SUPERSPACE or One thousand and one lessons in supersymmetry S. James Gates, Jr. Massachusetts Institute of Technology, Cambridge, Massachusetts (Present address: University of Maryland, College Park, Maryland) gatess@wam.umd.edu Marcus T. Grisaru Brandeis University, Waltham, Massachusetts (Present address: McGill University, Montreal, Quebec) grisaru@physics.mcgill. ca Martin Rocek State University of New York, Stony Brook, New York rocek@insti.physics.sunysb.edu Warren Siegel University of California, Berkeley, California (Present address: State University of New York) warren@wcgall.physics.sunysb.edu  Library of Congress Cataloging in Publication Data Main entry under title: Superspace : one thousand and one lessons in supersymmetry. (Frontiers in physics ; v. 58) Includes index. 1. Supersymmetry. 2. Quantum gravity. 3. Supergravity. I. Gates, S. J. II. Series. QC174.17.S9S97 1983 530.1'2 83-5986 ISBN 0-8053-3160-3 ISBN 0-8053-3160-1 (pbk.)  Superspace is the greatest invention since the wheel [1] . Preface Said T to 4, E, and T: "Let's write a review paper." Said k and "Great idea!" Said T: "Naaa." But a few days later T had produced a table of contents with 1001 items. E, ,W, and T wrote. Then didn't write. Then wrote again. The review grew; and grew; and grew. It became an outline for a book; it became a first draft; it became a second draft. It became a burden. It became agony. Tempers were lost; and hairs; and a few pounds (alas, quickly regained). They argued about ";" vs. ".", about "which" vs. "that", "~" vs. "^", "7y" vs. "F", "+" vs. "-". Made bad puns, drew pic- tures on the blackboard, were rude to their colleagues, neglected their duties. Bemoaned the paucity of letters in the Greek and Roman alphabets, of hours in the day, days in the week, weeks in the month. E, , I and T wrote and wrote. * * * This must stop; we want to get back to research, to our families, friends and stu- dents. We want to look at the sky again, go for walks, sleep at night. Write a second volume? Never! Well, in a couple of years? We beg our readers' indulgence. We have tried to present a subject that we like, that we think is important. We have tried to present our insights, our tools and our knowledge. Along the way, some errors and misconceptions have without doubt slipped in. There must be wrong statements, misprints, mistakes, awkward phrases, islands of incomprehensibility (but they started out as continents!). We could, probably we should, improve and improve. But we can no longer wait. Like climbers within sight of the summit we are rushing, casting aside caution, reaching towards the moment when we can shout "it's behind us". This is not a polished work. Without doubt some topics are treated better else- where. Without doubt we have left out topics that should have been included. Without doubt we have treated the subject from a personal point of view, emphasizing aspects that we are familiar with, and neglecting some that would have required studying others' work. Nevertheless, we hope this book will be useful, both to those new to the subject and to those who helped develop it. We have presented many topics that are not avail- able elsewhere, and many topics of interest also outside supersymmetry. We have [1]. A. Qop, A supersymmetric version of the leg, Gondwanaland predraw (January 10,000,000 B.C.), to be discovered.  included topics whose treatment is incomplete, and presented conclusions that are really only conjectures. In some cases, this reflects the state of the subject. Filling in the holes and proving the conjectures may be good research projects. Supersymmetry is the creation of many talented physicists. We would like to thank all our friends in the field, we have many, for their contributions to the subject, and beg their pardon for not presenting a list of references to their papers. Most of the work on this book was done while the four of us were at the California Institute of Technology, during the 1982-83 academic year. We would like to thank the Institute and the Physics Department for their hospitality and the use of their computer facilities, the NSF, DOE, the Fleischmann Foundation and the Fairchild Visiting Schol- ars Program for their support. Some of the work was done while M.T.G. and M.R. were visiting the Institute for Theoretical Physics at Santa Barbara. Finally, we would like to thank Richard Grisaru for the many hours he devoted to typing the equations in this book, Hyun Jean Kim for drawing the diagrams, and Anders Karlhede for carefully read- ing large parts of the manuscript and for his useful suggestions; and all the others who helped us. S.J.G., M.T.G., M.R., W.D.S. Pasadena, January 1983 August 2001: Free version released on web; corrections and bookmarks added.  Contents Preface 1. Introduction 1 2. A toy superspace 2.1. Notation and conventions 7 2.2. Supersymmetry and superfields 9 2.3. Scalar multiplet 15 2.4. Vector multiplet 18 2.5. Other global gauge multiplets 28 2.6. Supergravity 34 2.7. Quantum superspace 46 3. Representations of supersymmetry 3.1. Notation 54 3.2. The supersymmetry groups 62 3.3. Representations of supersymmetry 69 3.4. Covariant derivatives 83 3.5. Constrained superfields 89 3.6. Component expansions 92 3.7. Superintegration 97 3.8. Superfunctional differentiation and integration 101 3.9. Physical, auxiliary, and gauge components 108 3.10. Compensators 112 3.11. Projection operators 120 3.12. On-shell representations and superfields 138 3.13. Off-shell field strengths and prepotentials 147 4. Classical, global, simple (N = 1) superfields 4.1. The scalar multiplet 149 4.2. Yang-Mills gauge theories 159 4.3. Gauge-invariant models 178 4.4. Superforms 181 4.5. Other gauge multiplets 198 4.6. N-extended multiplets 216 5. Classical N =1 supergravity 5.1. Review of gravity 232 5.2. Prepotentials 244  5.3. Covariant approach 267 5.4. Solution to Bianchi identities 292 5.5. Actions 299 5.6. From superspace to components 315 5.7. DeSitter supersymmetry 335 6. Quantum global superfields 6.1. Introduction to supergraphs 337 6.2. Gauge fixing and ghosts 340 6.3. Supergraph rules 348 6.4. Examples 364 6.5. The background field method 373 6.6. Regularization 393 6.7. Anomalies in Yang-Mills currents 401 7. Quantum N = 1 supergravity 7.1. Introduction 408 7.2. Background-quantum splitting 410 7.3. Ghosts 420 7.4. Quantization 431 7.5. Supergravity supergraphs 438 7.6. Covariant Feynman rules 446 7.7. General properties of the effective action 452 7.8. Examples 460 7.9. Locally supersymmetric dimensional regularization 469 7.10. Anomalies 473 8. Breakdown 8.1. Introduction 496 8.2. Explicit breaking of global supersymmetry 500 8.3. Spontaneous breaking of global supersymmetry 506 8.4. Trace formulae from superspace 518 8.5. Nonlinear realizations 522 8.6. Super~iggs mechanism 527 8.7. Supergravity and symmetry breaking 529 Index 542  1. INTRODUCTION There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition; and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call, "the Twilight Zone." Rod Serling 1001: A superspace odyssey Symmetry principles, both global and local, are a fundamental feature of modern particle physics. At the classical and phenomenological level, global symmetries account for many of the (approximate) regularities we observe in nature, while local (gauge) symmetries "explain" and unify the interactions of the basic constituents of matter. At the quantum level symmetries (via Ward identities) facilitate the study of the ultraviolet behavior of field theory models and their renormalization. In particular, the construc- tion of models with local (internal) Yang-Mills symmetry that are asymptotically free has increased enormously our understanding of the quantum behavior of matter at short distances. If this understanding could be extended to the quantum behavior of gravita- tional interactions (quantum gravity) we would be close to a satisfactory description of micronature in terms of basic fermionic constituents forming multiplets of some unifica- tion group, and bosonic gauge particles responsible for their interactions. Even more satisfactory would be the existence in nature of a symmetry which unifies the bosons and the fermions, the constituents and the forces, into a single entity. Supersymmetry is the supreme symmetry: It unifies spacetime symmetries with internal symmetries, fermions with bosons, and (local supersymmetry) gravity with mat- ter. Under quite general assumptions it is the largest possible symmetry of the S- matrix. At the quantum level, renormalizable globally supersymmetric models exhibit improved ultraviolet behavior: Because of cancellations between fermionic and bosonic contributions quadratic divergences are absent; some supersymmetric models, in particu- lar maximally extended super-Yang-Mills theory, are the only known examples of four- dimensional field theories that are finite to all orders of perturbation theory. Locally  2 1. INTRODUCTION supersymmetric gravity (supergravity) may be the only way in which nature can recon- cile Einstein gravity and quantum theory. Although we do not know at present if it is a finite theory, quantum supergravity does exhibit less divergent short distance behavior than ordinary quantum gravity. Outside the realm of standard quantum field theory, it is believed that the only reasonable string theories (i.e., those with fermions and without quantum inconsistencies) are supersymmetric; these include models that may be finite (the maximally supersymmetric theories). At the present time there is no direct experimental evidence that supersymmetry is a fundamental symmetry of nature, but the current level of activity in the field indicates that many physicists share our belief that such evidence will eventually emerge. On the theoretical side, the symmetry makes it possible to build models with (super)natural hierarchies. On esthetic grounds, the idea of a superunified theory is very appealing. Even if supersymmetry and supergravity are not the ultimate theory, their study has increased our understanding of classical and quantum field theory, and they may be an important step in the understanding of some yet unknown, correct theory of nature. We mean by (Poincard) supersymmetry an extension of ordinary spacetime sym- metries obtained by adjoining N spinorial generators Q whose anticommutator yields a translation generator: { Q , Q } = P. This symmetry can be realized on ordinary fields (functions of spacetime) by transformations that mix bosons and fermions. Such realiza- tions suffice to study supersymmetry (one can write invariant actions, etc.) but are as cumbersome and inconvenient as doing vector calculus component by component. A compact alternative to this "component field" approach is given by the super- space--superfield approach. Superspace is an extension of ordinary spacetime to include extra anticommuting coordinates in the form of N two-component Weyl spinors 0. Superfields 'T(x, 0) are functions defined over this space. They can be expanded in a Taylor series with respect to the anticommuting coordinates 0; because the square of an anticommuting quantity vanishes, this series has only a finite number of terms. The coefficients obtained in this way are the ordinary component fields mentioned above. In superspace, supersymmetry is manifest: The supersymmetry algebra is represented by translations and rotations involving both the spacetime and the anticommuting coordi- nates. The transformations of the component fields follow from the Taylor expansion of the translated and rotated superfields. In particular, the transformations mixing bosons  1. INTRODUCTION 3 and fermions are constant translations of the 0 coordinates, and related rotations of 0 into the spacetime coordinate x. A further advantage of superfields is that they automatically include, in addition to the dynamical degrees of freedom, certain unphysical fields: (1) auxiliary fields (fields with nonderivative kinetic terms), needed classically for the off-shell closure of the super- symmetry algebra, and (2) compensating fields (fields that consist entirely of gauge degrees of freedom), which are used to enlarge the usual gauge transformations to an entire multiplet of transformations forming a representation of supersymmetry; together with the auxiliary fields, they allow the algebra to be field independent. The compen- sators are particularly important for quantization, since they permit the use of super- symmetric gauges, ghosts, Feynman graphs, and supersymmetric power-counting. Unfortunately, our present knowledge of off-shell extended (N > 1) supersymmetry is so limited that for most extended theories these unphysical fields, and thus also the corresponding superfields, are unknown. One could hope to find the unphysical compo- nents directly from superspace; the essential difficulty is that, in general, a superfield is a highly reducible representation of the supersymmetry algebra, and the problem becomes one of finding which representations permit the construction of consistent local actions. Therefore, except when discussing the features which are common to general superspace, we restrict ourselves in this volume to a discussion of simple (N = 1) superfield super- symmetry. We hope to treat extended superspace and other topics that need further development in a second (and hopefully last) volume. We introduce superfields in chapter 2 for the simpler world of three spacetime dimensions, where superfields are very similar to ordinary fields. We skip the discussion of nonsuperspace topics (background fields, gravity, etc.) which are covered in following chapters, and concentrate on a pedagogical treatment of superspace. We return to four dimensions in chapter 3, where we describe how supersymmetry is represented on super- fields, and discuss all general properties of free superfields (and their relation to ordinary fields). In chapter 4 we discuss simple (N =1) superfields in classical global supersym- metry. We include such topics as gauge-covariant derivatives, supersymmetric models, extended supersymmetry with unextended superfields, and superforms. In chapter 5 we extend the discussion to local supersymmetry (supergravity), relying heavily on the com- pensator approach. We discuss prepotentials and covariant derivatives, the construction  4 1. INTRODUCTION of actions, and show how to go from superspace to component results. The quantum aspects of global theories is the topic of chapter 6, which includes a discussion of the background field formalism, supersymmetric regularization, anomalies, and many exam- ples of supergraph calculations. In chapter 7 we make the corresponding analysis of quantum supergravity, including many of the novel features of the quantization proce- dure (various types of ghosts). Chapter 8 describes supersymmetry breaking, explicit and spontaneous, including the superHiggs mechanism and the use of nonlinear realiza- tions. We have not discussed component supersymmetry and supergravity, realistic superGUT models with or without supergravity, and some of the geometrical aspects of classical supergravity. For the first topic the reader may consult many of the excellent reviews and lecture notes. The second is one of the current areas of active research. It is our belief that superspace methods eventually will provide a framework for streamlin- ing the phenomenology, once we have better control of our tools. The third topic is attracting increased attention, but there are still many issues to be settled; there again, superspace methods should prove useful. We assume the reader has a knowledge of standard quantum field theory (sufficient to do Feynman graph calculations in QCD). We have tried to make this book as peda- gogical and encyclopedic as possible, but have omitted some straightforward algebraic details which are left to the reader as (necessary!) exercises.  1. INTRODUCTION 5 A hitchhiker's guide We are hoping, of course, that this book will be of interest to many people, with different interests and backgrounds. The graduate student who has completed a course in quantum field theory and wants to study superspace should: (1) Read an article or two reviewing component global supersymmetry and super- gravity. (2) Read chapter 2 for a quick and easy (?) introduction to superspace. Sections 1, 2, and 3 are straightforward. Section 4 introduces, in a simple setting, the concept of constrained covariant derivatives, and the solution of the constraints in terms of prepo- tentials. Section 5 could be skipped at first reading. Section 6 does for supergravity what section 4 did for Yang-Mills; superfield supergravity in three dimensions is decep- tively simple. Section 7 introduces quantization and Feynman rules in a simpler situa- tion than in four dimensions. (3) Study subsections 3.2.a-d on supersymmetry algebras, and sections 3.3.a, 3.3.b.1-b.3, 3.4.a,b, 3.5 and 3.6 on superfields, covariant derivatives, and component expansions. Study section 3.10 on compensators; we use them extensively in supergrav- ity. (4) Study section 4.1a on the scalar multiplet, and sections 4.2 and 4.3 on gauge theories, their prepotentials, covariant derivatives and solution of the constraints. A reading of sections 4.4.b, 4.4.c.1, 4.5.a and 4.5.e might be profitable. (5) Take a deep breath and slowly study section 5.1, which is our favorite approach to gravity, and sections 5.2 to 5.5 on supergravity; this is where the action is. For an inductive approach that starts with the prepotentials and constructs the covariant derivatives section 5.2 is sufficient, and one can then go directly to section 5.5. Alterna- tively, one could start with section 5.3, and a deductive approach based on constrained covariant derivatives, go through section 5.4 and again end at 5.5. (6) Study sections 6.1 through 6.4 on quantization and supergraphs. The topics in these sections should be fairly accessible. (7) Study sections 8.1-8.4. (8) Go back to the beginning and skip nothing this time.  6 1. INTRODUCTION Our particle physics colleagues who are familiar with global superspace should skim 3.1 for notation, 3.4-6 and 4.1, read 4.2 (no, you don't know it all), and get busy on chapter 5. The experts should look for serious mistakes. We would appreciate hearing about them. A brief guide to the literature A complete list of references is becoming increasingly difficult to compile, and we have not attempted to do so. However, the following (incomplete!) list of review articles and proceedings of various schools and conferences, and the references therein, are useful and should provide easy access to the journal literature: For global supersymmetry, the standard review articles are: P. Fayet and S. Ferrara, Supersymmetry, Physics Reports 32C (1977) 250. A. Salam and J. Strathdee, Fortschritte der Physik, 26 (1978) 5. For component supergravity, the standard review is P. van Nieuwenhuizen, Supergravity, Physics Reports 68 (1981) 189. The following Proceedings contain extensive and up-to-date lectures on many supersymmetry and supergravity topics: "Recent Developments in Gravitation" (Cargese 1978), eds. M. Levy and S. Deser, Plenum Press, N.Y. "Supergravity" (Stony Brook 1979), eds. D. Z. Freedman and P. van Nieuwen- huizen, North-Holland, Amsterdam. "Topics in Quantum Field Theory and Gauge Theories" (Salamanca), Phys. 77, Springer Verlag, Berlin. "Superspace and Supergravity"(Cambridge 1980), eds. S. W. Hawking and M. Roeek, Cambridge University Press, Cambridge. "Supersymmetry and Supergravity '81" (Trieste), eds. S. Ferrara, J. G. Taylor and P. van Nieuwenhuizen, Cambridge University Press, Cambridge. "Supersymmetry and Supergravity '82" (Trieste), eds. S. Ferrara, J. G. Taylor and P. van Nieuwenhuizen, World Scientific Publishing Co., Singapore.  