory the effective action is = + rI where rA is given by equation (7.4.33) and rl is some unknown superfunctional. Because of the super Weyl invariance of rl,the effective equations read 6
-(sSG rA + r,)= o
SH
+
41.L.Buchbinder and S.M. Kuzenko, PAYS. Lett. 202B 233, 1988. Copyright © 1998 IOP Publishing Ltd
(7.4.36a)
Efective Action in Curved Superspace
621
(7.4.36b) Here the first equation is still unknown. We may suppose, nevertheless, that there are conformally flat solutions ( Wppv=0) for the full system of equations. Then, only the second equation remains essential. So, let us consider the equation for the chiral compensator choosing the supergravity action with cosmological term (6.1.17) in the role of SSG.With W.pt = 0, equation (7.4.36b)reads 3 1 -( R-,U)+- (G2-4R) K2 4
C
b(G"G,+ 2RE)-- g 2 R 4
A particular solution is obtained by setting G,=O and R=const, R being subject to the constraint
(7.4.38) and corresponds to the anti-de Sitter supergeometry. 7.5. Quantum equivalence in superspace
7.5.1. Problem of quantum equivalence One of the specific features of superspace, as compared to ordinary space-time, is that any on-shell supersymmetry representation admits, in general, several off-shell superfield realizations. Of course, there exist examples of different field theories possessing equivalent dynamics, but superspace provides far more possibilities. The most impressive example is the existence of infinitely many superfield formulations for Einstein supergravity (minimal (n= -+), new minimal (n = O), non-mininal (n # -+,0)) instead of the unique generally accepted formulation for Einstein gravity. All these formulations were shown in Chapter 6 to be equivalent at the classical level. So, in supersymmetric field theory we inevitably run into a problem of quantum equivalence for classically equivalent realizations. By quantum equivalence is meant equivalence of S-matrices. For theories in external (super)fields, quantum equivalence can be understood as the coincidence of in-out vacuum amplitudes modulo an irrelevant factor expressed via boundary values of background (super)fields at infinity. It is clear that the problem of quantum equivalence is non-trivial in general and deserves special consideration in any concrete case. Of principal importance is the study of quantum equivalence between different supergravity formulations. Unfortunately this is still an open problem because of the non-renormalizability of (super)gravity.In the simpler case of renormalizable classically equivalent theories the study of their Copyright © 1998 IOP Publishing Ltd
628
Ideas and Methods of Supersymmetry and Supergravity
quantum equivalence can, in principle, be carried out in the framework of perturbation theory. In the present section we investigate three superfield realizations for the so-called non-conformal scalar multiplet which is normally described by a covariantly chiral scalar superfield 0 on the basis of the action superfunctional (’7.5.1)
This model is said to be non-conformal because the action involves the term
! S d s i E - 1(0’ +a’) 2 which breaks the super Weyl invariance present in the action (’7.4.1). Among other chiral models based on classical actions of the form
with < # f 1, the above model is selected because of its equivalence to the gauge theory of a covariantly chiral spinor superfield za, Bhx,= 0, which is described by the action (7.5.2) The classical equivalence of these theories has been established in subsection 6.3.4. One more realization for the non-conformal scalar multiplet follows from equation (’7.5.1) by expressing 0 via a real scalar superfield U in accordance with the rule 0 =-‘(G’-4R)U 4 Then we obtain the action l
U= 0
(7.5.3)
r
da~E-’U((~’-4R)+(~’-4~)’U.
(7.5.4)
Thus we have three classically equivalent models realized in terms of (i) chiral scalar (CSC), (ii) chiral spinor (CSP), and (iii) real scalar (RSC) superfields. Let us introduce into considerations the in-out vacuum amplitudes and effective actions for the above classically equivalent theories: (out/in)c,c = exP(ir,cCK PI) (outlin),,, = exp(il-,,[K
Copyright © 1998 IOP Publishing Ltd
491)
=exp(irRSC[% PI).