Contents of 2. A TOY SUPERSPACE 2.1. Notation and conventions 7 a. Index conventions 7 b. Superspace 8 2.2. Supersymmetry and superfields 9 a. Representations 9 b. Components by expansion 10 c. Actions and components by projection 11 d. Irreducible representations 13 2.3. Scalar multiplet 15 2.4. Vector multiplet 18 a. Abelian gauge theory 18 a.1. Gauge connections 18 a.2. Components 19 a.3. Constraints 20 a.4. Bianchi identities 22 a.5. Matter couplings 23 b. Nonabelian case 24 c. Gauge invariant masses 26 2.5. Other global gauge multiplets 28 a. Superforms: general case 28 b. Super 2-form 30 c. Spinor gauge superfield 32 2.6. Supergravity 34 a. Supercoordinate transformations 34 b. Lorentz transformations 35 c. Covariant derivatives 35 d. Gauge choices 37 d.1. A supersymmetric gauge 37 d.2. Wess-Zumino gauge 38 e. Field strengths 38 f. Bianchi identities 39 g. Actions 42 2.7. Quantum superspace 46 a. Scalar multiplet 46  a. 1. General formalism 46 a.2. Examples 49 b. Vector multiplet 52  2. A TOY SUPERSPACE 2.1. Notation and conventions This chapter presents a self-contained treatment of supersymmetry in three spacetime dimensions. Our main motivation for considering this case is simplicity. Irre- ducible representations of simple (N = 1) global supersymmetry are easier to obtain than in four dimensions: Scalar superfields (single, real functions of the superspace coor- dinates) provide one such representation, and all others are obtained by appending Lorentz or internal symmetry indices. In addition, the description of local supersymme- try (supergravity) is easier. a. Index conventions Our three-dimensional notation is as follows: In three-dimensional spacetime (with signature - + +) the Lorentz group is SL(2, R) (instead of SL(2, C)) and the cor- responding fundamental representation acts on a real (Majorana) two-component spinor a_(L+ -). In general we use spinor notation for all Lorentz representations, denot- ing spinor indices by Greek letters a, -,. .. , ,, v, -,--. Thus a vector (the three-dimen- sional representation) will be described by a symmetric second-rank spinor V" =/ (V++, V+-, V--) or a traceless second-rank spinor Vj/. (For comparison, in four dimensions we have spinors cJ, and a vector is given by a hermitian matrix Vc".) All our spinors will be anticommuting (Grassmann). Spinor indices are raised and lowered by the second-rank antisymmetric symbol C /, which is also used to define the "square" of a spinor: 0 - C = C = 0 =Cap, C?=1 =z+-(2.1.1) 2 We represent symmetrization and antisymmetrization of n indices by ( ) and [ ], respec- tively (without a factor of 2k). We often make use of the identity A[ B131 - - Ca/3 A"' B (2.1.2)  8 2. A TOY SUPERSPACE which follows from (2.1.1). We use Cad (instead of the customary real ep) to simplify the rules for hermitian conjugation. In particular, it makes #92 hermitian (recall @9a and 9, anticommute) and gives the conventional hermiticity properties to derivatives (see below). Note however that whereas @9" is real, @9a is imaginary. b. Superspace Superspace for simple supersymmetry is labeled by three spacetime coordinates x"" and two anticommuting spinor coordinates 0", denoted collectively by zM _(x"" , 9 ). They have the hermiticity properties (zM)t -= M. We define derivatives by 8,6" = {a , 6"} = 6," 1gvx"07 =[,x T] =26 ," (2.1.3a) so that the "momentum" operators have the hermiticity properties (i)t - (i&) ,(i&)t= + (i&,v) . (2.1.3b) and thus (i&M)t - j&M. (Definite) integration over a single anticommuting variable 7 is defined so that the integral is translationally invariant (see sec. 3.7); hence J d71 = 0 , I dy 7 = a constant which we take to be 1. We observe that a function fy(7) has a ter- minating Taylor series f (7) = f (0) + 7 f'(0) since {'y ,7} = 0 implies 72 = 0. Thus Sdy f(7) = f'(0) so that integration is equivalent to differentiation. For our spinorial coordinates J d0 &= 6aand hence J d6,/0a0= . (2.1.4) Therefore the double integral Jd20062 =- 1 , (2.1.5) and we can define the s-function $(0) =- 02 = - 6± 62 * * * We often use the notation X Ito indicate the quantity X evaluated at 0 0.  2.2. Supersymmetry and superfields 9 2.2. Supersymmetry and superfields a. Representations We define functions over superspace: i... (x, 0) where the dots stand for Lorentz (spinor) and/or internal symmetry indices. They transform in the usual way under the Poincard group with generators P,, (translations) and Map (Lorentz rotations). We grade (or make super) the Poincard algebra by introducing additional spinor supersym- metry generators Qa, satisfying the supersymmetry algebra P,, , Ppo] = 0 , (2.2.1a) {Q,,Q,} =2 P , (2.2.1b) [Q, , Pp] = 0 , (2.2.1c) as well as the usual commutation relations with M g. This algebra is realized on super- fields i... (x , 0) in terms of derivatives by: P = io0 , Q"+ = i (0 - "i ) ; (2.2.2a) $(x ",60) - exp[i( PAP + eQA)](x" + "" - 2 ", +E). (2.2.2b) Thus (APAp + eAQA generates a supercoordinate transformation xII V -t 1-t V 5() , O' - 6"+ E . (2.2.2c) 2 with real, constant parameters (P ,E. The reader can verify that (2.2.2) provides a representation of the algebra (2.2.1). We remark in particular that if the anticommutator (2.2.1b) vanished, Q would annihi- late all physical states (see sec. 3.3). We also note that because of (2.2.1a,c) and (2.2.2a), not only @ and functions of @, but also the space-time derivatives Q,,A carry a representation of supersymmetry (are superfields). However, because of (2.2.2a), this is not the case for the spinorial derivatives Q8 @. Supersymmetrically invariant derivatives can be defined by DM = (D, , D ) = (&, ,&,, +0"i&e ) (2.2.3) (2.2.3)  10 2. A TOY SUPERSPACE The set Dm (anti)commutes with the generators: [DM , P,] = [DM , Qv} = 0. We use [A , B} to denote a graded commutator: anticommutator if both A and B are fermionic, commutator otherwise. The covariant derivatives can also be defined by their graded commutation rela- tions {D, , Dv} = 2iD,, , [D, , Dv]= [D,, , DT] = 0 ; (2.2.4) or, more concisely: [Dm , DN}=TMNPDP T ,7107T = 16( 6v1 T rest = 0 (2.2.5) Thus, in the language of differential geometry, global superspace has torsion. The derivatives satisfy the further identities OIUOvU = 6v Q D D & = i[B + CvD2 DVDDv= 0 , D2D, D D2 = i& Dv (D2)2 = (2.2.6) They also satisfy the Leibnitz rule and can be integrated by parts when inside d3x d20 integrals (since they are a combination of x and 0 derivatives ). The following identity is useful I d3x d2O D(xe0) I d3x a2 (xe) I d3x ( D2 to 1 so that @, has canonical dimension (mass)1. (Although it is not immediately obvious which scalar should have canonical dimension, there is only one spinor.) Then A will have dimension (mass)y and will be the physical scalar partner of @9, whereas F has too high a dimen- sion to describe a canonical physical mode. Since a 0 integral is the same as a 0 derivative, f d20 has dimension (mass)1. Therefore, on dimensional grounds we expect the following expression to give the correct (massless) kinetic action for the scalar multiplet: Sk - - f d3x d20 (D$@)2 , (2.3.1) (recall that for any spinor @9a we have =2 ) bb. This expression is reminiscent of the kinetic action for an ordinary scalar field with the substitutions f d3x - d3x d20 and0a 0 - Da. The component expression can be obtained by explicit 0-expansion and integration. However, we prefer to use the alternative procedure (first integrating Do by parts): Ski 2 f d3x d20 D24 fdox D2[@ D2] Sfd3x (D2@ D2F + Do@F DaD2@ + @(D2)24) fdax (F2 + @"ioji/> + ADA) , (2.3.2)  16 2. A TOY SUPERSPACE where we have used the identities (2.2.6) and the definitions (2.2.13). The A and 9 kinetic terms are conventional, while F is clearly non-propagating. The auxiliary field F can be eliminated from the action by using its equation of motion F = 0 (or, in a functional integral, F can be trivially integrated out). The resulting action is still invariant under the bose-fermi transformations (2.2.1la,b) with F = 0; however, these are not supersymmetry transformations (not a representation of the supersymmetry algebra) except "on shell". The commutator of two such transforma- tions does not close (does not give a translation) except when @9a satisfies its field equa- tion. This "off-shell" non-closure of the algebra is typical of transformations from which auxiliary fields have been eliminated. Mass and interaction terms can be added to (2.3.1). A term S,= d3xd2 f () , (2.3.3) leads to a component action S1 fd3x D2f(4) d3x [f"(4) (Da$)2 + f'(4) D2 d3x [f"(A) 92 + f'(A) F] . (2.3.4) In a renormalizable model f(4) can be at most quartic. In particular, f (4) =1 m2 + A3 gives mass terms, Yukawa and cubic interaction terms. Together with the kinetic term, we obtain d3xd2O[ - } (Da$)2+} m2+ 1A3 fdox[~ (AD A+@"/fiojI@+ F2) + m(2 + AF) +A(A@2+ }A2F)] . (2.3.5) F can again be eliminated using its (algebraic) equation of motion, leading to a  2.3. Scalar multiplet 17 conventional mass term and quartic interactions for the scalar field A. More exotic kinetic actions are possible by using instead of (2.3.1) S'ki fd3x d2 Q ((, 4) , (a = DN , (2.3.6) where Q is some function such that 2 1 C . If we introduce more than one multiplet of scalar superfields, then, for example, we can obtain generalized super- symmetric nonlinear sigma models: S = - d3x d28g1(4b) ( Do@ ) (DaI) (2.3.7)  18 2. A TOY SUPERSPACE 2.4. Vector multiplet a. Abelian gauge theory In accordance with the discussion in sec. 2.2, a real spinor gauge superfield I, 1 1 with superhelicity h =- (h = 2 can be gauged away) will consist of components with 2 2 helicities 0' 2{ '1. It can be used to describe a massless gauge vector field and its fermionic partner. (In three dimensions, a gauge vector particle has one physical compo- nent of definite helicity.) The superfield can be introduced by analogy with scalar QED (the generalization to the nonabelian case is straightforward, and will be discussed below). Consider a complex scalar superfield (a doublet of real scalar superfields) trans- forming under a constant phase rotation 4 eK .(.41 The free Lagrangian |DG|2 is invariant under these transformations. a.1. Gauge connections We extend this to a local phase invariance with K a real scalar superfield depend- ing on x and 0, by covariantizing the spinor derivatives Da: Da V a = DaTF i fa , (2.4.2) when acting on 4 or 4, respectively. The spinor gauge potential (or connection) F. transforms in the usual way ora = DaK , (2.4.3) to ensure V'a eiK V - iK (24) This is required by (V @)'= eiK (V @), and guarantees that the Lagrangian IVG|2 is locally gauge invariant. (The coupling constant can be restored by rescaling 1,a gFa).  2.4. Vector multiplet 19 It is now straightforward, by analogy with QED, to find a gauge invariant field strength and action for the multiplet described by F, and to study its component cou- plings to the complex scalar multiplet contained in IV b2. However, both to understand its structure as an irreducible representation of supersymmetry, and as an introduction to more complicated gauge superfields (e.g. in supergravity), we first give a geometrical presentation. Although the actions we have considered do not contain the spacetime derivative 0,0, in other contexts we need the covariant object V aO = j - Fay , Fa- = o a/K , (2.4.5) introducing a distinct (vector) gauge potential superfield. The transformation 6F,1 of this connection is chosen to give: V'ay3= eiK V e-iK . (2.4.6) (From a geometric viewpoint, it is natural to introduce the vector connection; then Fa and F can be regarded as the components of a super 1-form FA (Fa, Fa); see sec. 2.5). However, we will find that F should not be independent, and can be expressed in terms of Fa. a.2. Components To get oriented, we examine the components of F in the Taylor series 0-expansion. They can be defined directly by using the spinor derivatives Da: Xa=Fa , B=}DF 2 a V - DF A jDIDF ..a V (a , 2 a| , (2.4.7a) and WaF = o|, p D"Fap| apPa=JD(aFg| , Ta = D2Fail . (2.4.7b) We have separated the components into irreducible representations of the Lorentz group, that is, traces (or antisymmetrized pieces, see (2.1.2)) and symmetrized pieces. We also  20 2. A TOY SUPERSPACE define the components of the gauge parameter K: w=K I, Ua=DaK , T=D2K I(2.4.8) The component gauge transformations for the components defined in (2.4.7) are found by repeatedly differentiating (2.4.3-5) with spinor derivatives Da. We find: 6xa = 6a , B = T SaI-= o , SA =0 , (2.4.9a) and SW aI3 = OCV1w , Spa = OCpan6 , 6bap,=80a) , STaI = ovapT . (2.4.9b) Note that x and B suffer arbitrary shifts as a consequence of a gauge transformation, and, in particular, can be gauged completely away; the gauge x = B = 0 is called Wess- Zumino gauge, and explicitly breaks supersymmetry. However, this gauge is useful since it reveals the physical content of the ,a multiplet. Examination of the components that remain reveals several peculiar features: There are two component gauge potentials V,/ and W,/ for only one gauge symmetry, 3 and there is a high dimension spin - field @9p,. These problems will be resolved below when we express F in terms of Ia. We can also find supersymmetric Lorentz gauges by fixing DaFa; such gauges are useful for quantization (see sec. 2.7). Furthermore, in three dimensions it is possible to choose a supersymmetric light-cone gauge F+ = 0. (In the abelian case the gauge trans- formation takes the simple form K = D,(i&++)-1 F+.) Eq. (2.4.14) below implies that in this gauge the superfield F++ also vanishes. The remaining components in this gauge are x_, V,__ V__, and A_, with V,, 0 and A,~4,_ a.3. Constraints To understand how the vector connection Fa can be expressed in terms of the spinor connection Fa, recall the (anti)commutation relations for the ordinary derivatives are:  2.4. Vector multiplet 21 [DM ,DN}TMNPDP . (2.4.10) For the covariant derivatives VA =(Va, Va,) the graded commutation relations can be written (from (2.4.2) and (2.4.5) we see that the torsion TABC is unmodified): [VA,VB}TABCVC-iFAB (2.4.11) The field strengths FAB are invariant (F'AB = FAB) due to the covariance of the deriva- tives VA. Observe that the field strengths are antihermitian matrices, FAB = - FBA, so that the symmetric field strength Fa1 is imaginary while the antisymmetric field strength Fag, is real. Examining a particular equation from (2.4.11), we find: {V,V } = 2iV - iFa = 2i&Bag + 2Fag - iFag . (2.4.12) The superfield F was introduced to covariantize the space-time derivative &,'. How- ever, it is clear that an alternative choice is F' = F- F since F is covariant (a field strength). The new covariant space-time derivative will then satisfy (we drop the primes) {V, , V4} = 2iVa, , (2.4.13) with the new space-time connection satisfying (after substituting in 2.4.12 the explicit forms VA =DA - iFA) a= 2 D(aF) . (2.4.14) Thus the conventional constraint Fag = 0 , (2.4.15) imposed on the system (2.4.11) has allowed the vector potential to be expressed in terms of the spinor potential. This solves both the problem of two gauge fields WC/ , V cand the problem of the higher spin and dimension components @h, , T l: The gauge fields are identified with each other (Way Vag), and the extra components are expressed as derivatives of familiar lower spin and dimension fields (see 2.4.7). The independent com- ponents that remain in Wess-Zumino gauge after the constraint is imposed are Va and  22 2. A TOY SUPERSPACE We stress the importance of the constraint (2.4.15) on the objects defined in (2.4.11). Unconstrained field strengths in general lead to reducible representations of supersymmetry (i.e., the spinor and vector potentials), and the constraints are needed to ensure irreducibility. a.4. Bianchi identities In ordinary field theories, the field strengths satisfy Bianchi identities because they are expressed in terms of the potentials; they are identities and carry no information. For gauge theories described by covariant derivatives, the Bianchi identities are just Jacobi identities: [V[A,[VB,Vc)}}O ,(2.4.16) (where [) is the graded antisymmetrization symbol, identical to the usual antisym- metrization symbol but with an extra factor of (-1) for each pair of interchanged fermionic indices). However, once we impose constraints such as (2.4.13,15) on some of the field strengths, the Bianchi identities imply constraints on other field strengths. For example, the identity 0 =[ Va , { V, , Vy } ] + [ V, , { Vy , Va } ] + [ V, , { Va , V, } ] 1 [V(aV/ , V1)}(2.4.17) gives (using the constraint (2.4.13,15)) 0 [V(a , v,3)]=- i F(a,/3 . (2.4.18) Thus the totally symmetric part of F vanishes. In general, we can decompose F into irreducible representations of the Lorentz group: Fa_ =1F(a,y) - 1Ca FE s (2.4.19) (where indices between I...| , e.g., in this case o, are not included in the symmetriza- tion). Hence the only remaining piece is: Fa~gy = iCa, W,) , (2.4.20a) where we introduce the superfield strength Wa. We can compute F a , in terms of l'a  2.4. Vector multiplet 23 and find Wa = 2De Da P . (2.4.20b) The superfield Wa is the only independent gauge invariant field strength, and is constrained by DaWa = 0, which follows from the Bianchi identity (2.4.16). This implies that only one Lorentz component of Wa is independent. The field strength 1 describes the physical degrees of freedom: one helicity - and one helicity 1 mode. Thus W. is a suitable object for constructing an action. Indeed, if we start with S gf dx d2 W2 f dxd2 (DDaP3)2 , (2.4.21) we can compute the component action S gfd3xD2W2I2f dx[ W" D2Wa - (D"W ) (DaWp) I fdx[A_2faij .(2.4.22) Here (cf. 2.4.7) Aa = W a while f a/3= DaW/31l= D13Wa is the spinor form of the usual field strength Faj?"| = (&air?" - ?rya||) = 68(a(? f113) - i - [& DFra) - ibD( a )]| . (2.4.23) To derive the above component action we have used the Bianchi identity DaWa,= 0, and its consequence D2W a = ij/W3. a.5. Matter couplings We now examine the component Lagrangian describing the coupling to a complex scalar multiplet. We could start with  24 2. A TOY SUPERSPACE - -fd3xD2 [(Da + i)1] [(Da - i7)a>] , (2.4.24) and work out the Lagrangian in terms of components defined by projection. However, a more efficient procedure, which leads to physically equivalent results, is to define covari- ant components of 1 by covariant projection A = (x,O)| F = V24D(x, 8)| (2.4.25) These components are not equal to the ordinary ones but can be obtained by a (gauge- field dependent) field redefinition and provide an equally valid description of the theory. We can also use J d3x d2O -fd3x D2| =fd3xV2 , (2.4.26) when acting on an invariant and hence S dx V2[V2] -d d3X [V2V24C + VDaVa24 + (v2)2~] d3x [FF + /"(ioj + Vj)@y + (i >aAaA + h. c.) + A(&ay - i Va3)2 A]. (2.4.27) We have used the commutation relations of the covariant derivatives and in particular VaV2 = iV jV3 + iWa , V2Va - - iVcjV,3 - 2iWa , (V2)2 L= - jiWVa, where LIis the covariant d'Alembertian (covariantized with l'at3). b. Nonabelian case We now briefly consider the nonabelian case: For a multiplet of scalar superfields transforming as I' = eiK @, where K =KIT, and Ti are generators of the Lie algebra, we introduce covariant spinor derivatives Va precisely as for the abelian case (2.4.2). We define ja =FQi Ti so that  2.4. Vector multiplet 25 Va= Da - Fa = Da - iFe T .(2.4.28) The spinor connection now transforms as oFa=VaK=DK-i[F,,K] , (2.4.29) leaving (2.4.4) unmodified. The vector connection is again constrained by requiring Fag = 0; in other words, we have Va _=-({fVc, ,V/3} , (2.4.30a) F2 = - iZ- [D FO)g f - Fa, F}]1 . (2.4.30b) The form of the action (2.4.21) is unmodified (except that we must also take a trace over group indices). The constraint (2.4.30) implies that the Bianchi identities have nontriv- ial consequences, and allows us to "solve" (2.4.17) for the nonabelian case as in (2.4.18,19,20a). Thus, we obtain [Vc, , V ] = Ca(i3W,) (2.4.31a) in terms of the nonabelian form of the covariant field strength W: W =D DaFj-([F,DF[] - [F{F,,Fa}1 . (2.4.31b) The field strength transforms covariantly: W' = eiKW ae-iK. The remaining Bianchi identity is [{Va ,V },V,,,] - { V(a , [Vp) ,V ] } = 0 . (2.4.32a) Contracting indices we find [{V , V/3}, V a] = {V C, [V's, Va]}. However, [{V", V'}, V ] = 2i[V/ , V a] = 0 and hence, using (2.4.31a), 0 = {V(Ca, [vp) , Va ]} = -6{ V", Wa } . (2.4.32b) The full implication of the Bianchi identities is thus: { Va, V4 2iVa, (2.4.33a) [ Va , 4 CatjjWy) , { V" , Wa } =0 (2.4.33b) [ Vag, V7" ] = iot>( fAQ)) , f o ={ V~a , W,) } . (2.4.33c)  26 2. A TOY SUPERSPACE The components of the multiplet can be defined in analogy to (2.4.7) by projec- tions of F: Xa=Fa , B=}DTF 2 (2.4.34) Va =-Fop| , Aa = Wa| c. Gauge invariant masses A curious feature which this theory has, and which makes it rather different from four dimensional Yang-Mills theory, is the existence of a gauge-invariant mass term: In the abelian case the Bianchi identity D(Wa = 0 can be used to prove the invariance of Sm A1fd3xd2[ mFWc] (2.4.35) In components this action contains the usual gauge invariant mass term for three-dimen- sional electrodynamics: mfd3x V/Ua Vj -?= m fd3x V/fa , (2.4.36) which is gauge invariant as a consequence of the usual component Bianchi identity O& 3f = 0. The superfield equations which result from (2.4.21,35) are: i&a/ W/ + m Wa = 0 , (2.4.37) which describes an irreducible multiplet of mass m. The Bianchi identity DaWa =0 implies that only one Lorentz component of W is independent. For the nonabelian case, the mass term is somewhat more complicated because the field strength W is covariant rather than invariant: Sm =tr 1fd3x d20 ~m(FaWa + ~{F" , F# } Dia + 12{ F4, F0 } { F,,F0})  2.4. Vector multiplet 27 t 1 f d3X d2Ol2 m7(W l- p/3F ])(2.4.38) The field equations, however, are the covariantizations of (2.4.37): iV a /3W/3+mWav=0O.(2.4.39)  28 2. A TOY SUPERSPACE 2.5. Other global gauge multiplets a. Superforms: general case The gauge multiplets discussed in the last section may be described completely in terms of geometric quantities. The gauge potentials FA = (F,, Fa0) which covariantize the derivatives DA with respect to local phase rotations of the matter superfields consti- tute a super 1-form. We define super p-forms as tensors with p covariant supervector indices (i.e., supervector subscripts) that have total graded antisymmetry with respect to these indices (i.e., are symmetric in any pair of spinor indices, antisymmetric in a vector pair or in a mixed pair). For example, the field strength FAB = (Fad, Fa,y, Fa0'-) con- stitutes a super 2-form. In terms of supervector notation the gauge transformation for FA (from (2.4.3) and (2.4.5)) takes the form IA = DAK . (2.5.1) The field strength defined in (2.3.6) when expressed in terms of the gauge potential can be written as FAB - D [AIB) - TAB CC (2.5.2) The gauge transformation law certainly takes the familiar form, but even in the abelian case, the field strength has an unfamiliar nonderivative term. One way to understand how this term arises is to make a change of basis for the components of a supervector. We can expand DA in terms of partial derivatives by introducing a matrix, EAM, such that DA=EAMOM, M ( pu,&j) Fr ui 1!jQ(JIv) 1 E AM [].(2.5.3) This matrix is the flat vielbein; its inverse is  2.5. Other global gauge multiplets 29 6 a -1 jQ(a /i3~) EMA = (2.5.4) 01- (" / 3) If we define FM by FA = EAMFM, then oFM OM K . (2.5.5) Similarly, if we define FMN by FAB ()A(B+N)EB N EAM FMN , (2.5.6a) then FMN &[MFN) . (2.5.6b) (In the Grassmann parity factor (-)A(B+N) the superscripts A , B , and N are equal to one when these indices refer to spinorial indices and zero otherwise.) We thus see that the nonderivative term in the field strength is absent when the components of this supertensor are referred to a different coordinate basis. Furthermore, in this basis the Bianchi identities take the simple form &[M FNP) - . (2.5.7) The generalization to higher-rank graded antisymmetric tensors (superforms) is now evident. There is a basis in which the gauge transformation, field strength, and Bianchi identities take the forms oM1..M ( [M1K M2 .Mp), M1...M (p-[ M1KM2...Mp+) 0 = &[M1FM2..M±+2) . (2.5.8) We simply multiply these by suitable powers of the flat vielbein and appropriate Grass- mann parity factors to obtain A1...A) D[AK A...Ap) - (p)!T[A1 A|BKBA...Ap)  30 2. A TOY SUPERSPACE F,.1.Di 1A+1- __TA B Al.. A~l P! [A 2 ..A - 2 p- 1)! T[Al A2 B A . pl 0 ( DpFl! D[AFA2.-.Av+2) - Tp! [AlA2BFBA3..A+2) . (2.5.9) (The | 's indicate that all of the indices are graded antisymmetric except the B 's.) b. Super 2-form We now discuss in detail the case of a super 2-form gauge superfield FAB with gauge transformation bra,p = D (a K ) - 2i Ka/ oFaO'Y = DaK- -0 &3YKa ofaFy1-s = OapKy - y6KaI . (2.5.10) The field strength for FAB is a super 3-form: 1 F,,=- (D(aFy) + 2iF(a,/3y)) Fa,13- =D (aFI3)>yS + O yFai3 - 2i1ali,'Y Fa,13-6 =DaF13-,6 + &Scra,/N - &0/yFa,6E Fa13,-A &a/3F-yS,<+ &c~Faii,-Y6 + O yFc,a/ . (2.5.11) All of these equations are contained in the concise supervector notation in (2.5.9). The gauge superfield rA was subject to constraints that allowed one part (Fr,) to be expressed as a function of the remaining part. This is a general feature of supersym- metric gauge theories; constraints are needed to ensure irreducibility. For the tensor gauge multiplet we impose the constraints Fa,,,y =O , Fa,,,7 = i % og C G- T a,j7G , (2.5.12) which, as we show below, allow us to express all covariant quantities in terms of the sin- gle real scalar superfield C. These constraints can be solved as follows: we first observe that in the field strengths Fe~ always appears in the combination Da Up y + 2i F(a~gy).  2.5. Other global gauge multiplets 31 Therefore, without changing the field strengths we can redefine Fra, by absorbing D(, F3) into it. Thus I, disappears from the field strengths which means it could be set to zero from the beginning (equivalently, we can make it zero by a gauge transforma- tion). The first constraint now implies that the totally symmetric part of F,,ya is zero and hence we can write r,,gy= i Ca(g GY) in terms of a spinor superfield Iy. The remaining equations and constraints can be used now to express F C,3 and the other field strengths in terms of Ia. We find a solution ra =0, F iy ZCap 4D-,y) G = - Dc4a . .(2.5.13) Thus the constraints allow FAB to be expressed in terms of a spinor superfield a. (The general solution of the constraints is a gauge transform (2.5.10) of (2.5.13).) The quantity G is by definition a field strength; hence the gauge variation of Ga must leave G invariant. This implies that the gauge variation of Ga must be (see (2.2.6)) 6a=1D/DA , (2.5.14) 2 where Ag is an arbitrary spinor gauge parameter. This gauge transformation is of course consistent with what remains of (2.5.10) after the gauge choice (2.5.13). We expect the physical degrees of freedom to appear in the (only independent) field strength G. Since this is a scalar superfield, it must describe a scalar and a spinor, and I1 (or FAB) provides a variant representation of the supersymmetry algebra nor- mally described by the scalar superfield k. In fact Go contains components with helici- 1n 1 1 ties 0' 2 ' 1 just like the vector multiplet, but now the - , 1 components are auxiliary fields. (Ia = # + OaA + O0eap - 602a). For Ia with canonical dimension (mass>i, on dimensional grounds the gauge invariant action must be given by S = - }d3Xd2 (DC)2 . (2.5.15) Written in this form we see that in terms of the components of C, the action has the  32 2. A TOY SUPERSPACE same form as in (2.3.2). The only differences arise because G is expressed in terms of Qa. We find that only the auxiliary field F is modified; it is replaced by a field F'. An explicit computation of this quantity yields F'= - D2D"a|' = io"Da4D,=|3 &"Va| , Vap 2 iD abp). (2.5.16) In place of F the divergence of a vector appears. To see that this vector field really is a gauge field, we compute its variation under the gauge transformation (2.5.14): Va, = 7(, [ D1) A, + D1 Al)] . (2.5.17) This is not the transformation of an ordinary gauge vector (see (2.4.9)), but rather that of a second-rank antisymmetric tensor (in three dimensions a second-rank antisymmetric tensor is the same Lorentz representation as a vector). This is the component gauge field that appears at lowest order in 0 in I'agg in eq. (2.5.13). A field of this type has no dynamics in three dimensions. c. Spinor gauge superfield Superforms are not the only gauge multiplets one can study, but the pattern for other cases is similar. In general, (nonvariant) supersymmetric gauge multiplets can be described by spinor superfields carrying additional internal-symmetry group indices. (In a particular case, the additional index can be a spinor index: see below.) Such super- fields contain component gauge fields and, as in the Yang-Mills case, their gauge trans- formations are determined by the 0 = 0 part of the superfield gauge parameter (cf. (2.4.9)). The gauge superfield thus takes the form of the component field with a vector index replaced by a spinor index, and the transformation law takes the form of the com- ponent transformation law with the vector derivative replaced by a spinor derivative. 3 For example, to describe a multiplet containing a spin - component gauge field, we introduce a spinor gauge superfield with an additional spinor group index: aQ" Do (2.5.18) The field strength has the same form as the vector multiplet field strength but with a spinor group index:  2.5. Other global gauge multiplets 33 W/ = D a1bf .(2.5.19) (We can, of course, introduce a supervector potential lM ' in exact analogy with the abelian vector multiplet. The field strength here simply has an additional spinor index. The constraints are exactly the same as for the vector multiplet, i.e., FaA? =0.) In three dimensions massless fields of spin greater than 1 have no dynamical degrees of freedom. The kinetic term for this multiplet is analogous to the mass term for the vector multiplet: S d 3xd20 W ba a/ . (2.5.20) .3 This action describes component fields which are all auxiliary: a spin - gauge field (ah, a vector, and a scalar, as can be verified by expanding in components. The invariance of the action in (2.5.20) is not manifest: It depends on the Bianchi identity DaWa/3= 0. The explicit appearance of the superfield Iag is a general feature of super- symmetric gauge theories; it is not always possible to write the superspace action for a gauge theory in terms of field strengths alone.  34 2. A TOY SUPERSPACE 2.6. Supergravity a. Supercoordinate transformations Supergravity, the supersymmetric generalization of gravity, is the gauge theory of the supertranslations. The global transformations with constant parameters c"", E gen- erated by P,, and Q are replaced by local ones parametrized by the supervector KM(x, 0) = (K", K"). For a scalar superfield I(x, 0) we define the transformation TI(z) - I'I(z) =eiK I(z)=eiK 4I(z) e-iK , (2.6.1) where K = KM iDM= K" i&,v + K" iD . (2.6.2) (To exhibit the global supersymmetry, it is convenient to write K in terms of D rather than Q (or &,). This amounts to a redefinition of K"). The second form of the transformation of T can be shown to be equivalent to the first by comparing terms in a power series expansion of the two forms and noting that iK TI= [iK, T]. It is easy to see that (2.6.1) is a general coordinate transformation in superspace: eiKT-(zeiK =T -iKe-KK)i; defining z'=e-iKzeiK, (2.6.1) becomes '(z') =_T(z). We may expect, by analogy to the Yang-Mills case, to introduce a gauge superfield HaM with (linearized) transformation laws (HaM = D. KM(2.6.3) (we introduce Ha M as well, but a constraint will relate it to HaM) and define covariant derivatives by analogy to (2.4.28): EA = DA + HAM DM = EAM DM (2.6.4) EAM is the vielbein. The potentials Ha", H a" have a large number of components among which we identify, according to the discussion following equation (2.5.17), a sec- ond-rank tensor (the "dreibein", minus its flat-space part) describing the graviton and a 3 spin - field describing the gravitino, whose gauge parameters are the 0 0 parts of the 2 vector and spinor gauge superparameters KM|. Other components will describe gauge degrees of freedom and auxiliary fields.  2.6. Supergravity 35 b. Lorentz transformations The local supertranslations introduced so far include Lorentz transformations of a scalar superfield, acting on the coordinates zM - (x', 0). To define their action on spinor superfields it is necessary to introduce the concept of tangent space and local frames attached at each point zM and local Lorentz transformations acting on the indices of such superfields 1,(...(ZM). (In chapter 5 we discuss the reasons for this pro- cedure.) The enlarged full local group is defined by W ()...(x, 0) -+ 'a,1...