Effective Action in Curved Superspace
629
To have equivalence at the quantum level, the renormalized effective actions must possess the same local dependence on the supergravity prepotentials. In other words, the average supercurrents as well as the average supertraces arising in the theories under examination must coincide. The CSC model is non-gauge, and the corresponding effective action rcsc = rtR) is formally given by
rcsc=-i Tr In H(R) 2
(7.5.5)
(see also Sections 7.2 and 7.3). As for the models (7.5.2) and (7.5.4), they are gauge theories with linearly dependent generators (see subsection 4.5.1). The action (7.5.2) is invariant under the gauge transformations (7.5.6)
the gauge parameter being an unconstrained real scalar superfield defined modulo arbitrary shifts of the form I/+ I/*+
V+A+A
G&A=O.
(7.5.7)
Next, the action (7.5.4) remains unchanged under gauge transformations (7.5.8)
as a consequence of equation (5.5.11). Here the gauge parameters qa is a covariantly chiral spinor superfield defined modulo arbitrary shifts of the form (7.5.6), and in its turn V is defined modulo arbitrary shifts (7.4.7). So, the CSP and RSC models are reducible gauge theories with the first and second, respectively, stages of reducibility. Hence, it is required to give proper before proceeding further. definitions for rapand rRsC As is known, the Faddeev-Popov approach is inapplicable to quantize reducible gauge theories. It is the Batalin-Vilkovisky quantization method' which leads to correct Feynman rules in the Lagrangian approach for arbitrary finitely reducible gauge theories. In the case of Abelian finitely reducible gauge theories, the Batalin-Vilkovisky method turns out to be equivalent to the quantization procedure of A. Schwarz2, the latter being the first known technique developed to quantize reducible theories. Since the CSP and RSP models are Abelian gauge theories, their quantization can be fulfilled in the framework of the Schwarz procedure. In order to illustrate this technique, we first discuss the quantization of the more familiar gauge theory of a second-rank antisymmetric tensor field in a curved space-time. 'LA. Batalin and G.A. Vilkovisky, Phys. Lett. 120B 166, 1983; Phys. Rev. D 28 2567, 1983. 'AS. Schwarz, Lett. Math. Phys. 2 247, 1976; Commun. Math. Phys. 67 1, 1979. Copyright © 1998 IOP Publishing Ltd
630
Ideas and Methods of Supersymmetry and Supergravity
7.5.2. Gauge antisymmetric tensor field The gauge theory of antisymmetric tensor field Aab(x)coupled to an external gravitational field is determined by the action
A[S]
=A2 [d'x
e-'L"(A)L,(A)
(7.5.9)
1 La(A)=-&abcdvbAcd. 2
This theory is known to be classically equivalent to the theory of a scalar field q(x) minimally coupled to gravity
'S
S [ q ] = -- d4x e-'VaqVaq. 2
(7.5.10)
Both models are of interest to us since the former arises at the component level in the model (7.5.2) and the latter in the model (7.5.1). The action S [ A ] is invariant under the following gauge transformations Af;b =
-k v[aAb]
(7.5.11)
the gauge parameter being defined modulo arbitrary shifts of the form 61,=V,Y. So, the model under consideration is a reducible gauge theory with the first stage of reducibility. In the role of gauge-fixing functions we choose K,(A)= VbAab
(7.5.12)
which is a transversal vector field, VaKa(A)= 0
(7.5. 3)
and transforms under (7.5.1 1) according to the law 6Ka(A)=VbV[b1(,]~~~3.b.