(x, 0) = eiK -(x,0) eiK , (2.6.5) where now K = KM iDM + K)o iMj" . (2.6.6) Here the superfield K)1 parametrizes the local Lorentz transformations and the Lorentz generators Mg" act on each tangent space index as indicated by [X7 M/, Wa'] = X)a/ , (2.6.7) for arbitrary Xj7. Ma/ is symmetric, i.e., M)o is traceless (which makes it equivalent to a vector in three dimensions). Thus, X,) is an element of the Lorentz algebra SL(2, R) (i.e., SO(2, 1)). Therefore, the parameter matrix K)1 is also traceless. From now on we must distinguish tangent space and world indices; to do this, we denote the former by letters from the beginning of the alphabet, and the latter by letters from the middle of the alphabet. By definition, the former transform with K)1 whereas the latter transform with KM. c. Covariant derivatives Having introduced local Lorentz transformations acting on spinor indices, we now define covariant spinor derivatives by Va =EM DM + GIac7 M/j , (2.6.8) as well as vector derivatives Vag. However, just as in the Yang-Mills case, we impose a conventional constraint that defines Va i{Va, V,} , (2.6.9)  36 2. A TOY SUPERSPACE The connection coefficients 'A, which appear in VA = EAM DM + 'A/-' M,/ , (2.6.10) and act as gauge fields for the Lorentz group, will be determined in terms of HaM by imposing further suitable constraints. The covariant derivatives transform by VA > VA'= eiK VA e-iK . (2.6.11a) All fields W... (as opposed to the operator V) transform as '... = eK - iK ... (2.6. b) when all indices are flat (tangent space); we always choose to use flat indices. We can use the vielbein EAM (and its inverse EMA) to convert between world and tangent space indices. For example, if TM is a world supervector, PA = EAM M is a tangent space supervector. The superderivative EA = EAM DM is to be understood as a tangent space super- vector. On the other hand, DM transforms under the local translations (supercoordinate transformations), and this induces transformations of EAM with respect to its world index (in this case, M). We can exhibit this, and verify that (2.6.6) describes the famil- iar local Lorentz and general coordinate transformations, by considering the infinitesimal version of (2.6.11): oVA = [iK, VA] , (2.6.12) which implies (SEAM = EANDNKM - KNDN EAM - EANKP T PNM - KABEBM o4A,"y= EA Kj - KMDM A-, - KAB 4B,, - K, 41A + KjgA-,y VAK, - KMDM A - KABIB ,(2.6.13) where TMN' is the torsion of flat, global superspace (2.4.10), and Kalobai-' E -1L(V AfI , Q) , (2.6.40) in analogy to ordinary gravity. Thus, the action for the scalar multiplet described by eq. (2.3.5) takes the covariantized form S4)=Jd3x d20 E-1 [ - } (Va$)2 + 2 m@2 + 3 . (2.6.41) For vector gauge multiplets the simple prescription of replacing flat derivatives DA by gravitationally covariant ones VA is sufficient to convert global actions into local actions, if we include the Yang-Mills generators in the covariant derivatives, so that they are covariant with respect to both supergravity and super-Yang-Mills invariances. How- ever, such a procedure is not sufficient for more general gauge multiplets, and in particu- lar the superforms of sec. 2.5. On the other hand, it is possible to formulate all gauge theories within the superform framework, at least at the abelian level (which is all that is relevant for p-forms for p > 1). Additional terms due to the geometry of the space will automatically appear in the definitions of field strengths. Specifically, the curved- space formulation of superforms is obtained as follows: The definitions (2.5.8) hold in arbitrary superspaces, independent of any metric structure. Converting (2.5.8) to a tan- gent-space basis with the curved space EAM, we obtain equations that differ from (2.5.9) only by the replacement of the flat-space covariant derivatives DA with the curved-space ones VA. To illustrate this, let us return to the abelian vector multiplet, now in the presence of supergravity. The field strength for the vector multiplet is a 2-form:  44 2. A TOY SUPERSPACE Fad~ VaF,3 + Vj3Ia -2iFa, F a / V'- _ V -777a3- Ta/'EF F V4Tyj-Vyla-TaEE (2.6.42) We again impose the constraint F( = 0, which implies Fa,=iC W , W 2=VV1F7+ RFa ; (2.6.43) where we have used (2.6.30) substituted into (2.4.33). Comparing this to the global field strength defined in (2.4.20), we see that a new term proportional to R appears. The extra term in W. is necessary for gauge invariance due to the identity V VV = i [Va, Vag]. In the global limit the commutator vanishes, but in the local case it gives a contribution that is precisely canceled by the contribution of the R term. These results can also be obtained by use of derivatives that are covariant with respect to both supergravity and super-Yang-Mills. We turn now to the action for the gauge fields of local supersymmetry. We expect to construct it out of the field strengths GC , and R. By dimensional analysis (noting that r has dimensions (mass) - in three dimensions), we deduce for the Poincard super- gravity action the supersymmetric generalization of the Einstein-Hilbert action: SSG 2 d3xd2oE-1 R . (2.6.44) We can check that (2.6.44) leads to the correct component action as follows: d20 E-1 R ~ V2R ~3 -r (see (2.6.33)), and thus the gravitational part of the action is correct. We can also add a supersymmetric cosmological term Scosmd3x d20 E1 , (2.6.45) which leads to an equations of motion R =A , GCp = 0. The only solution to this equa- tion (in three dimensions) is empty anti-deSitter space: From (2.6.33), r =2A2 , Waj~s 0. Higher-derivative actions are possible by using other functions of GCp and R. For example, the analog of the gauge-invariant mass term for the Yang-Mills multiplet exists  2.6. Supergravity 45 here and is obtained by the replacements in (2.4.38) (along with, of course, f d3x d20 -- f d 3x d20 E-1): FA2T Z- 4'Aj3- M Wjl T2 +Gaj'' M /j+ 3 (V/3 R)iMcj1 (2.6.46) This gives Il -mass= f d3x d20 E-14Icj b( G(II + 2 a'b 1 V 6,114b) (2.6.47)  46 2. A TOY SUPERSPACE 2.7. Quantum superspace a. Scalar multiplet In this section we discuss the derivation of the Feynman rules for three-dimen- sional superfield perturbation theory. Since the starting point, the superfield action, is so much like a component (ordinary field theory) action, it is possible to read off the rules for doing Feynman supergraphs almost by inspection. However, as an introduction to the four-dimensional case we use the full machinery of the functional integral. After deriving the rules we apply them to some one-loop graphs. The manipulations that we perform on the graphs are typical and illustrate the manner in which superfields handle the cancellations and other simplifications due to supersymmetry. For more details, we refer the reader to the four-dimensional discussion in chapter 6. a.1. General formalism The Feynman rules for the scalar superfield can be read directly from the Lagrangian: The propagator is defined by the quadratic terms, and the vertices by the interactions. The propagator is an operator in both x and 0 space, and at the vertices we integrate over both x and 0. By Fourier transformation we change the x integration to loop-momentum integration, but we leave the 0 integration alone. (0 can also be Fourier transformed, but this causes little change in the rules: see sec. 6.3.) We now derive the rules from the functional integral. We begin by considering the generating functional for the massive scalar superfield 1 with arbitrary self-interaction Z( J )= 1D exp doxd20 [2 D2 + 2m2 + f (b) + JC ] Z() fD rx ~x2 f1DQF exp [So(Q) + SINT(~ =exp [ SINT ( j) ]1D exp [f (D2 +m) + JQ] . (2.7.1) In the usual fashion we complete the square, do the (functional) Gaussian integral over @, and obtain  2.7. Quantum superspace 47 Z(J) = exp [SINT() exp[_fd3xd2OJ J] (2.7.2) exp 2 D2+m Using eq.(2.2.6) we can write 1 D2 - m D2+m D-m2 (Note D2 behaves just as 0 in conventional field theory.) We obtain, in momentum space, the following Feynman rules: Propagator: ___ ____ dok 1J 6'D2 -m I d20 - ~,)J(-k,O) 6J(k, e) 6J(-k, e') 1(2c)3 de2 J k2 D+ m2 J- m2 flo-0') .(2.7.4) Vertices: An interaction term, e.g. f d3xd2O 4DoD# -.-.- , gives a vertex with 4 lines leaving it, with the appropriate operators Da, D3, etc. acting on the corresponding lines, and an integral over d20. The operators Da which appear in the propagators, or are coming from a vertex and act on a specific propagator with momentum k leaving that vertex, depend on that momentum: Da +O0ka .(2.7.5) In addition we have loop-momentum integrals to perform. In general we find it convenient to calculate the effective action. It is obtained in standard fashion by a Legendre transformation on the generating functional for con- nected supergraphs W (J) and it consists of a sum of one-particle-irreducible contribu- tions obtained by amputating external line propagators, replacing them by external field factors I(p, O ), and integrating over pi , Q . Therefore, it will have the form x (27r)36(p3 H Hd2O Qpropagators Qvertices (2.7.6) loops internal vertices  48 2. A TOY SUPERSPACE As we have already mentioned, all of this can be read directly from the action, by anal- ogy with the derivation of the usual Feynman rules. The integrand in the effective action is a priori a nonlocal function of the x's (non- polynomial in the p's) and of the 01, - - - 0n . However, we can manipulate the 0-integra- tions so as to exhibit it explicitly as a functional of the <'s all evaluated at a single com- mon 0 as follows: A general multiloop integral consists of vertices labeled i, i + 1, con- nected by propagators which contain factors o(0 - 02+1) with operators D, acting on them. Consider a particular loop in the diagram and examine one line of that loop. The factors of D can be combined by using the result ("transfer" rule): Da(02, k)o(0z - 02+1) - Da(O+1, -k)6(02 - 02+1) , (2.7.7) as well as the rules of eq.(2.2.6), after which we have at most two factors of D acting at one end of the line. At the vertex where this end is attached these D's can be integrated by parts onto the other lines (or external fields) using the Leibnitz rule (and some care with minus signs since the D's anticommute). Then the particular 6-function no longer has any derivatives acting on it and can be used to do the 02 integration, thus effectively "shrinking" the (2, Oi+1) line to a point in 0-space. We can repeat this procedure on each line of the loop, integrating by parts one at a time and shrinking. This will gener- ate a sum of terms, from the integration by parts. The procedure stops when in each term we are left with exactly two lines, one with b(01 - 0m) which is free of any deriva- tives, and one with 5(m - 01) which may carry zero, one, or two derivatives. We now use the rules (which follow from the definition b2(0) =- 02) 62(01i_2m)D(0m1) =0) , 62 1- ~2 ( 0-61)= 0, (S2(01 - Dm) D262 (m-01 2 (01 - 8m). (2.7.8) Thus, in those terms where we are left with no D or one D we get zero, while in the terms in which we have a D2 acting on one of the s-functions, multiplied by the other s-function, we use the above result. We are left with the single s-function, which we can use to do one more 0 integration, thus finally reducing the 0-space loop to a point.  2.7. Quantum superspace 49 The procedure can be repeated loop by loop, until the whole multiloop diagram has been reduced to one point in 0-space, giving a contribution to the effective action F(Q) f d3p1... d3pd2 J (27)3n 2 X G(pl, ... pn) 4b(pl 8) ... DO'4b(pi 8) ... D244 (p , 8) ... , (2.7.9) where G is obtained by doing ordinary loop-momentum integrals, with some momentum factors in the numerators coming from anticommutators of D's arising in the previous manipulation. a.2. Examples We give now two examples, in a massless model with 3 interactions, to show how the 0 manipulation works. The first one is the calculation of a self-energy correction represented by the graph in Fig. 2.7.1 k 0(-p, 0') k+p Fig. 2.7.1 f dop dok D26(0-0') D26('-0) 172 ]=(2)3 d20d20'1(-p, e') (p, 0) .(27)3D(p ) The terms involving 0 can be manipulated as follows, using integration by parts: D26( - 0') D26(0' -0) 1(p,0) - ID~o(0 - 0') [DaD26(0' - 0)(p,0)+ D26(0' - 0)DaI (p,0)] (2.7.10)  50 2. A TOY SUPERSPACE = 6(0 - 0')[([D 2 )26(0' - 0)4(p, 0) + D"D 26(o' -)Da@b(p,0) + D 26(o'-O)D2I(p,0)] . (2.7.11) However, using (D2)2 = - k2 and DaD2 = k"#D, we see that according to the rules in eq. (2.7.8) only the last term contributes. We find /d3p d2f dk1 2= (2)3d2OI (-pO0)D21(p, ) .()3 k2(k+p)2 (2.7.12) Doing the integration by parts explicitly can become rather tedious and it is preferable to perform it by indicating D's and moving them directly on the graphs. We show this in Fig. 2.7.2: D2 Da+D2 D2 D2 D2 D2 Da D2 Fig. 2.7.2 Only the last diagram gives a contribution. One further rule is useful in this procedure: In general, after integration by parts, various D-factors end up in different places in the final expression and one has to worry about minus signs introduced in moving them past each other. The overall sign can be fixed at the end by realizing that we start with a particular ordering of the D's and we can examine what happened to this ordering at the end of the calculation. For example, we may start with an expression such as D2 ---D2---D2--- = } DoD- - -}DID,---} D7D,--- and end up with D". - - -' D. --"-D7- - - D. --"-D - - - D13-"-"- where the various D's act on different fields. The overall sign can obviously be determined by just counting the number of transpositions. For example, in the case above we would end up with a plus sign. Note that this rule also applies if factors such as kay arise, provided one pays attention to the manner in which they were produced (e.g., at which end of the line were the D's acting? Did it come from DaDa or from DaD?). Our second example is the three-point diagram below, which we manipulate as shown in the sequence of Fig. 2.7.3:  2.7. Quantum superspace 51 D2 D2 AD2 D Da D2DaD2 D+1D2 DO // D2 + D2 + ]D2 Da Fig. 2.7.3 At the first stage we have integrated by parts the D2 off the bottom line and immedi- ately replaced (D2)2 by = - k2 . At the second stage we have integrated by parts the D2 off the right side, but kept only those terms that are not zero: The bottom line has already been shrunk to a point by the corresponding 8-function (but we need not indi- cate this explicitly; any line that has no D's on it can be considered as having been shrunk) and in the end we keep only terms with exactly two factors of D in the loop. For the middle diagram this means using DaD2D13 =Dak3yD7= - k gaD2+ a term with no D's which may be dropped. The integrand in the effective action can be written then as f d3k 1 3k 1 I 2(p3,8)[- MP1, 6)%P2, )k 2 (21r)3 k2(k +p1)2(k - p3)2 - DaI(pi, )DihI(p2, )kap + D21(pI, )D21 (p2, 0)] , (2.7.13) and only the k-momentum integral remains to be done. In general, the loop-momentum integrals may have to be regularized. The proce- dure we use, which is guaranteed to preserve supersymmetry, is to do all the D-manipu- lations first, until we reduce the effective action to an integral over a single 0 of an expression that is a product of superfields, and therefore manifestly supersymmetric. The remaining loop-momentum integrals may then be regularized in any manner  52 2. A TOY SUPERSPACE whatsoever, e.g., by using dimensional regularization. We shall discuss the issues involved in this kind of regularization in sec. 6.6. An alternative procedure, somewhat cumbersome in its application but better understood, is supersymmetric Pauli-Villars regularization. In three dimensions this is applicable even to gauge theories, since gauge invariant mass terms exist. b. Vector multiplet Nothing new is encountered in the derivation or application of the Feynman rules. However, the derivation must be preceded by quantization, i.e., introduction of gauge-fixing terms and Faddeev-Popov ghosts, which we now discuss. We begin with the classical action SC = trfd2xd2 W 2 . (2.7.14) g j The gauge invariance is of, = VaK and, by direct analogy with the ordinary Yang-Mills case, we can choose the gauge-fixing function F - Dc a . We use an averaging proce- dure which leads to a gauge-fixing term without dimensional parameters, FD2F, and obtain, for the quadratic action, S=2 = tr f d3xd2O [ ( DAD 17>) (1 D7DaFy) -11 f32 1 1F)D2(} DD ) tr f d3xd2[ +(1 +)FDF + -(1- )FTai D2 ] . (2.7.15) 2 (2g J 2 a 2 a Various choices of the gauge parameter a are possible: The choice a - 1 gives the kinetic term1 Fti jD2F , while the choice a = 1 gives - 1 FE, which results in the 2 2 simplest propagator. The Faddeev-Popov action is simply SFP 32 '(,6 } D!Va c(x,O) , (2.7.16) with two scalar multiplet ghosts. (Note that in a background-field formulation of the  2.7. Quantum superspace 53 theory, similar to the one we discuss in sec. 6.5, one would replace the operator D2 in the gauge fixing term by the background-covariant operator V2, and this would give rise to a third, Nielsen-Kallosh, ghost as well.) The Feynman rules are now straightforward to obtain. The ghost propagator is conventional, following from the quadratic ghost kinetic term c'D2c, while the gauge field propagator is 62(o - 0') .(2.7.17) Vertices can be read off from the interaction terms. The gauge-field self-interactions (in the nonabelian case) are g2-LINT = - D , [ F , Dyre ] - 1D7Dar [ F/ , { F/ , Fa}] - [ F7, Drys] [ F/', D,3fa ] + 2 [ F7, D,a ] [ F/3,{ F/3, F }] + ; [FY{F7,{F1}] [1/,{13,}Fa}] , (2.7.18) those of the ghosts are 2LINT = - i 2 c'Da [Fa , c] , (2.7.19) 2 while those of a complex scalar field are 2LNT (V2 - D2) 1 case is called extended supersymmetry. Central charges can arise only in the case of extended (N > 1) supersymmetry. The supersymmetry generators Q act as "square roots" of the momentum generators P. d. Positivity of the energy A direct consequence of the algebra is the positivity of the energy in supersym- metric theories. The simplest way to understand this result is to note that the total energy E can be written as 6 - 1 -P+ _P-) ! pa 1 " 2 (3.2.9)  3.2. The supersymmetry groups 65 Since P can be obtained from the anticommutator of spinor charges, we have =- 2N {Qa , Qa} = {Qa , (Qaa)t} (3.2.10) (we use Q = - (Qaa)t). The right hand side of eq. (3.2.10) is manifestly non-negative: For any operator A and any state |@9>, < @|{A , AI}|@b > = (< @|lAln >< n|Atf| > + < @|lA fn >< n|A l@ >) Hence, E is also nonnegative. Further, if supersymmetry is unbroken, Q must annihilate the vacuum; in this case, (3.2.10) leads to the conclusion that the vacuum energy van- ishes. Although this argument is formal, it can be made more precise; indeed, it is possi- ble to characterize supersymmetric theories by the condition that the vacuum energy vanish. e. Superconformal algebra For massless theories, Haag, Jopuszanski, and Sohnius showed what form exten- sions of the conformal group can take: The generators of the superconformal groups consist of the generators of the conformal group (Pa, Jag , J , K , A) (these are the generators of the Poincard algebra, the special conformal boost generators, and the dila- tion generator), 2N spinor generators (Qaa, S"") (and their hermitian conjugates -Q., -5a with a total of 8N components), and N2 further bosonic charges (A, Tab) where T a= 0. The algebra has structure constants defined by the following (anti)com- mutators: {Qaa , Q b} =a bP_ , {s"a, S-} =.a KEa , (3.2.12a) {Qaa , Sbo} = iob(J/1 +loi3A) - 6,a/6ab(1 - +)A + 265jTab (3.2.12b) [Ta6 , S'7] - 1 (e6a'3b7 - ogbS'7) ,(3.2.12c) 2 Na  66 3. REPRESENTATIONS OF SUPERSYMMETRY [A ,S'7]= 1S'7- ,[0 ,SC'7]_=- 2i1S'7,(3.2.12d) 2 2 [JciVSC'1] =_- oSa|' ,) [Paz, SC'7 - - J'Qc. , (3.2.12e) [Ta bQcyy] = - (ocbQa, - a Qcy) , (3.2.12f) 1 1 [A ,Q c] =-21'7 O 7 O' '(3.2.12g) [J/ , Qcyl -2 io Qca) , [Kod , Qy] = 5 c , (3.2.12h) [Tab , Tcd = (oadTcb - ocbTad) , (3.2.12i) 2 [A , Ka] =- iKaa , [A,Pa1 =iPa= , (3.2.12j) [JcjK7] - }ia , [Jci, P z] = } ioP - , (3.2.12k) [Jag, J7 ] =- ib(- 'J) ,(3.2.121) [PaK, K' ] =i+J + o )J) + ofoA) =(Ja + oakA) . (3.2.12m) All other (anti)commutators vanish or are found by hermitian conjugation. The superconformal algebra contains the super-Poincard algebra as a subalgebra; however, in the superconformal case, there are no central charges (this is a direct conse- quence of the Jacobi identities). In the same way that the supersymmetry generators Q act as "square roots" of the translation generators P, the S-supersymmetry generators S act as "square roots" of the special conformal generators K. The new bosonic charges A and Tab generate phase rotations of the spinors (axial or ! rotations) and SU(N) trans- formations respectively (all but the SO(N) subgroup of the SU(N) is axial). For N =4, the axial charge A drops out of the {Q , S} anticommutator whereas the [Q , A] and [S ,A] commutators are N independent. The normalization of A is chosen such that Tab + o5ab A generates U(N) (e.g., [Tab + oab A ,c,] 2 c cbay)-  3.2. The supersymmetry groups 67 f. Super-deSitter algebra Finally, we turn to the supersymmetric extension of the deSitter algebra. The generators of this algebra are the generators of the deSitter algebra (P, J, J), spinorial generators (Q, Q), and- N(N - 1) bosonic SO(N) charges Tab= - Tba. They can be 2 constructed out of the superconformal algebra (just as the super-Poincard algebra is a subalgebra of the superconformal algebra, so is the super-deSitter algebra). We can define the generators of the super-deSitter algebra as the following linear combinations of the superconformal generators: P =& P +|A|2 Ka , Qac =Qaa+ A abS bcv ac/ = Ja , Tab = c[b Ta" , (3.2.13) where, since we break SU(N) to SO(N), we have lowered the isospin indices of the superconformal generators with a kronecker delta. (We could also formally maintain SU(N) invariance by using instead Aab satisfying Aab =ba and AacA ~Cab, with Aab Ahab in an appropriate SU(N) frame.) Thus we find the following algebra: {Qac , Q6I} = 2A(-ioabJap + CacTab) , (3.2.14a) {Q , b}l=X bjP .aa(3.2.14b) Q C [a ab /3 ,(3.2. 14c) [Jo=,Py] =1 iC P ) , (3.2.14e) Pa, Pg] =- i2|A l2(C.gJcvg + CcvpJf ) , (3.2.14f) [JP , Y] = - 1 i 7 .(3.2.14g) [Tab, Qcy] = -octaUb , (3.2. 14h)  68 3. REPRESENTATIONS OF SUPERSYMMETRY [Tab , Tcdl = (ob[eTd a - oa[c Tdlb) (3.2.14i) This algebra, in contrast to the superconformal and super-Poincard cases, depends on a dimensional constant A. Physically, |Al2 is the curvature of the deSitter space. (Actu- ally, the sign is such that the relevant space is the space of constant negative curvature, or anti-deSitter space. This is a consequence of supersymmetry: The algebra deter- mines the relative sign in the combination P +|Al2K above.)  3.3. Representations of supersymmetry 69 3.3. Representations of supersymmetry a. Particle representations Before discussing field representations of supersymmetry, we study the particle content of Poincard supersymmetric theories. We analyze representations of the super- symmetry group in terms of representations of its Poincard subgroup. Because P2 is a Casimir operator of supersymmetry (it commutes with all the generators), all elements of a given irreducible representation will have the same mass. a.1. Massless representations We first consider massless representations. We then can choose a Lorentz frame where the only nonvanishing component of the momentum pa is p+. In this frame the anticommutation relations of the supersymmetry generators are {fQav , Qb±} = 0 , {Qav , Q> } = pe ab, {Qa-,Q=b-}0 , {Qa_,Q b}'=0 {fQa±, Qb-} =-0 , {Q a± , Qab} = 0 . (3.3.1) Since the anticommutator of Qa_ with its hermitian conjugate vanishes, Qa_ must van- ish identically on all physical states: From (3.2.11) we have the result that 0 = < @|I{A , Af}|@ >= (I < n|Af l@ > 12 + < n| Al@ > 12) 7z -< n|Al~ > = = 0 . (3.3.2) On the other hand, Qav and its hermitian conjugate satisfy the standard anticommuta- tion relations for annihilation and creation operators, up to normalization factors (with the exception of the case p,=0, which in this frame means pa=0 and describes the physical vacuum). We can thus consider a state, the Clifford vacuum |C>, which is annihilated by all the annihilation operators Qav (or construct such a state from a given state by operating on it with a sufficient number of annihilation operators) and generate all other states by action of the creation operators Q". Since, as usual, an annihilation operator acting on any state produces another with one less creation operator acting on  70 3. REPRESENTATIONS OF SUPERSYMMETRY the Clifford vacuum, this set of states is closed under the action of the supersymmetry generators, and thus forms a representation of the supersymmetry algebra. Further- more, if the Clifford vacuum is an irreducible representation of the Poincard group, this set of states is an irreducible representation of the supersymmetry group, since any attempt to reduce the representation by imposing a constraint on a state (or a linear combination of states) would also constrain the Clifford vacuum (after applying an appropriate number of annihilation operators; see also sec. 3.8.a). The Clifford vacuum may also carry representations of isospin and other internal symmetry groups. The Clifford vacuum, being an irreducible representation of the Poincard group, is 1 also an eigenstate of helicity. In this frame, Qa" has helicity - -, thus determining the helicities of the other states in terms of that of the Clifford vacuum. (In general frames, 1 1 the helicity - 1 component of Q3 is the creation operator, and the helicity + - compo- 2 2 nent, which is the linearly independent Lorentz component of PaaQa vanishes: {pcQa ., P" Q b} = "P2P =0, since p2 = 0 in the massless case.) The representa- tions of the states under isospin are also determined from the transformation properties of the Clifford vacuum and the Q's: We take the tensor product of the Clifford vac- uum's representation with that of the creation operators (namely, that formed by multi- plying the representations of the individual operators and antisymmetrizing). As examples, we consider the cases of the massless scalar multiplet (N = 1, 2), super-Yang-Mills (N=1, ... , 4), and supergravity (N=1, ... , 8), defined by Clifford 1 vacua which are isoscalars and have helicity + -, +1, and +2, respectively. (In the 2 scalar and Yang-Mills cases, the states may carry a representation of a separate internal symmetry group.) The states are listed in Table 3.3.1. Each state is totally antisym- metric in the isospin indices, and thus, for a given N, states with more than N isospin 1 1 N indices vanish. The scalar multiplet contains helicities ( ,... , --aN), super Yang-Mills N9 N contains heiite (1-.. ,1 , and supergravity contains helicities (2, . . . , 2 2 -). In addition, any representation of an internal symmetry group that commutes with super- symmetry (such as the gauge group of super Yang-Mills) carried by the Clifford vacuum is carried by all states (so in super Yang-Mills all states are in the adjoint representation of the gauge group). Thus the total number of states in a massless representation is 2Nk, where k is the number of states in the Clifford vacuum.  3.3. Representations of supersymmetry 71 helicity scalar multiplet super-Yang-Mills supergravity +2 =|C > +3/2 +1 =|C > a +1/2 b=|C >"abc -1/2 "< abc ,ab'de -1 0 is not allowed (just as for Z = 0 we never have P2 > 0). a.3. Casimir operators We can construct other Casimir operators than P2. We first define the supersym- metric generalization of the Pauli-Lubanski vector W - =i(P Jc)- PaJ ") - -[Qaa, Qag] , (3.3.4) where the last term is absent in the nonsupersymmetric case. This vector is not invari- ant under supersymmetry transformations, but satisfies W1 1- [ Q_]= PaQ/ [Wa,,Q1 1 P Q .(3.3.5) As a result, PgqWy commutes with Qa, and thus its square P2W2 _ I (P.- W)2 com- - 4 mutes with all the generators of the super-Poincard algebra and is a Casimir operator. In the massive case this Casimir operator defines a quantum number s, the superspin. The generalization of the nonsupersymmetric relation W2 =m2s(s + 1) is  3.3. Representations of supersymmetry 73 P2W2 - ( (P.W2= - m4s(s + 1) . (3.3.6) In the massless case, not only P2 = 0, but also PaaQaa = PoaQa.= 0, and hence P - W = P[aW = 0. However, using the generator A of the superconformal group (3.2.12), we can construct an object that commutes with Q and Q: Wa - APa. Thus we can define a quantum number A, the superhelicity, that generalizes helicity A0 (defined by Wa = AoPa): Wa - APa =APa . (3.3.7) We also can construct supersymmetry invariant generalizations of the axial genera- tor A and of the SU(N) generators: W P2A+ P"(Qaa[Q, Qa* Wab _ P2Tab + I pa"([Q Q - , Q%3] N Gab ' QC,]) . (3.3.8) In the massive case, the superchiral charge and the superisospin quantum numbers can then be defined as the usual Casimir operators of the modified group generators -m-2W5 , - m-2Wab. In the massless case, we define the operators W5 =P A+ [Q a 4 a T + ([Qav Qb l N a[Qc' QC.]) .(3.3.9) These commute with Q and Q when the condition PodQaa = 0 holds, which is precisely the massless case. Since PMW 5 = PW ab = 0, we can find matrix representations g5 , gab such that Ws =g5 Pc, Wabc= gab Pc . (3.3.10) The superchiral charge is g5, and superisospin quantum numbers can be defined from the traceless matrices gqb. All supersymmetrically invariant operators that we have con- structed can be reexpressed in terms of covariant derivatives defined in sec. 3.4.a; see sec 3.4.d.  74 3. REPRESENTATIONS OF SUPERSYMMETRY b. Representations on superfields We turn now to field (off-shell) representations of the supersymmetry algebras. These can be described in superspace, which is an extension of spacetime to include extra anticommuting coordinates. To discover the action of supersymmetry transforma- tions on superspace, we use the method of induced representations. We discuss only simple N = 1 supersymmetry for the moment. b.1. Superspace Ordinary spacetime can be defined as the coset space (Poincard group)/(Lorentz group). Similarly, global flat superspace can be defined as the coset space (super-Poincard group)/(Lorentz group): Its points are the orbits which the Lorentz group sweeps out in the super-Poincard group. Relative to some origin, this coset space can be parametrized as: i fxa/3 P .8 8" 0*Q h(x, 0, 0) =e( (3.3.11) where x, 0, 0 are the coordinates of superspace: x is the coordinate of spacetime, and 0, 0 are new fermionic spinor coordinates. The "hat" on P and Q indicates that they are abstract group generators, not to be confused with the differential operators P and Q used to represent them below. The statistics of 0, 0 are determined by those of Q, Q: {0,=} = {0,o} = {Q,=} = [0,x] = [0, P] = 0 , (3.3.12) etc., that is, 0, 0 are Grassmann parameters. b.2. Action of generators on superspace We define the action of the super-Poincard group on superspace by left multiplica- tion: h(x', 0', 0') = 1-h(x, 0, 0) mod SO(3, 1) (3.3.13) where y is a group element, and "mod SO(3, 1)" means that any terms involving Lorentz generators are to be pushed through to the right and then dropped. To find the action of the generators (J, P, Q) on superspace, we consider  3.3. Representations of supersymmetry 75 g ( e- ') , e P , e- (3.3.14) respectively. Using the Baker-Hausdorff theorem (eAeB eA+B+[AB] f [A, [A, B]] =[B, [A, B]] = 0) to rearrange the exponents, we find: J & J: x'" [ew]j [es]Wlj x , '"= []" ew]j0 , 0' / [j e] Q&Q: zX'= -Z -xii(c4 + E-0),0'=86+cE , 8' 8 +E-. (3.3.15) 2 Thus the generators are realized as coordinate transformations in superspace. The Lorentz group acts reducibly: Under its action the x's and 0's do not transform into each other. b.3. Action of generators on superfields To get representations of supersymmetry on physical fields, we consider superfields Wes... (x, 0, 0): (generalized) multispinor functions over superspace. Under the supersym- metry algebra they are defined to transform as coordinate scalars and Lorentz multi- spinors. They may also be in a matrix representation of an internal symmetry group. The simplest case is a scalar superfield, which transforms as: 1'(x', 0', 0') =1D(x, 0, 0) or, infinitesimally, 6 =V(z) - 1(z) =- zMM -(z). Using (3.3.13), we write the trans- formation as 64= - [(E"aQ + E-i) , 1] =[(E"Q, + E- Qg) , 4], etc. Hence, just as in the ordinary Poincard case, the generators Q, etc., are represented by differential opera- tors Q, etc.: J1 Ja i -(xjOQ> + 60(c3)) - iMap, where Map generates the matrix Lorentz transformations of the superfield 'II:  76 3. REPRESENTATIONS OF SUPERSYMMETRY 1 [MoVy.. 2 C(aT).... For future use, we write Q and Q as Q,= eue2 , QC-22 g '= e 2Ug el ,(3.3.17a) where U = 8i3 . . (3.3.17b) Finally, from the relation {Q, Q} = P, we conclude that the dimension of 0 and 0 is (m)-. b.4. Extended supersymmetry We now generalize to extended Poincard supersymmetry. In principle, the results we present could be derived by methods similar to the above, or by using a systematic differential geometry procedure. In practice the simplest procedure is to start with the N = 1 Poincard results and generalize them by dimensional analysis and U(N) symme- try. For general N, superspace has coordinates zA -= (x , aa , Oaa ) ( , O , e ). Superfields 1T,/3 ...ab... (x, 0,06) transform as multispinors and isospinors, and as coordinate scalars. Including central charges, the super-Poincard generators act on superfields as the following differential operators: Qav Z(a - -ai. - 2(3.3.18a) 2 a# 2 Q"a =i("; - 2"iOg(- }(2b "Z) , (3.3.18b) Ja = - i (x(2A + 0a(a A )) - i/ , (3.3.18c) J -i-(73 + Oa(&6 ) - i , (3.3.18d) Pa ~g. (3.3.18e) Central charges are discussed in section 4.6.  3.3. Representations of supersymmetry 77 b.5. CPT in superspace Poincard supersymmetry is compatible with the discrete invariances CP (charge conjugation x parity) and T (time reversal). We begin by reviewing C, P, and T in ordinary spacetime. We describe the transformations as acting on c-number fields, i.e., we use the functional integral formalism, rather than acting on q-number fields or Hilbert space states. Under a reflection with respect to an arbitrary (but not lightlike) axis ua, (u = u, u2 = 1) the coordinates transform as x'a = R(u)xa _- -2 a x = X" - u-2 uau - X , R2 = I (3.3.19) (u - x changes sign, while the components of x orthogonal to u are unchanged.) T then acts on the coordinates as R(5a0) while a space reflection can be represented by R(oa1)R(ba2)R(j(a3) (in terms of a timelike vector ga°, and three orthogonal spacelike vectors oat, i = 1, 2, 3). We define the action of the discrete symmetries on a real scalar field by #'(x') = #(x). The action on a Weyl spinor is 'a (X') = ~i (x) , ' (x') = inaf 9(x) / (X) = u2@ (x) .(3.3.20) Since this transformation involves complex conjugation, we interpret R as giving CP and T. Indeed, since under complex conjugation e-x e+ZP, we have p'a _ (pa _-a-2 an - p). Therefore p0 changes sign for spacelike u, and this is consis- tent with our interpretation. The combined transformation CPT is simply x - - x and the fields transform without any factors (except for irrelevant phases). The transforma- tion of an arbitrary Lorentz representation is obtained by treating each spinor index as in (3.3.20). The definition of C, and thus P and CT, requires the existence of an additional, internal, discrete symmetry, e.g., a symmetry involving only sign changes: For the pho- ton field CA a =-A a; for a pair of real scalars, C#$1 =+ #1i, C#$2 =- #~2 gives  78 3. REPRESENTATIONS OF SUPERSYMMETRY C(#1 + i#2) = (#1 + i#2)t. For a pair of spinors, C/1" =2', C/2' =914 gives, for the Dirac spinor ($19,529), the transformation C($1a, 2) =(/2a,/1a)t i.e., complex con- jugation times a matrix. Therefore, C generally involves complex conjugation of a field, as do CP and T, whereas P and CT do not. (However, note that the definition of com- plex conjugation depends on the definition of the fields, e.g., combining #1 and #2 as #1 + i#2-) The generalization to superspace is straightforward: In addition to the transforma- tion R(u)x given above, we have (as for any spinor) 0'a c= io ua , 'ate= ina 6a (3.3.21) A real scalar superfield and a Weyl spinor superfield thus transform as the corresponding component fields, but now with all superspace coordinates transforming under R(u). To preserve the chirality of a superspace or superfield (see below), we define R(u) to always complex conjugate the superfields. We thus have, e.g., (M; + !6)],(3334c) ib[~ (tb + ba ( - )Y) -b (M + Q )1 , (33.3d 24N 2+[-,&))-i ,(233e S (Z + Z +X~ aa X0 a3eb b~ 1 ia3cvc , bcv )Q acv a b.\ +42 N 2 2~u  82 3. REPRESENTATIONS OF SUPERSYMMETRY - i(x + iQa/ a ) Mp" - i - ij IaaaO)/3Mj - xaid - 2OaaO (tab + o(ab(1 - +)Y) . (3.3.34f) Here d is the matrix piece of the generator A; its eigenvalue is the canonical dimension d. Similarly, Y ,tab are the matrix pieces of the axial generator A and the SU(N) gener- 1 ators Tab; the eigenvalue of Y is - w. The terms in S, S proportional to Y and tab do 2 not follow from the inversion (3.3.33), but are determined by the commutation relations and (3.3.34a,b). b.8. Super-deSitter representations To construct the generators of the super-deSitter algebra, we use the expressions for the conformal generators and take the linear combinations prescribed in (3.2.13). To summarize, for general N, in each of the cases we have considered the genera- tors act as differential operators. In addition the superfields may carry a nontrivial matrix representation of all the generators except for P and Q in the Poincard and deSitter cases, and P, Q, K, and S in the superconformal case. They may also carry a representation of some arbitrary internal symmetry group.  