(7.5. 4)
Adding to S [ A ] the gauge-breaking term -,-
'J
sG,[A] = -- d ' ~e-'Ka(A)Ka(A) 2
(7.5.15)
we arrive at the non-gauge action s [ A ] + s a , [ A ] =tSd'x f ? - ' A a b ~ 2 A a b
(7.5.16)
+ 2Wc[aAb]c
(7.5.17)
where
- 2*[&,d being an antisymmetric tensor field d'Alembertian. 0 2 Aab
0
= VcVcAab
Copyright © 1998 IOP Publishing Ltd
Efective Action in Curved Superspace
631
Now, it is worth making some general remarks concerning the Schwarz quantization prescription. Let us recall that the central idea in the Faddeev-Popov construction was separation of the gauge group volume from the naive path integral. From the technical point view, this was done by attracting the delta-function of a gauge-fixing function. The chief observation made by Schwarz was that basic elements of the Faddeev-Popov construction can be preserved, but must be properly modified, in the case of Abelian reducible gauge theories. First, it is necessary to modify the notion of gauge group volume, since some of the gauge parameters do not appear in the transformation law of the gauge fields. Second, one must extend the notion of functional delta-function to the case when its argument is a constrained field. The point is that admissible gauge-fixing functions in reducible gauge theories should obey some constraints. We proceed to the quantization of the theory (7.5.9). Because of the constraint (7.5.13), the delta-function S4[K,(A)] is ill-defined (S4[K,(A)] S[O], where S4[ ...I and S[ ...I denote the vector and scalar functional deltafunctions). As a result, the standard Faddeev-Popov construction does not work. Let us try to define a generalized delta-function J4[B,], where B,(x) is an arbitrary transverse vector field, V"B, = 0. We start with the formal Fourier-representation
-
r
S4[B,]=
J
/
(J
r
9 V e x p i d4xe-lV"B,
\
(7.5.18)
Owing to the transversality of B,, the exponential in equation (7.5.18) is invariant under the transformations
v,-i v:=
V,+V,Y
(7.5.19)
with arbitrary Y, hence the path integral is ill-defined. However, making use of the Faddeev-Popov prescription allows us to extract from (7.5.18) the integral over the gauge group (7.5.19) in the same fashion as in electrodynamics. Then one arrives at the following well-defined object
4B4 (s
8 [ ]= 9 V e x p
i d4xe-'V"B,
)SIVaV,]Det(no)
(7.5.20)
where no= V"V, is a scalar field d'Alembertian. With the help of the identity
s s
'
S[Va V,] = 9cp exp i d4x e- V"V,cp J4[B,] can be rewritten in the form
(s
J4[B,] = 9cp S4[B, Copyright © 1998 IOP Publishing Ltd
+ V,cp]Det( no).
(7.5.21)
632
Ideas and Methods of Supersymmetry and Supergravity
Next, we turn to finding a proper integration measure 9 p j . for the group of gauge transformations (7.5.11). Let r[A] be a functional of the gauge field. Naive path integral Z[A] =J 9 2 F[A'.] over the gauge group is seen to be degenerate, since AA=A" for 1L=i,+V,Y, Y being arbitrary. In order to extract the Y-group volume from Z[A], we can again make use of the Faddeev-Popov prcscription. This leads to the following definition of integrals over the A-group: Z[A] = j9pAF[AA]
9 p A = S[V"A,]Det( o0)91.
(7.5.22)
Once S4[K,(A)] and 9 p Ahave been defined, we can follow the standard Faddeev-Popov procedure to quantize the theory (7.5.9). First, we introduce the Faddeev-Popov determinant A:
(7.5.23) This can be expressed in the form
A = Det( 1)Det-2(oo)
(7.5.24)
where o1is the vector field d'Alembertian defined by 01
v,= VbV, vu- B,bv,.