3.4. Covariant derivatives 83 3.4. Covariant derivatives In ordinary flat spacetime, the usual coordinate derivative &a is translation invariant: the translation generator Pa, which is represented by &O, commutes with itself. In supersymmetric theories, the supertranslation generator QC has a nontrivial anticommutator, and hence is not invariant under supertranslations; a simple computa- tion reveals that the fermionic coordinate derivatives &a, &3 are not invariant either. There is, however, a simple way to construct derivatives that are invariant under super- symmetry transformations generated by Qa , Qg (and are covariant under Lorentz, chiral, and isospin rotations generated by Jag , J., A, and Tab). a. Construction In the preceding section we used the method of induced representations to find the action of the super-Poincard generators in superspace. The same method can be used to find covariant derivatives. We define the operators Dc and D& by the equation (eE+E)(eiX+O+Q) (eix+O+Q)(ei +Q) (3.4.1) The anticommutator of Q with D can be examined as follows: (ei(EQ + Q))(e'D + (D )(e(EQ + Q))(ei(xP + oj+ OQ)) (eiEQ)Q)(( QE ))(ix+QO)(i~+Q) e - 466 +(E i(+X)+ OQ + ) 0() ( e) ( e)(C(2( + eQ +eQ z Q)) (e'D +D)( ei(xP+OQ + eQ)) .(3.4.2) Thus the D's are invariant under supertranslations (and also under ordinary transla- tions): {Q, D} ={Q, D} =[P, D] =0 . (3.4.3) We can use the Baker-Hausdorff theorem, (3.4.1), and (3.3.11,13) to compute the explicit forms of the D's from the Q's. We find Da=-Q +eaPa , DU=-iQ +OeP. (3.44) (3.4.4)  84 3. REPRESENTATIONS OF SUPERSYMMETRY For N = 1, when acting on superfields, they have the form Da = O 2 Oia g , U s= 53 + 2} BOag ,(3.4.5) 2 2(34) and are covariant generalizations of the ordinary spinor derivative O(, O. For general N, with central charges, the covariant derivatives have the form: Da = D O = l + 8-vao + -ObcvZb , U. =DU" = 2+ }08-O bc+ "b . (3.4.6) They can be rewritten using eU as: Da = e-(2_ +1O2aZba)e2U D. =C2Uo .+ e ba -2U(3.4.7) 2 Consequently, just as the generators Q simplify in the chiral (antichiral) representation, the covariant derivatives have the simple but asymmetric form: Da(+) e-U(& +}ObcZ a)eU , (+) = + "Zab D(-)=O+}ObZa , o(-)b=eU + } 8 ")e-ab.((3.4.8) In any representation, they have the following (anti)commutation relations: {Da , D1} = CapZab , {Da , D.} =i . (3.4.9) It is also possible to derive deSitter covariant derivatives by these methods. How- ever, there is an easier, more useful, and more physical way to derive them within the framework of supergravity, since deSitter space is simply a curved space with constant curvature. This will be described in sec. 5.7. b. Algebraic relations The covariant derivatives satisfy a number of useful algebraic relations. For N =1, the only possible power of D is D2 -1 DaDa. (Because of anticommutativity 2 higher powers vanish: (D)3 =0.) From the anticommutation relations we also have  3.4. Covariant derivatives 85 [Do, D21 =i&D& , D2D2D2 = DD2 D -D = b3"D2 , D202 = - 1 . (3.4.10) For N > 1 we have similar relations; for vanishing central charges: D"~l.... = DCV1 ---.Dan D nan + 1 .2N C2N'''x1 D" D" - 1 C D"n+1'''G2N 1...i (2N - n)!lJ 2N'''l D2 N -n 1... D" ...F [/Q. [ . * . - --ocpnf D2N (D"... )f = UD..' (D2N-nDTi...It)nD2N-n ...41 D2NO2N (1)N U2ND2NI52N _= NI52N.(3.4.11) It is often necessary to reduce the product of D's or D's with respect to SU(N), as well as with respect to SL(2, C). For each, the reduction is done by symmetrizing and antisymmetrizing the indices. Specifically, we find the irreducible representations as fol- lows: A product DD.... DA is totally antisymmetric in its combined indices since the D's anticommute; however, antisymmetry in a, #_ implies opposite symmetries between a, b and a, ,3, (one pair symmetric, the other antisymmetric), and hence a Young tableau for the SU(N) indices is paired with the same Young tableau reflected about the diagonal for the SL(2, C) indices. The latter is actually an SU(2) tableau since if we have only D's then only undotted indices appear, and has at most two rows. (Actually, for SU(2) a column of 2 is equivalent to a column of 0, and hence the SL(2C) tableau can be reduced to a single row.) Therefore, the only SU(N) tableaux that appear have two columns or less. The SL(2, C) representation can be read directly from the SU(N) tableau (if we keep columns of height N): The general SU(N) tableau consists of a first  86 3. REPRESENTATIONS OF SUPERSYMMETRY column of height p and a second of height q, where p + q is the number of D's; the cor- responding SL(2C) representation is a (p - q)-index totally symmetric undotted spinor. Therefore this representation of SL(2, C)®SU(N) has dimensionality (p - q +1) p( )(N ) . (3.4.12) p + 1 p q c. Geometry of flat superspace The covariant derivatives define the geometry of "flat" superspace. We write them as a supervector: DA = (Da , D ,&a8) .(3.4.13) In general, in flat or curved space, a covariant derivative can be written in terms of coor- dinate derivatives Om &-BZM and connections FA: DA = DAM&M + FA(M) + FA(T) + FA(Z) . (3.4.14) The connections are the Lorentz connection FA(M) =AAM j + FA Mg j , (3.4.15a) isospin connection FA(T) = PAcTc , (3.4.15b) and central charge connection FA(Z) = (PAcZ c + FabcZ) . (3.4.15c) The Lorentz generators M act only on tangent space indices. (Although the distinction is unimportant in flat space, we distinguish "curved", or coordinate indices M, N,--- from covariant or tangent space indices A, B,.-.-.. In curved superspace we usually write the covariant derivatives as VA= EAMDM + fA, DM = oMDA, i.e., we use the flat superspace covariant derivatives instead of coordinate derivatives: see chapter 5 for details.) In flat superspace, in the vector representation, from (3.4.6) we find the flat viel- bein  3.4. Covariant derivatives 87 o 0 ioa 1oa"O MP D AM m 0 6(1- !U.-6armpIIA(3.4.16) 0 0 6,a'- and the flat central charge connection FAc=- 10(C O0b/a1,0,0) FAbc -= - (0 , C . " o c3" , 0) , (3.4.17) terms of covariant torsions T ABC, curvatures R AB(M), and field strengths F AB(T n F AB(Z): [DA , DB} TABCDC + RAB (M) + FAB(T) + FAB(Z) (3.4.18) From (3.4.16-17), we find that flat superspace has nonvanishing torsion T j = oab a 036 (3.4.19) and nonvanishing central charge field strength Fcd Caja[cObd , FZcd = C "(ca. , b,(3.4.20) all other torsions, curvatures, and field strengths vanishing. Hence flat superspace has a nontrivial geometry. d. Casimir operators The complete set of operators that commute with Pa, Q, and Qg (and trans- form covariantly under J(1 and J(") is {DA , Maj , M)g , Y , tab , d}. (Except for DA, which is only covariant with respect to the super-Poincard algebra, all these operators are covariant with respect to the entire superconformal algebra. Note that the matrix operators M , Y , t , d act only on tangent space indices.) Thus the Casimir operators (group invariants) can all be expressed in terms of these operators. Following the dis- cussion of subsec. 3.3.a.3, it is sufficient to construct:  88 88 3. REPRESENTATIONS OF SUPERSYMMETRY P[aWb] = P[a b] a 2 DacsDa " z (&Mj & M.~ a Me Wa - APa fa + YZta (3.4.21) 4 W5 =m2Y- az ra[ D a, 1 b[b D c . N a[a ] (3.4.22) Waba aqb a ([Dajb 1 bba[DcD"] W5a a ac- ~[~ (3.4.23) where we have used P2 m2 for W a b, and p"aQ, &aa Da = &a& &a =O0for Wabc (the massless case: see subsec. 3.3.a.3).  3.5. Constrained superfields 89 3.5. Constrained superfields The existence of covariant derivatives allows us to consider constrained super- fields; the simplest (and for many applications the most useful) is a chiral superfield defined by DUI = 0 (3.5.1) We observe that the constraint (3.5.1) implies that on a chiral superfield DG= 0 and therefore {D, D}@ = 0 - Z4= 0. In a chiral representation, the constraint is simply the statement that b(+) is inde- pendent of 0, that is (+)(x, 0, 0) - (+)(x, 0). Therefore, in a vector representation, 1(x,08,08) = e2UI(+)(x,6)eiU = I(+)(x(+),0) , (3.5.2) where x(+) is the chiral coordinate of (3.3.27b). Alternatively, one can write a chiral superfield in terms of a general superfield by using D2N+1=0: D 2N41(x,06,0 ) (3.5.3) This form of the solution to the constraint (3.5.1) is valid in any representation. It is the most general possible; see sec. 3.11. Similarly, we can define antichiral superfields; these are annihilated by Da. Note that 1, the hermitian conjugate of a chiral superfield 1, is antichiral. These superfields may carry external indices. * * * The supersymmetry generators are represented much more simply when they act on chiral superfields, particularly in the chiral representation (3.3.25), than when they act on general superfields. For the super-Poincard case we have: Q- =i(& -O>pZb) , QO =6"8& , Pa = _8 J =- i (xa&89 + Qat~ap) - iMap J. l a g --iM ., (3.5.4)  90 3. REPRESENTATIONS OF SUPERSYMMETRY where Zab4 = 0 (as explained above) but Zab is unrestricted. If we think of Zab as a partial derivative with respect to complex coordinates ("b, i.e., Za= i then a chiral p p p ab ab~ superfield is a function of x , 0, ( and is independent of 0 , (. In the superconformal case, Zab must vanish, and, for consistency with the algebra, a chiral superfield must have no dotted indices (i.e., M..>= 0). On chiral superfields, the inversion (3.3.33) takes the form H14"''(x ,0) =_x- f2df ... - (x', 0') = x-2df ... -a. (X, ' ) fcl = i4x)-1xa , = x-2 ,a - 2xaggaa ; (3.5.5) (note that d- = 0 and hence d = d+). The generators of the superconformal algebra are now just (3.5.4), = -x"Daa ,3sa - =_apGc" , Ka=psG a";(3.5.6a) with G = Ji + <[A + 2(xaO + 2 - N)] A~(Xaha + 10c + N ) 11 A=- (+2-N+ d 2 A = 628 - (4 - N)-1 Nd, 2 T =bab ([8 ,al )c] 5a- [o %,a]) . (3.5.6b) 4 N The commutator algebra is, of course, unchanged. Note that the expression for A con- tains a term (1 - N)-1d; this implies that for N = 4, either d vanishes, or the axial 4 charge must be dropped from the algebra (see sec. 3.2.e). The only known N = 4 theo- ries are consistent with this fact: N =4 Yang-Mills has no axial charge and N =4 con- formal supergravity has d =0. We further note that consistency of the algebra forbids the addition of the matrix operator tab to Tab in the case of conformal chiral superfields. This means that conformal chiral superfields must be isosinglets, i.e., cannot carry exter- nal isospin indices.  3.5. Constrained superfields 91 * * * For N = 1, a complex field satisfying the constraint 92E = 0 is called a linear superfield. A real linear superfield satisfies the constraint D2G = 2G = 0. While such objects appear in some theories, they are less useful for describing interacting particle multiplets than chiral superfields. A complex linear superfield can always be written as E = DST,, whereas a real linear superfield can be written as G = DcD2W + h. c..  92 3. REPRESENTATIONS OF SUPERSYMMETRY 3.6. Component expansions a. 0-expansions Because the square of any anticommuting number vanishes, any function of a finite number of anticommuting variables has a terminating Taylor expansion with respect to them. This allows us to expand a superfield in terms of a finite number of ordinary spacetime dependent fields, or components. For general N, there are 4N inde- 4N4N 4 pendent anticommuting numbers in 0, and thus ( =)24N components in an uncon- strained scalar superfield. For example, for N = 1, a real scalar superfield has the expansion V=C+0xa+0ay -02M-O2M + OcAa - O2Oaga - 2g9c; + 02e2D' (3.6.1) with 16 real components. Similarly, a chiral scalar superfield in vector representation has the expansion: S=e>(A + 8 - 02F)e-U =A+0"@av-02F+2i 2"OOaA + i 202e('Oa + 4o2o2DA (3.6.2) 2 4 with 4 independent complex components. These expansions become complicated for N > 1 superfields but fortunately are not needed. However, we give some examples to familiarize the reader with the compo- nent content of such superfields. For instance, for N =2, in addition to carrying Lorentz spinor indices, superfields are representations of SU(2). A real scalar-isoscalar superfield has the expansion V(x,0,O) =C(x ) + Q0xa + 007 - Q2cvf3JpJ - O2abMab  3.6. Component expansions 93 + 04N + ON + - + 02c432c4hap + ... + 0404D'(x) (3.6.3) where Was =- 0, while a chiral scalar isospinor superfield has the expansion (in the chi- ral representation) I = EcF -aZ 6F = i(c - Q + -Q)D2 | ( E " DD24b (3.6.8)  3.6. Component expansions 95 Explicit computation of the components shows that, in this particular case, the components in the 0-expansion are identical to those defined by projection. This is not necessarily the case: For superfields that are not chiral, some components are defined with both D's and D's; for these components, there is an ambiguity stemming from how the D's and D's are ordered. For example, the 020 component of a real scalar superfield V could be defined as D2DV I, D DD V I, or DD2V I. These definitions differ only by spacetime derivatives of components lower down in the 0-expansion (defined with fewer D's). In general, they will also differ from components defined by 0-expansions by the same derivative terms. These differences are just field redefinitions and have no physical significance. Usually, one particular definition of components is preferable. For example, one model that we will consider (see sec. 4.2.a) depends on a real scalar superfield V which transforms as V'= V + i (A - A) under a gauge transformation that leaves all the physics invariant (here A is a chiral field). In this case, if possible, we select components that are gauge invariant; in the example above, D2DV| is the preferred choice. If the superfield carries an external Lorentz index, the separation into components requires reduction with respect to the Lorentz group. Thus, for example, a chiral spinor superfield has the expansion in the chiral representation (where it only depends on 0): a+)(x,8) =A + 8 (Cj3aD' + f ) - 02xC . (3.6.9) Using projections, we would define the components by D' = 1D 4a" 2a fad =D(aeb/3| , ya =D2@a l. (3.6.10) For N > 1 a similar definition of components by projection is possible. In this case, in addition to reduction with respect to the external Lorentz indices, one can fur- ther reduce with respect to SU(N) indices.  96 3. REPRESENTATIONS OF SUPERSYMMETRY The projection method is also convenient for finding components of a product of superfields. For example, the product 4 =_#152 is chiral, and has components 1 =4b~ 2| = A1A2 , DaF =D ) +1( + 1(Da2)| = 1iaA2 + A12a D2| =(D241I)2| + (DN1i)(D$@2)| + 1(D2 2)| = F1A2 + /i1 2a + A1F2 (3.6.11) Similarly, the components of the product 4' =_152 can be worked out in a straightfor- ward manner, using the Leibnitz rule for derivatives. c. The transformation superfield The transformations of Poincard supersymmetry (translations and Q-supersymme- try transformations) are parametrized by a 4-vector , and a spinor Ea respectively. It is possible to view these, along with the parameter r of "R-symmetry" transformations generated by A in (3.2.12, 3.3.34a), as components of an x-independent real superfield ( (a=2[Da , Da](1 , e aiD , !r }Da2Da(| , (3.6.12) and to write the supersymmetry transformations in terms of ( and the covariant derivatives DA: j ( aPq + EavQa + E Q + 2rA)4' - [(iD2Da()Da + (- iD2D()D + (} [ljy , Do]()6 1 - + iw (- Do D2 Da()]4 , (3.6.13) 1 where - w is the eigenvalue of the operator Y (the matrix part of the axial generator A). These transformations are invariant under "gauge transformations" o( i i(A- A), A chi- ral and x-independent. Consequently, they depend only on (g es, and the component r. The R-transformations with parameter r are axial rotations W'(x, 0, 0) = e-"w(x, eir0, e-ir) . (3.6.14)  3.6. Component expansions 97 3.7. Superintegration a. Berezin integral To construct manifestly supersymmetrically invariant actions, it is useful to have a notion of (definite) integration with respect to 0. The essential properties we require of the Berezin integral are translation invariance and linearity. Consider a 1-dimensional anticommuting space; then the most general form a function can take is a + Ob. The most general form that the integral can take has the same form: I dO' (a + Ob) = A + OB where A , B are functions of a, b. Imposing linearity and invari- ance under translations 0' 0'+ E leads uniquely to the conclusion that I dO (a + Ob) b. The normalization of the integral is arbitrary. We choose I dO 0 1 (3.7.1) and, as we found above, dO 1 = 0 . (3.7.2) We can define a 6-function: We require Jd6O(O - ')(a + Ob)= a + 'b (3.7.3) and find 6(e-0') =0 -0' (3.7.4) These concepts generalize in an obvious way to higher dimensional anticommuting spaces; for N-extended supersymmetry, J d2NO d2NO picks out the highest 0 component of the integrand, and a s-function has the form - AN'g _ (g0- Q)2N( - _f-)2N . (3.7.5) We define 64+4Nz - z_ 4~( - xf)4N (0 - 0'). We thus have Jd4+4Nz 4+4Nz -  98 3. REPRESENTATIONS OF SUPERSYMMETRY fd4xd4NO 4(x - /X)64N(O - O')4'(X,>0) 'T'(z,) (3.7.6) We note that all the properties of the Berezin integral can be characterized by say- ing it is identical to differentiation: / deb f (e) = 0,3 f(8) . (3.7.7) This has an important consequence in the context of supersymmetry: Because super- space actions are integrated over spacetime as well as over 0, any spacetime total deriva- tive added to the integrand is irrelevant (modulo boundary terms). Consequently, inside a spacetime integral, in the absence of central charges we can replace J d01 =/ by D3. This allows us to expand superspace actions directly in terms of components defined by projection (see chap. 4, where we consider specific models). Inside superspace integrals, we can integrate D by parts, because J d4Ng 02N-2N 0(since2N+1=0). Since supersymmetry variations are also total derivatives (in superspace), we have J d4xd2N9 Q Q fJd4xd2NQ4! = 0, and thus for any general superfield T the follow- ing is a supersymmetry invariant: S = fd4xd4NO q .