(7.5.25)
To prove equation (7.5.24), we rewrite the right-hand side of equation (7.5.23) as follows: f-
I-
for %,b the operator given in (7.5.14). By construction, this expression does not depend on external scalar fields p and y, so we can integrate the right-hand side of equation (7.5.26) over p and y with the weight exp( -i d4x e - 'py). Making also the shift 1, -+ 1,- (1/ &,(A) in the path integral, one obtains (7.5.24). With the aid of the representation (7.5.21), it is possible to rewrite equation (7.5.23) in one more useful form: n
j
A - = 9 p A9 q S4[K,(A')
+ V, cp -i,]Det( U,)
(7.5.27)
where 5, is an arbitrary external vector field. This expression does not depend on the transversal part of by its very construction. The longitudinal part of arises in expression (7.5.27) as a result of the shift cp q - V"(l/ nl)c,.
e,
e,
Copyright © 1998 IOP Publishing Ltd
-+
Efective Action in Curved Superspace
633
The next step in the Faddeev-Popov construction is to extract the gauge group volume. In our case this consists of inserting into the naive path integral
the following unit 1= A
s
s
9AeiSMI
9 p A 9 q64[K,(AA)+V,cP-S,]det(00)
along with the replacement of the integration variable A --+A-*. This leads to the effective action eirA
+ V , cP - S.]ADet( no)exp(iS[A]).
= s 9 A 9cp S4[K,(A)
(7.5.28)
-*
Since rAdoes not depend on la,it is in our power to integrate over (, with the weight exp( d4x e - 'S'). Then we obtain e i r A = s 9 A 9 qA Det(oo)exp i(S[A]
d'xe-'V'cpV,q}. (7.5.29)
Now, equations (7.5.15) and (7.5.23) allow us to obtain the final expression for the effective action
T A= l-,+-
1
2
{Tr In o2-2 Tr In
o1+ 2 Tr In 0,)
(7.5.30)
where i r,=-Trln 2
no
(7.5.31)
is the effective action of the scalar field model (7.5.10).
7.5.3. Quantization of the chiral spinor model We proceed by constructing the effective action for the gauge superfield theory with classical action (7.5.2)on the basis of the quantization technique described above. An admissible gauge-fixing function in the theory under consideration can be chosen in the form 1
W(X) = - ( W X @- B&p) 2 Copyright © 1998 IOP Publishing Ltd
(7.5.32)
634
Ideas and Methods of Supersymmetry and Supergravity
W(x) being a covariantly linear real scalar superfield, (G2-4R)H(x)=O W(x)= W(x).
(7.5.33)
The transformation law W(xY)=
(7.5.34)
W(1[)-b(P-4Rpav 8
tells us that the requirement W(x")=O fixes V modulo aribitrary shifts (7.5.7). Adding the gauge-breaking term
sSB[xl
s
= d8X
E-
'(w(x))2
(7.5.35)
to the classical action S [ x ] , we obtain the followingnon-gauge superfunctional
where (i&)
1 =-(92-4R)(92-6R)~, a - 16
(7.5.37) The operator %c is a d'Alembertian on the space of covariantly chiral spinor superfields. Let 6[. . .], 6, [. . .] and 6- [. . .] denote the functional delta-functions on spaces of unconstrained real scalar, covariantly chiral and antichiral scalar superfields, respectively. Because of constraint (7.5.33),the expression S[W(x)] turns out to be ill-defined, S[W[x]] 6 + [Old - [O]. Following the Schwarz .] for covariantly approach, we must look for generalized delta-functions linear real scalars. Let L be such a superfield, (B2 - 4R)L = 0, [I= C. Then the path integral
-
a,.
(7.5.38) is seen to be degenerate, since the exponential remains unchanged under the Copyright © 1998 IOP Publishing Ltd
635
Efective Action in Curved Superspace gauge transformations
II-+II"=II+A+A
gkA=O.
(7.5.39)
This is exactly the gauge invariance of the massless vector multiplet model. Therefore, the gauge group volume can be extracted from equation (7.5.38) in the same fashion as was done in subsection 7.3.1. As a result, we obtain the following well-defined superfunctional
1
1
-4 R ) n 6 - - (g2- 4R)II Det(H(R))
[:
x exp( i I d % E - lIIlL)
(7 5.40) a
which can be called a functional delta-function of a covariantly linear scalar superfield. Using the identity
where the integration is performed over chiral(0) and antichiral (a)variables, equation (7.5.40) can be rewritten in the form &L] = I9590 S[L
+ @ + a] Det(H(R)).
(7.5.41)
The next major step is to define an integration measure 9 p v over a group of gauge transformations (7.5.6). Let F[x] be a functional of the gauge The naive path integral f 9 V F [ x ' ] is degenerate, since superfields xa, 9 [ f ]does not change under arbitrary A-transformations (7.5.7). Extracting the A-group volume from the above path integral, with the aid of the results of subsection 7.3.1, we arrive at
zk.
(7.5.42)
We set up equation (7.5.42) as the definition of integration over the gauge group (7.5.6). Further consideration is a simple repetition of the Faddeev-Popov prescription. One introduces the Faddeev-Popov determinant A defined by A-'
s
s
9 p v RK(x")] = '9/.hv9@9@ 6 [ K ( x v ) + @ + 5 -
Copyright © 1998 IOP Publishing Ltd
U]Det(H(R)) (7.5.43)
636
Ideas and Methods of Supersymmetry and Supergravity
U being an external real scalar superfield (as is readily seen, A does not depend on U). Then one inserts the unit
s
1= A -
into the naive path integral
s
9 p v RK(x")]
eiWI
G J I ~ ~
s
and separates the gauge group volume 9 p v .This leads to the effective action
s
eircsP= B i 9 ~ 9 G 9 O A Det(H(R))d[K(x)+O+G- U]eis[X1. (7.5-44)
It remains to integrate the right-hand side over U with the weight exp(i d8z E - 'U'). As a result, one obtains
s
eirCSP = A Det(H(R)) g j g x @ a @ ei(s[Xl+SGBCX] +2s(R)[o]), (7.5.45) Direct calculation of the Faddeev-Popov determinant gives
A = Det(
e))Det
- 3(H(R))
(7.5.46)
utR)being the scalar superfield d'Alembertian (7.3.8)with Y = R . Now, using equations (7.5.1) and (7.5.36), the effective action can be expressed in the form r a p = rcsc
+x
(7.5.47)
where 1
X = -- (Tr, In 2c +Tr - In %,+ 2 Tr In otR)-4Tr In H(R))(7.5.48) 2
rcscbeing the effective action (7.5.5) of the chiral scalar model (7.51).
As for the real scalar model (7.5.4), its quantization requires more tedious consideration. Omitting details, which can be found in our paper3, we present the final expression for the effective action r R s P = rcsc
-2 x
where X is exactly the superfunctional given by equation (7.5.48).
31.L. Buchbinder and S.M. Kuzenko, Nucl. Phys. B 308 162, 1988. Copyright © 1998 IOP Publishing Ltd
(7.5.49)
Efective Action in Curved Superspace
637
7.5.4. Analysis of quantum equivalence Comparing equations (7.5.47) and (7.5.49) we see that the effective actions rat, rcsp abd rRsp do not coincide. However, this fact does not yet mean absence of equivalence at the quantum level. It may appear that X is a constant locally independent of the supergravity prepotentials. If this is the case,
6X 6X ---_ - -0 6W
(7.5.50)
6q
then the models under consideration will remain equivalent in the quantum theory. In this subsection, we are going to give a proof of equation (7.5.50). We show first of all that X has no divergent part. We begin by rewriting equation (7.5.48) in terms of Green’s functions. Let us introduce the Feynman superpropagator
ccOp’,
associated with the chiral spinor d’Alembertian 9,(7.5.37), 2cc,(Z,
zr)= - 06+(z, Zl).
(7.5.51)
Its antichiral analogue will be denoted by ca.Taking into account equations (7.2.30) and (7.3.18), the superfunctional X can be rewritten as follows: i X=-(Tr+ lncc-2Tr+ lnG,+Tr- lnca-2Tr- lnG, 2
+4Tr In G, - 2Tr In GtR)).