(3.7.8) In the case of chiral superfields we can define invariants in the chiral representation by S =) d4xd2N9 4D,(3.7.9) since 4 is a function of only x" and W. In fact, this definition is representation indepen- dent, since the operator U used to change representations is a spacetime derivative, so only the 1 part of e2 contributes to S4. Furthermore, if we express 1 in terms of a gen- eral superfield 'I by @ D 2N4I, we find S f = d4xd2N0 D 2N f d4xd4N0 ,(71) since Dg f dOg when inside a d4x integral.  3.7. Superintegration 99 Similarly, the chiral delta (X - x')2N(O _ 0)- _x)()N( following form in arbitrary representations: function, which we define as Qf)2N in the chiral representation, takes the 52N 4+4N(z - z/) (3.7.11) which is equivalent in the chiral representation (Da gives f d4xd2N -[D2N 4+4N(z = o), and in general representations - z')1D(Z) _ I d4 xd4N 4+4N(z - z')4(z) b(z) (3.7.12) b. Dimensions Since the Berezin integral acts like a derivative (3.7.7), it also scales like a deriva- tive; thus it has dimension [J dO]= [D]. However, from (3.4.9), we see that the dimen- sions of D.~m2, and consequently, a general integral has dimension J d4x d4Ng m2N-4 and a chiral integral has dimension J d4x d2NgmN-4. In particular, for N = 1, we have J d4x d48 Jd8z~m-2 andfd4x d20 =Jd6z ~m-3. c. Superdeterminants Finally, we use superspace integrals to define superdeterminants (Berezinians). Consider a (k, n) by (k, n) dimensional supermatrix M with a k by k dimensional even- even part A, a k by n dimensional even-odd part B, an n by k dimensional odd-even part C, and an n by n dimensional odd-odd part D: A M =C B) D (3.7.13) where the entries of A, D are bosonic and those of B, C are fermionic. We define the superdeterminant by analogy with the usual determinant:  100 3. REPRESENTATIONS OF SUPERSYMMETRY (sdet M)-1 = K dk x dk x' d"e d"6' e-z"tM z (3.7.14a) where z'' = (x' 8) , z - = e , (3.7.14b) and K is a normalization factor chosen to ensure that sdet(1) = 1. The exponent x'Ax + x'BO + 0'Cx + 0'DO can be written, after shifts of integration variables either in x or in 0, in two equivalent forms: x'Ax + 0'(D - C A-1 B)8 or x'(A - B D-1 C)x + 0'DO. Integration over the bosonic variables gives us an inverse determinant factor, and integration over the fermionic variables gives a determinant fac- tor. We obtain sdet M in terms of ordinary determinants: det A det(A - BD-1C) (3715) sdet(M) =.(..5 det(D - CA-1B) det D This formula has a number of useful properties. Just as with the ordinary deter- minant, the superdeterminant of the product of several supermatrices is equal to the product of the superdeterminants of the supermatrices. Furthermore, in (sdet M) = str(ln M) , (3.7.16a) where the supertrace of a supermatrix M is the trace of the even-even matrix A minus the trace of the odd-odd matrix D: strM = trA - trD (3.7.16b) An arbitrary infinitesimal variation of M induces a variation of the superdeterminant: o(sdet M) = Sexp[str(ln M)] (sdet M)str(M-1oM) (3.7.17)  3.8. Superfunctional differentiation and integration 101 3.8. Superfunctional differentiation and integration a. Differentiation In this section we discuss functional calculus for superfields. We begin by review- ing functional differentiation for component fields: By analogy with ordinary differentia- tion, functional differentiation of a functional F of a field A can be defined as SF[A]-. F[A + 6S,A] - F[A] (3.8.1) oA(x) e-O where 66,,A(x') = co*(x - x') . (3.8.2) This is not the same as dividing 8F by SA. The derivative can also be defined for arbi- trary variations by a Taylor expansion: F[ A+ A]= F[A] + (1A, ,1 $A ) +O((6A)2) , (3.8.3) where the product (,) of two arbitrary functions is given by (C, B) f d4x C(x)B(x) . (3.8.4) In particular, from (3.8.2) we find (SEA , B) = eB(x) . (3.8.5) This definition allows a convenient prescription for generalized differentiation. For example, in curved space, where the invariant product is (C, B) f d4x gi2 CB, the normalization (8A, B) =eB(x) corresponds to the functional variation 5A(x') = eg-12(x)64(x - x'). Generally, a choice of SeX is equivalent to a choice of the product (, ). In particular, for (3.8.2,4) we have the functional derivative 5A(x) =Ax' 6'(x - x') . (3.8.6) In curved space, using the invariant product, we would obtain g-/2(x)64(x - x'). Note that the inner product is not always symmetric: In (C, B), C transforms contragredi- ently to B. For example, if A is a covariant vector, the quantity on the left-hand side of  102 3. REPRESENTATIONS OF SUPERSYMMETRY the inner-product is a covariant vector, while that on the right is a contravariant vector; 5F if A is an isospinor, is a complex-conjugate isospinor; etc. In superspace, the definitions for general superfields are analogous. The product (4', ') is J d4+4Nz 4'(z>Ti'(z) Jfd4xd4NO 4T(x, )T'(x, 0), and thus 64(Z) 641W(z') S4+4N(z - z,) 4 4Xf)4N(0 - 0') (3.8.7) (Appropriate modifications will be made in curved superspace.) However, for chiral superfields we have (4 ,'i = d4+2N z Sd4 xd2N Qbf (3.8.8) since k and k' essentially depend on only x" and ", defined in terms of the chiral delta function: not 8". The variation is therefore S 6(z') = eD2N64+4N( -z/) (3.8.9) so that 6D(z') g2Ng4+4N (z z') (3.8.10) and the complex conjugate relation o@(z') D2N64+4N (z z') (3.8.11) (Again, appropriate modifications will be made in curved superspace.) Furthermore, variations of chiral integrals give the expected result s fz') d4xd2NO f(e(z)) -ffd4xd4N ( /d4 xd2Ng ff((z))D2N64+4N(z - z,) - z') f'(4b (Z )) (3.8.12) When the functional differentiation is on an expression appearing in a chiral integral with d2N0, the D2N can always be used to convert it to a d4N0 integral, after which the full 6-function can be used as in (3.8.12).  3.8. Superfunctional differentiation and integration 103 This result can also be obtained by expressing 4 in terms of a general superfield, as 4 = U2NWj: we have oI@(z) S2Np(z - 2N -N44 8 (z) = ( 2N o 4+4N( - z) . (3.8.13) 6 6- We can thus identify with for 4 D2N4. These definitions can be analyzed in terms of components and correspond to ordi- nary functional differentiation of the component fields. We cannot define functional dif- ferentiation for constrained superfields other than chiral or antichiral ones. For exam- ple, for a linear superfield T (which can be written as T =Dada) there is no functional derivative which is both linear and a scalar. b. Integration In chapters 5 and 6 we discuss quantization of superfield theories by means of functional integration. We need to define only integrals of Gaussians, as all other func- tional integrals in perturbation theory are defined in terms of these by introducing sources and differentiating with respect to them. The basic integrals are I 4DVe.dXd2V=1 , (3.8.14a) I l 'e d gx 2N 42 Ji e. N 2 1 ,(3.8.14b) D4 e.d4x2NO 2 = 1 ,(3.8.14c) where, e.g., ]DV = 1 DVZ, for VZ the components of V. Because a superfield has the same number of bose and fermi components, many factors that appear in ordinary func- tional integrals cancel for superfields. Thus we can make any change of variables that does not involve both explicit 0's and 's without generating any Jacobian factor, because unless the bosons and fermions mix nontrivially, the superdeterminant (3.7.14) is equal to one. For example, a change of variables V -~ f (V , X) where X is an arbi- trary external superfield generates no Jacobian factor; the same is true for the change of  104 3. REPRESENTATIONS OF SUPERSYMMETRY variables V -- LIV as long as L is a purely bosonic operator. Nontrivial Jacobian determinants arise for changes of variables such as V - D2V or V -- LIV where L is background covariant, e.g., in supergravity or super-Yang-Mills theory, and hence con- tains spinor derivatives. To prove the preceding assertions, we consider the case with one 0; the general case can be proven by choosing one particular 0 and proceeding inductively. We expand the superfield with respect to 0 as V = A + 0i/; similarly, we expand the arbitrary external superfield as X = C + 9x. Then we can expand the new variable f (V , X) as f(V , X) - f(A, C) + 0[i/fy(A, C)| + xfx(A, C)|] (3.8.15) where fy|= fA QA , etc. The Jacobian of this transformation is __f fA(AC fA xfAc de~g sdet = sdet A ) fAA+X det(f A) 1 (3.8.16) 8V0 fA( A,C) 2 det(fA) In particular, the external superfield X can be a nonlocal operator such as l-1. An immediate consequence of the preceding result is that superfield S-functions 8(V - V') fJo(Vi - V'Z) (3.8.17a) are invariant under "0-nonmixing" changes of variables: 6(f (V)) = 6(V - cZ) . (3.8.17b) In general, if nontrivial operators appear in the actions, the functional integrals are no longer constant. We first introduce the following convenient notation: Jdxfd4NgVt fd2Ngt 2NOt where V, 1 A+)2. We can perform similar redefinitions to separate arbitrary fields into physical, aux- iliary, and gauge components. Any original component that transforms under a gauge transformation with a a+ or a nonderivative term corresponds to a gauge component of the redefined field. Any component that transforms with a &_ term corresponds to an auxiliary component. Of the remaining components, some will be auxiliary and some physical (depending on the action), organized in a way that preserves the "transverse" SO(2) Lorentz covariance. For the known fields appearing in interacting theories, the components with highest spin are physical and the rest (when there are any: i.e., for 3 physical spin 2 or -) are auxiliary. These arguments can be applied in all dimensions. An example that illustrates the separation between physical and auxiliary (but not gauge) components without the use of nonlocal, noncovariant redefinitions is that of a massive spinor field: I = "iB;@a- -m(#"a +fif) .(3.9.7) Since @9 and I9 may be considered as independent fields in the functional integral (and,  110 3. REPRESENTATIONS OF SUPERSYMMETRY in fact, must be considered independent locally after Wick rotation to Euclidean space), we can make the following nonunitary (but local and covariant) redefinition: a > 2ba - z- mi2aag@" . (3.9.8) The Lagrangian becomes 1 1 <)(D - m2)9a - } m/bja . (3.9.9) We thus find that @9 represents two physical polarizations, while 9 contains two auxiliary components. The same analysis can be made for the simplest supersymmetric multiplet: the massive scalar multiplet, described by a chiral scalar superfield (see section 4.1). The action is S = d'xde O-2m (fd4xd2O2+fd4xd2O2) . (3.9.10) We now redefine 4D-- 4+!D2 ; (3.9.11) M and, using J d28= D2 we obtain the action S Jfd4xd2O 1(D - m2) - m fd4xd2O2 . (3.9.12) (Note that the redefinition of 4 preserves its antichirality Dad =0.) Now 4 contains only physical and 4 contains only auxiliary components; each contains two Bose compo- nents and two Fermi. As can be checked using the component expansion of @, the origi- nal action (3.9.10) contains the spinor Lagrangian of (3.9.7), whereas (3.9.12) contains the Lagrangian (3.9.9). It also contains two scalars and two pseudoscalars, one of each being a physical field (with kinetic operator LI- in2) and the other an auxiliary field (with kinetic operator 1). For more detail of the component analysis, see sec. 4.1.  3.9. Physical, auxiliary, and gauge components 111 As we discuss in sec. 4.1, auxiliary fields are needed in interacting supersymmetric theories for several reasons: (1) They facilitate the construction of actions, since without them the kinetic and various interaction terms are not separately supersymmetric; (2) because of this, actions without auxiliary fields have supersymmetry transformations that are nonlinear and coupling dependent, and make difficult the application of super- symmetry Ward identities (e.g., to prove renormalizability); and (3) auxiliary fields are necessary for manifestly supersymmetric quantization. Compensating fields (see follow- ing section) are also necessary for the latter two reasons. Although they disappear from the classical action, they appear in supersymmetric gauge-fixing terms.  112 3. REPRESENTATIONS OF SUPERSYMMETRY 3.10. Compensators In our subsequent discussions, we will often use "compensating" fields or compen- sators. These are fields that enter a theory in such a way that they can be algebraically gauged away. Thus, in a certain sense, they are trivial: The theory can always be writ- ten without them. However, they frequently simplify the structure of the theory; in par- ticular, they can be used to write nonlinearly realized symmetries in a linear way. This is often important for quantization. Another application, which is particularly relevant to supergravity, arises in situations where one knows how to write invariant actions for systems transforming under a certain symmetry group G (e.g., the superconformal group): If one wants to write actions for systems transforming only under a subgroup H (e.g., the super-Poincard group), one can enlarge the symmetry of such systems to the full group by introducing compensators. After writing the action for the systems with the enlarged symmetry, one simply chooses a gauge, thus breaking the symmetry of the action down to the subgroup H. A simple example in ordinary field theory is "fake" scalar electrodynamics. The usual kinetic action for a complex scalar z (x) S = 2] Zd4x aOz (3.10.1) has a global U(1) symmetry: z' = ed z. This symmetry can be gauged trivially by intro- ducing a real compensating scalar #, assumed to transform under a local U(1) transfor- mation as #' # - A. We can then construct a covariant derivative Va = e-ZOOae2 &= + ia§5 that can be used to define a locally U(1) invariant action S 2 Zd4x EVav z (3.10.2) 2- Fake spinor electrodynamics can be obtained by an obvious generalization. a. Stueckelberg formalism In the previous example, the compensator served no useful purpose. The Stueck- elberg formalism provides a familiar example of a compensator that simplifies the theory. We begin with the Lagrangian for a massive vector Aa: 1]L= -FabF ab - m2(Aa)2 , (3.10.3) 8 -  3.10. Compensators 113 Fab &[aAl .(3.10.4) The propagator for this theory is: 1 1 Dab - (ba b- 22 &2a) (3.10.5) We can recast the theory in an improved form by introducing a U(1) compensator # that makes the action (3.10.3) gauge invariant. We define A'a = A +1 6,#(3.10.6) where A'a and # transform under U(1) gauge transformations: SA'a &= aA , 60#= mA . (3.10.7) In terms of these fields, the gauge invariant Lagrangian is (dropping the prime): E =a- _Fa Fa - m2(Aa )2 - m#6TAa - (Oa#)2 . (3.10.8) We now choose a gauge by adding the gauge fixing term EGF - 4 (&-Aa 2mb)2 (3.10.9) and find: IL- + ]L-GF = 1 Aah(D - m2)Aa + 1$( 1-22)§ . (3.10.10) 2+ G - _ The propagators can be trivially read off from (3.10.10): for A_ Dab - ra(D - m2-1, and for #, D = - (E - m2)-1. They have better high energy behavior than (3.10.5). Thus, by introducing the compensator #, we have simplified the structure of the theory. We note that the compensator decouples whenever Aa is cou- pled to a conserved source (i.e., in a gauge invariant way). b. CP(1) model Another familiar example is the CP(1) nonlinear o--model, which describes the Goldstone bosons of an SU(2) gauge theory spontaneously broken down to U(1). It con- sists of a real scalar field p and a complex field y subject to the constraint  114 3. REPRESENTATIONS OF SUPERSYMMETRY ly|2 + p2 = 1 (3.10.11) The group SU(2) can be realized nonlinearly on these fields by -1 6p= - (/(y + 13W) 8y = - 2iay - Op - } p-1(#y - /g)y. (3.10.12) where a, 0, and / are the (constant) parameters of the global SU(2) transformations. These transformations leave the Lagrangian 1L = - [(&aP)2 + |&aY|2 + 4 (&qy)2] (3.10.13) invariant, but because the transformations are nonlinear this is far from obvious. We can give a description of the theory where the SU(2) is represented linearly by introducing a local U(1) invariance which is realized by a compensating field #. Under this local U(1), # transforms as #'(x) = #(x) - A(x) . (3.10.14) We define fields zi by z= ep , z2 = ey . (3.10.15) Because of this definition they transform under the local U(1) as z' = ei zi . (3.10.16) The constraint (3.10.11) becomes Izil2 + Iz22 = (3.10.17) Ignoring the constraint the SU(2) acts linearly on these fields (see below): -z iavzi + /3z2 oz2 = - iavz2 - /3zi . (3.10.18) The complicated nonlinear transformations (3.10.12) arise in the following manner: when we fix the U(1) gauge zi = zi=p (3.10.19)  3.10. Compensators 115 the linear SU(2) transformations (3.10.18) do not preserve the condition (3.10.19). Thus we must add a "gauge-restoring" U(1) transformation with parameter 1- 1 1 iA(x) - p-1(6zi - 681)=- icy - p-1(/3z2 - /Iz2) . (3.10.20) The combined linear SU(2) transformation and gauge transformation (3.10.16) with nonlinear parameter (3.10.20) preserves the gauge condition (3.10.19) and are equivalent to (3.10.12). To write an action invariant under both the global SU(2) and the local U(1) trans- formations we need a covariant derivative for the latter. By analogy with our first example we could write Va =e-ZO -e =& a + iaO . (3.10.21) A manifestly SU(2) invariant choice in terms of the new variables is 2 1 = as + i2a#O- -2g sy .