(7.5.52)
The structure of the Green’s functions G, and Gty) were discussed in Sections 7.2 and 7.3. In complete analogy with our previous consideration of G,, one can develop a proper-time representation for the chiral spinor superprogator c, and calculate the corresponding coefficients i i and 3; at coincident points. Then one obtains tr H“,z, z ) = 2aE(z, z )
(7.5.53)
tr i;(z, z ) = 2af(z, z ) - WaBYWaBr +-41 (a2-4R)RR. In the framework of the L-regularization scheme, the divergent part of X Copyright © 1998 IOP Publishing Ltd
638
Ideas and Methods of Supersymmetry and Supergravity
reads
+-
s d 8 zE -
32n2
{f
z)-2a;(z, z)] +c.c.
[tr
I
+ 4a2(z,2 ) - 2aLR)(z,z ) .
(7.5.54)
Making use of equations (7.3.33), (7.5.53) and omitting obvious total derivatives, one arrives at (7.5.55) 9 being the topological invariant (5.6.61). As a result, Xdivis a constant
with respect to arbitrary infinitesimal displacements of the supergravity prepotentials. Now, we give a formal proof of equation (7.5.50). Let us consider the local non-degenerate change of variables U = (D
+ a + -21 (Yx, +
V= Y
+ T +-2i (Px, -Gap)
(7.5.56)
$&?a)
where U and V are real scalar superfields, (B and Y covariantly chiral scalar superfields and x, a covariantly chiral spinor superfield. It turns out that the corresponding Jacobian is equal to J ( U , VI@,Y, x)= Det(H("))Det;"'(2,)Det I
(7.5.57)
This relation can be proved by making the above change of variables in the path integral s B U 9 V erp(
Sds,
E-'(U' - V ' ) ) .
Then we consider the inverse change of variables 1
0= -(9' - 4R) [@' - 4R) + (9' - 41?)] U 16 Ut"'
1
1 Y =-(9' - 4R)[(g2 - 4R) (9' - 4R)]
+
16
x
1 -
1
= --(9'-4Rpa-(U-iV). U " 8
Copyright © 1998 IOP Publishing Ltd
t"'
1 V 0 I"'
(7.5.58)
Efective Action in Curved Superspace
639
The corresponding Jacobian reads
J(@, Y,
XI
U, V )= Det(H(R))Det - '(
LR))
(7.5.59)
which can be proved by carrying out the replacement of variables (7.5.58) in the path integral
It follows from equations (7.5.57) and (7.5.59) that the superfunctional (7.5.48) must vanish modulo boundary terms at infinity, the latter being always ignored in (super)field redefinitions. The above analysis gives strong grounds in defence of equation (7.5.50) and, hence, quantum equivalence, of the three superfield models under consideration. As an instructive exercise, we suggest that the reader find a field analogue of the superfield definition (7.5.56), which makes it possible to establish formal coincidence of the effective actions (7.5.30) and (7.5.31).
Copyright © 1998 IOP Publishing Ltd
Bibliography
Pioneering papers on supersymmetry
Yu. A. Gol’fand and E. P. Likhtman. Extension of the algebra of PoincarC group generators and violation of P-invariance. JETP Lett. 13 323, 1971. D. V. Volkov and V. P. Akulov. Possible universal neutrino interaction. JETP Lett. 16 438, 1972. J. Wess and B. Zumino. Supergauge transformations in four dimensions. Nucl. Phys. B 70 39, 1974. J. Wess and B. Zumino. A Lagrangian model invariant under supergauge transformations. Phys. Lett. 49B 52, 1974. Pioneering papers on supergravity D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara. Progress towards a theory of supergravity. Phys. Rev. D 13 3214, 1976. S . Deser and B.Zumino. Consistent supergravity. Phys. Lett. 62B 335, 1976.
Basic reviews and books on supersymmetry and supergravity V. I. Ogievetsky and L. Mezincescu. Boson-fermion symmetries and superfields. SOV. Phys.-Usp. 18 960, 1976. P. Fayet and S.Ferrara. Supersymmetry. Phys. Rep. 32 249, 1977. A.Salam and J. Strathdee. Supersymmetry and Superfields. Fortschr. Phys. 26 57, 1978. P. van Nieuwenhuizen. Supergravity. Phys. Rep. 68 189, 1981. H. P. Nilles. Supersymmetry, supergravity and particle physics. Phys. Rep. 110 1, 1984. H. E. Haber and G. L. Kane. The search for supersymmetry: probing physics beyond the standard model. Phys. Rep. 117 75, 1985. E. S . Fradkin and A. A. Tseytlin. Conformal supergravity. Phys. Rep. 119 233, 1985. M. F. Sohnius. Introducing supersymmetry. Phys. Rep. 128 39, 1985. M.J. Duff,B.E. W. Nilsson and C. N. Pope. Kaluza-Klein supergravity. Phys. 640 Copyright © 1998 IOP Publishing Ltd
Bibliography
64 1
Rep. 130 1 , 1986. N. Dragon, U. Ellwanger and M. Schmidt. Supersymmetry and Supergravity. Prog. Particle Nucl. Phys. 18 1, 1987. S. J. Gates, M. T. Grisaru, M. R d e k and W. Siegel. Superspace or One Thousand and One Lessons in Supersymmetry. Benjamin Cummings, Reading, MA, 1983. J. Wess and J. Bagger. Supersymmetry and Supergravity. Princeton University Press, Princeton, NJ, 1991 (Second Edition, Revised and Corrected). P. West. Introduction to Supersymmetry and Supergravity. World Scientific, Singapore, 1992 (Extended Second Edition). P. Freund. Introduction to Supersymmetry. Cambridge University Press, Cambridge, 1986. R. N. Mohapatra, Wnijication and Supersymmetry. Springer-Verlag, New York, 1986. 0. Piquet and K. Sibold. Renormalized Supersymmetry: The Perturbation Theory of N = 1 supersymmetric Theories in Flat Space-Time. Birkhauser, Boston, 1986. H. J. W. Muller-Kirsten and A. Wiedemann. Supersymmetry: An Introduction with Conceptual and Calculational Details. World Scientific, Singapore, 1987. M. Muller. Consistent Classical Supergravity Theories. Springer-Verlag, New York, 1989. L. Castellani, R. D’ Auria and P. Fre. Supergravity and Superstrings: A Geometric Perspective. World Scientific, Singapore, 1991. P. Fre and P. Soriani. The N = 2 Wonderland: From Calabi-Yau Manifolds to Topological Field Theories. World Scientific, Singapore, 1995.
Field theory in superspace A. Salam and J. Strathdee. Super-gauge transformations. Nucl. Phys. B 76 477, 1974. S. Ferrara, J. Wess and B.Zumino. Supergauge multiplets and superfields. Phys. Lett. 51B 239, 1974. Supergauge invariant extension of quantum J. Wess and B. Zumino. electrodynamics. Nucl. Phys. B 78 1, 1974. A. Salam and J. Strathdee. Super-symmetry and non-Abelian gauges. Phys. Lett. 51B 353, 1974. S. Ferrara and B. Zumino. Supergauge invariant Yang-Mills theories. Nucl. Phys. B 79 413, 1974. A. Salam and J. Strathdee. Superfields and Fermi-Bose symmetry. Phys. Rev. D 11 1521, 1975. S. Ferrara and B. Zumino. Transformation properties of the supercurrent. Nucl. Phys. B 87 207, 1975. E. Sokatchev. Projection operators and supplementary conditions for superfields
Copyright © 1998 IOP Publishing Ltd
642
Bibliography
with arbitrary spin. Nucl. Phys. B 99 96, 1975. V. I. Ogievetsky and E. S . Sokatchev. The supercurrent. Sov. J. Nucl. Phys. 28423, 1978.
W. Siegel. Gauge spinor superfield as scalar multiplet. Phys. Lett. 85B 333, 1979.
M. T. Grisaru, M. RoEek and W. Siegel. Improved methods for supergraphs. Nucl. Phys. B 159 429, 1979. M. T. Grisaru and W. Siegel. Supergraphity (I). Background field formalism. Nucl. Phys. B 187 149, 1981. S . J. Gates and W. Siegel. Variant superfield representations. Nucl. Phys. B 187 389, 1981. M. T. Grisaru and W. Siegel. Supergraphity (11). Manifestly covariant rules and higher-loop finiteness. Nucl. Phys. B 201 292, 1982. B. de Wit and M. RoEek. Improved tensor multiplets. Phys. Lett. 1WB 439, 1982. E. A. Ivanov. Intristic geometry of the N = 1 supersymmetric Yang-Mills theory. J. Phys. A: Math. Gen. 16 2571, 1983. B. B. Deo and S . J. Gates. Comments on nonminimal N = 1 scalar multiplets. Nucl. Phys. B 254 187, 1985.
Superfield supergravity J. Wess and B. Zumino. Superspace formulation of supergarvity. Phys. Lett. 66B 361, 1977. J. Wess and B. Zumino. Superfield Lagrangian for supergarvity. Phys. Lett. 74B 51, 1978. V. Ogievetsky and E. Sokatchev. Structure of supergravity group. Phys. Lett. 79B 222, 1978. P. S . Howe and R. W. Tucker. Scale invariance in superspace. Phys. Lett. SOB 138, 1978.
R. Grimm, J. Wess and B. Zumino. A complete solution of the Bianchi identities in superspace with supergravity constraints. Nucl. Phys. B 152 255, 1979. W. Siegel and S.J. Gates. Superfield supergravity. Nucl. Phys. B 147 77, 1979. V. I. Ogievetsky and E. S.Sokatchev. The simplest Einstein supergarvity group. Sov. J. Nucl. Phys. 31 140, 1980. V. I. Ogievetsky and E. S . Sokatchev. The gravitational axial-vector superfield and the formalism of differential geometry. Sov. J. Nucl. Phys. 31 424, 1980. V. I. Ogievetsky and E. S . Sokatchev. The normal gauge in supergarvity. Sov. J. Nucl. Phys. 32 443, 1980. V. I. Ogievetsky and E. S . Sokatchev. Torsion and curvature in terms of the axial superfield. Sov. J. Nucl. Pliys. 32 447, 1980. V. I. Ogievetsky and E. S . Sokatchev. Equations of motion for the axial gravitational superfield. Sov. J. Nucl. Phys. 32 589, 1980.
Copyright © 1998 IOP Publishing Ltd
Bibliography
643
E. A. Ivanov and A. S. Sorin. Superfield formulation of OSp(1,4) supersymmetry. J. Phys. A: Math. Gen. 13 1159, 1980. S . J. Gates and W. Siegel. Understanding constraints in superspace formulations of supergravity. Nucl. Phys. B 163 519, 1980. S . J. Gates, K. S. Stelle and P. C. West. Algebraic origins of superspace constraints in supergravity. Nucl. Phys. B 169 347, 1980. P. S. Howe, K.S. Stelle and P. K.Townsend. The vanishing volume of N = 1 superspace. Phys. Lett. 107B 420, 1981. S. J. Gates, M. R d e k and W. Siegel. Solution to constraints for n = 0 supergravity. Nucl. Phys. B 198 113, 1982.
Supermathematics F. A. Berezin. The Method of Second Quantization. Academic Press, New York, 1966. F. A. Berezin. Introduction to Superanalysis. Reidel, Dordrecht, 1987. B. S . De Witt. Supermangolds. Cambridge University Press, Cambridge, 1986.
Copyright © 1998 IOP Publishing Ltd