(3.10.22) This differs from (3.10.21) by the U(1) gauge invariant term Jaay; one is always free to change a covariant derivative by adding covariant terms to the connection. (This is sim- ilar to adding contortion to the Lorentz connection in (super)gravity; see sec. 5.3.a.3.) Then a manifestly covariant Lagrangian is 1= - Vazi l2 |8&zi l2 - - (za5zi)2 . (3.10.23) In the gauge (3.10.19) this Lagrangian becomes that of (3.10.13). We consider now another application of compensators: The constraint (3.10.17) is awkward: It makes the transformations (3.10.18) implicitly nonlinear. We can avoid this by introducing a second compensating field. We observe that neither the constraint nor the Lagrangian are invariant under scale transformations. However, we can intro- duce a scale invariance into the theory by writing z2 = e- ZZ (3.10.24)  116 3. REPRESENTATIONS OF SUPERSYMMETRY in terms of new fields Zi and the compensator ((x). The constraint and the action, written in terms of Zi, C, will be invariant under the scale transformations Z' = eTZ , =(+r. (3.10.25) The SU(2) transformations of Zt are now the (truly) linear transformations (3.10.18). The U(1) and the scale transformations can be combined into a single com- plex scale transformation with parameter ST= + iA (3.10.26) Z' = eZ , =(+ . (3.10.27) The constraint (3.10.17) becomes ZZ-=e2C (3.10.28) where we write ZZ Z1l2 + Z2|2. In terms of the new variables the Lagrangian is 1= - |Va(e-CZi)|2 - |(e-Zi)|2 - 4 e4c(ZiaZi)2 . (3.10.29) Substituting for ( the solution of the constraint (3.10.28), a manifestly SU(2) invariant procedure, leads to 7 a Z 1 ( _ 1 Z ,| 2 =- 4i (ZZ)2 - 1(o Z-iZ-)-(ai) (TZk) (3.10.30) ZZ Z Z 2 This last form of the Lagrangian is expressed in terms of unconstrained fields Ze only. It is manifestly globally SU(2) invariant and also invariant under the local complex scale transformations (3.10.27). We can use this invariance to choose a convenient gauge. For example, we can choose the gauge Zi1 1; or we can choose a gauge in which we obtain (3.10.13). Once we choose a gauge, the SU(2) transformations become nonlinear again. These two compensators allowed us to realize a global symmetry (SU(2)) of the system linearly. However, they play different roles: #5(x), the U(1) compensator, gauges  3.10. Compensators 117 a global symmetry of the system, whereas ((x), the scale compensator introduces an altogether new symmetry. For the U(1) invariance we introduced a connection, whereas for the scale invariance we introduced ((x) directly, without a connection. In the former case, the connection consisted of a pure gauge part, and a covariant part chosen to make it manifestly covariant under a symmetry (SU(2)) of the system; had we tried to intro- duce # (x) directly, we would have found it difficult to maintain the SU(2) invariance. In the case of the scale transformations no such difficulties arise, and a connection is unnec- essary. As we shall see, both kinds of compensators appear in supersymmetric theories. c. Coset spaces Compensators also simplify the description of more general nonlinear u-models. We consider a model with fields y(x) that are points of a coset space G/H; they trans- form nonlinearly under the global action of a group G, but linearly with respect to a subgroup H. By introducing local transformations of the subgroup H via compensators #(x), we realize G linearly, and thus easily find an invariant action. The generators of G are T, S, where S are the generators of H and T are the remaining generators, with T , S antihermitian. Since H is a subgroup, the generators S close under commutation: [S, S]~S . (3.10.31) We require in addition that the generators T carry a representation of the H, that is [T, S]~T . (3.10.32) (This is always true when the structure constants are totally antisymmetric, since then the absence of [S, S] T terms implies the absence of [T, S] S terms.) We could write y(x) = eC(x)T mod H, but instead we introduce compensating fields #(x), and define fields z(x) that are elements of the whole group C: z = e(()T ed(x)S e (3.10.33) (where defined by QalC > =0: instead of |C>, Q, and Q with  3.11. Projection operators 121 QC> = 0, we have 0.) Explicitly, we write for each index Thus, for example, a vector Wa decomposes in the following manner:  122 3. REPRESENTATIONS OF SUPERSYMMETRY T "a W iIcvy z Q" 1 (Ca -YqIJ + T'(~ =Q "1 -A (AC +Q [(1L + nT)p]_ , (3.11.2) where 11L and 11T are the longitudinal and transverse projection operators for a four-vec- tor. The projections can be written in terms of field strengths S and F< : (flL1T)a:_:QD1O& SS =1 O 'q (UTW) _ - - 87F y, F 6 . 0 "T .(3.11.3) 2 2 (ao The field strengths are themselves irreducible representations of the Poincard group. The projections 1L 11L1 and W1T _11TT are invariant under gauge transformations Sj = UfTX and 6T = ULX respectively. The field strengths have the same gauge invari- ance as the projections: OWLa _--1O = 0 implies S = 0, and similarly SWT -Q-18736FcY = 0 implies OF y = 0. a.2. Super-Poincard projectors Projections of superfields can be written in terms of field strengths in superspace as well. We will find that projections of a general superfield can be expressed in terms of chiral field strengths with gauge invariances determined by the projection operators. Thus, for a superfield with decomposition TI= (1 U7H )T, any single term Wn =,T has a gauge invariance SW =S HI7xi. Each projection can be written in the form i#n W,=D2- (a where the chiral field strengths I(4 - D2ND"WT are Poincard and SU(N) irreducible and have the same gauge invariance as 'I': 0 =W =D2N-h (h) implies o@(4 0 because 1 and hence oI are irreducible.  3.11. Projection operators 123 The same index conversion used in (3.11.1) can be used to define the operation of rest-frame conjugation on a component field or general superfield I . by aa 3iA-1/l"'2s al...a~i+1.''2s l Az /i+1 12s. 2s i +11 i''1 (3.11.4) For example, we have: = , =a=HAa , IH AiH .(3.11.5) We extend this to chiral superfields and define a rest-frame conjugation operator K which preserves chirality, by using an extra factor D-ND2N to convert the antichiral (complex conjugated chiral) superfield back to a chiral one (and similarly for antichiral superfields). We define K /i..ai bi = 2N a-2 -Nb1...b, a1 .a2/3 '2 "'i a 2/~~'32±a KF ... = D2ND-72N4F ,K(4..)=(KV) K2 1= 1 [ 1±+K)2 11+K),(3.11.6) where D4...= Da... = 0. For example, for an N = 1 chiral spinor Ga, K 4D /"3g ,(3.11.7) We can define self-conjugacy or reality under K if we restrict ourselves to superfields N. that are real representations of SU(N) with smax = s + integral (the latter is required 2 to insure that only integral powers of LIappear). The reality condition is Kk... =+... and the splitting of a chiral superfield into real and imaginary parts is simply 1 In the previous example, if we impose the reality condition K~a = G, contract both sides with Do and use the antichirality of 53, we find the equivalent condition: DN'a =b (3=11.) (3.11.9)  124 3. REPRESENTATIONS OF SUPERSYMMETRY These "real" chiral superfields appear in many models of interest. For example, isoscalar "real" chiral superfields with 2 - N undotted spinor indices describe N < 2 Yang-Mills gauge multiplets. Similar superfields with 4 - N undotted spinor indices describe the conformal field strength of N < 4 supergravity. To decompose a general superfield into irreducible representations, we first expand it in terms of chiral superfields. In the chiral representation (D =&U) a Taylor series in 0 gives 2N 1 W(x, 0,0) -Z1'''!<(4) ..(x, 0) n=0 (3.11.10a) where N: KD2N Dni... h W01'b'b = (-1)2 j(n-N)2ND2N-n1....lC # Cana..an -(3.11.24) or KD2NDn a-1 2N-ntfbb (-)2mD j(n-N)D2N Ca1/1 ... Ca2N-n/2N-nD2N-ni 41'a'a2N-n , (3.11.25) where 2s extra Weyl spinor indices, and extra isospinor indices, reduced as in step 1, are implicit on T. For n < N: KD2NDN2N-n bi... 2n (N-n)D2Nn.1.(3.11.26) or KD2N D2N-n 1...Jbi... _ 2sn (NnD2N D_ -. n al... (3.11.27) As an example of this simplification, we consider the N = 1 chiral field above (3.11.7) for the special case when it is a field strength of a real superfield V: N: U = (2N1- n -N D2N-n1 2N-n± - -D2NDn 1 2N-n (n> 12Nn2 +K] DN ~12~ 3.11.29) If bisection is not possible: 1 I,= either of the above with (1+ K) dropped , (3.11.30) where the 1PZ are SU(N)®Poincard projectors acting on the explicit indices (including those of the superfield). We have chosen the particular forms of HI from (3.11.19,21-23) that minimize the number of indices that the 11P act on. The chiral expansion, besides its simplicity, has the advantage that the chiral field strengths appear explicitly, and the superspin and superisospin of the representation onto which 1 projects are those of the chiral field strength. b. Examples b.1. N=O We begin by giving a few Poincard projection operators. The procedure for find- ing them was discussed in considerable detail in subsec. 3.11.a.1, so here we simply list results. Scalars and spinors are irreducible (no bisection is possible for a spinor: N 1. s + 2 - is not an integer). A (real) vector decomposes into a spin 1 and a spin 0 pro- 2 2 jection (see (3.11.2,3)). For a spinor-vector = A03c13,,we have: V~/,y~I '-) + 1( b(wsj6CI3y + S/(~~C' y+ ~C S1Ij and hence .* _(-3+T1i+nL<)p where fl36bs ,Ds<6W. (3.11.31)  3.11. Projection operators12 129 For a real two-index tensor h ac =OA' shc 3,, we have h h y + (C~q + C~(q>)+ C Cq + CyaC >rj +1~(C~yh ~C~+ C ha (3. 11.32) where qa{3 1 ( h(a a3) + h(av,rna3)c + h(ca, 6X1 + h(a11<1,)a) r -1h (a i3) 24 (a ,f) i1hE q 4 E , (3. 11.33) Therefore the complete decomposition of the the two-index tensor is given by h .-(12s + ~+11L0 + flTO + "'iA++fllA )haCJ-I where the projectors are labeled by the spin (2, 1, 0), the symmetric and antisymmetric part of ha b (S and A), longitudinal and transverse parts (L and T), and self-dual and anti-self-dual parts (+ and -). The explicit form of the projection operators is FIlsha/I 11 Los 11 Tos 111,A h Q ;I W S 2u y W c @ y, 1 4! (a /01 3 )c - 1 &a(3[066) sh (a . + 6)e/) D5 - =CdcbaD5daV/N- + (CiD5acb- - Ca-yD5[ablc0) -6q/ =C1aD6 ab + D6 [ab~ra/ They satisfy the following algebraic relations (3. 11.49) D2ab -D2ba D 6ab =D6ba 4 aa a3=D 4 [b[cbl Daa3 ab Ca CD [ablca CabcdD 5[abl c a -0 (3. 11.50) All SL(2, C) indices on the Dn's are totally symmetric. We also have D4 = ~1C 45.Q-8a** 4  136 136 3. REPRESENTATIONS OF SUPERSYMMETRY D4'.Q-4 1 Ca1...k8 D4 4!Q5.fi and these imply --_1-y - Cdcba D 4 a3 + 1 (CaCCCdeD4[e] [abl - Ca Cb-Cbcef D 4[e [adl) + (Ca/ICeacdD4 ebUyS - Ca-yCeabdD4ec3 + Ca6CcabcD4ed3-) as can be verified by substituting explicit values for the indices. We consider now a complex scalar N= 4 superfield WI and find first (3. 11.51) (3. 11.52) 114 QW4D8D8 D4D(ID8D7 ~ [18,0,1 =DQ4D8D8 '2' D-4D aD8D7 p2,0,10 =DQ-452ab D8D6 ab p2,1,6 2/ 1 [abbl .8D ] 11 3 Q D4D3.."D85a 3,2,4 acj3 '20 1DQ4 D3 [ablc.D 8 D5 320 3! a [ablc 114,2,1 -_ D4D4 .. ,Y.DSD4(1; p,1,15-Dija .DDba 114,0,20' _QE4D4 Cd11abD D1 c][abl 115 3 Q-Da(3~YD8D5a a(3'Y 11Q- 4D3 D8D5 [ablca 5,2,20 3! [ablca 116,0,1 =:::Q-LI4D2 ab19SD6ab  3.11. Projection operators 137 ,1,6 1 Q-4D2 [ 8Dablcb]43 DD,[a , (3.11.53) 2 where the superisospin quantum number here refers to the dimensionality of the SU(4) representation. The only projectors that need bisection are the real representations of SU(4): the 1, 6, 15, and 20'. We find: 0,0,1t4D81 (D± - D2 U801W- 4S(D8W -E 2 ,-4D82 (D8+ (D 4 CD)D 11 2l+ - -DY ." DSD4a I(4W ± 4) QI 4D2[b]""D8 (D6ca ± 1 m 2 [cd&f,±(j ab bc 2 -4/2 D82 x[abl 'bCdL D2 61 ab~cta (D36[ak J± CbdD[cdla4) f4,1,15±' D4ab Dbaa (±) > 2 4,0,20'i 4 [cd[ab]DS/54 [cd] 1 (4'+ ) , (3.11.54) and a total of 22 irreducible representations. The 6 is a real representation only if we use a "duality" transformation in the rest-frame conjugation (3.11.4): C(acdX [cd. This occurs for rank - N antisymmetric tensors of SU(N) when N is a multiple of 4.  138 3. REPRESENTATIONS OF SUPERSYMMETRY 3.12. On-shell representations and superfields In section 3.9 we discussed irreducible representations of off-shell supersymmetry in terms of superfields; here we give the corresponding analysis of on-shell representa- tions. We first discuss the description of on-shell physical components by means of field strengths. We then describe a (non-Lorentz-covariant) subgroup of supersymmetry, which we call on-shell supersymmetry, under which (reducible or irreducible) off-shell representations of ordinary (or off-shell) supersymmetry decompose into multiplets that contain only one of the three types of components discussed in sec. 3.9. By considering representations of this smaller group in terms of on-shell superfields (defined in a super- space which is a non-Lorentz-covariant subspace of the original superspace), we can con- centrate on just the physical components, and thus on the physical content of the theory. a. Field strengths For simplicity we restrict ourselves to massless fields. (Massive fields may be treated similarly.) It is more convenient to describe the physical components in terms of field strengths rather than gauge fields: Every irreducible representation of the Lorentz group, when considered as a field strength, satisfies certain unique constraints (Bianchi identities) plus field equations, and corresponds to a unique nontrivial irreducible repre- sentation of the Poincard group (a zero mass single helicity state). On the other hand, a given irreducible representation of the Lorentz group, when considered as a gauge field, may correspond to several representations of the Poincard group, depending on the form of its gauge transformation. Specifically, any field strength 1...a2Ai ...2B, totally symmetric in its 2A undotted indices and in its 2B dotted indices, has mass dimension A + B + 1 and satisfies the con- straints plus field equations DC1 /9-Aac(i q1 -0n (12l The Klein-Gordon equation (3.12.1b) projects onto the mass zero representation, while (3.12.1a) project onto the helicity A - B state. The Klein-Gordon equation is a conse- quence of the others except when A B B0. To solve these equations we go to  3.12. On-shell representations and superfields 139 momentum space: Then (3.12.1b) sets p2 to 0 (i.e., 9~o(p2)), and we may choose the Lorentz frame p++ = p+ =0, p- 0. In this frame (3.12.la) states that only one com- 1 ponent of is nonvanishing: ... Since each "+" index has a helicity - and each 2 "+" has helicity -,- the total helicity of @ is A - B, and of its complex conjugate 2' B - A. In the cases where A = B we may choose b real (since it has an equal number of dotted and undotted indices), so that it describes a single state of helicity 0. The most familiar examples of field strengths have B = 0: A = 0 is the usual 1 description of a scalar, A = a Weyl spinor, A= 1 describes a vector (e.g., the photon), 2 3 A - the gravitino, and the case A = 2 is the Weyl tensor of the graviton. Since we are describing only the on-shell components, we do not see field strengths that vanish on shell: e.g., in gravity the Ricci tensor vanishes by the equations of motion, leaving the Weyl tensor as the only nonvanishing part of the Riemann curvature tensor. (This hap- pens because, although these theories are irreducible on shell, they may be reducible off shell; i.e., the field equations may eliminate Poincard representations not eliminated by (off-shell) constraints.) The most familiar example of A, B#0 is the field strength of 1 1 the second-rank antisymmetric tensor gauge field: (A, B) =(2 ') (see sec. 4.4.c). Some 232 less familiar examples are the spin-4 representation of spin , (A, B) = (1, ), the spin-2 representation of spin 0, (A, B) = (1, 1), and the higher-derivative representation of spin 1, (A, B) =(4 2). Generally, the off-shell theory contains maximum spin indicated by the indices of b : A + B. Although the analogous analysis for supersymmetric multiplets is not yet com- pletely understood, the on-shell content of superfields can be analyzed by component projection. In particular, a complete superfield analysis has been made of on-shell multi- plets that contain only component field strengths of type (A, 0). This is sufficient to describe all on-shell multiplets: Theories with field strengths (A, B) describe the same on-shell helicity states as theories with (A - B, 0), and are physically equivalent. They only differ by their auxiliary field content. Furthermore, type (A, 0) theories allow the most general interactions, whereas theories with B # 0 fields are generally more restricted in the form of their self-interactions and interactions with external fields. (In some cases, they cannot even couple to gravity.)  140 3. REPRESENTATIONS OF SUPERSYMMETRY Before discussing the general case, we consider a specific example in detail. The 3 multiplet of N = 2 supergravity (see sec. 3.3.a.1) with helicities 2, - 1, is described by 2' component field strengths @9ag1(x), "a (x), "b9 (x). They have dimension 3,, respectively, and satisfy the component Bianchi identities and field equations (3.12.1). We introduce a superfield strength F(0) ab. (x, 0) that contains the lowest dimension com- ponent field strength at the 0 = 0 level: F(0)aba (X, 0)|= CabF(0)a,(x, 0) @/abag,(X) . (3.12.2) We require that all the higher components of F(O) are component field strengths of the theory (or their spacetime derivatives; superfield strengths contain no gauge components and, on shell, no auxiliary fields). Thus, for example, we must have D a(,y F(0) ab)| 0, whereas Cc(dcDF(0)ab)a/ = DF( )"ba = 0. Since a superfield that vanishes at 0 = 0 vanishes identically (as follows from the supersymmetry transformations, e.g., (3.6.5-6)) Cc(dcD F(0)ab) = =DF( )abag= 0. From these arguments it follows that the superfield equations and Bianchi identities are: DF(0)aba/ = 0 , D aF()"ba = "[aF ,b ,y 0) c (1) c43,y D6DF(O) aba = 6dc F(2)ag3-y, DED DF(0)abag = 0 ; (3.12.3) where F(1)(x, 0) and F(2)(x, 0) are superfields containing the field strengths 9b"a/(x) and /cys(x) at the 0 = 0 level. By applying powers of D, and D to these equations we recover the component field equations and Bianchi identities, and verify that F(0)"bag contains no extra components. Generalization to the rest of the supermultiplets in Table 3.12.1 is straightforward: We introduce a set of superfields which at 0=0 are the component field strengths (as in (3.12.1)) that describe the states appearing in Table 3.3.1: These superfields satisfy a set of Bianchi identities plus field equations (as in the example (3.12.3)) that are uniquely determined by dimensional analysis and Lorentz ®SU(N) covariance.  3.12. On-shell representations and superfields 141 helicity scalar multiplet super-Yang-Mills supergravity +2 FaOYs +3/2 Faa ' +1 Fap Fab +1/2 Fa FaceFabc, 0 Fa Fab Fabcd -1/2 Fabs Fabc Fabcde -1 Fabcd Fabcdef -3/2 Fabcdefg -2 Fabcdefgh Table 3.12.1. Field strengths in theories of physical interest We now consider arbitrary supermultiplets of type (A, 0). There are two cases: For an on-shell multiplet with lowest spin s = 0, the superfield strength has the form N F )a1...a., < m < N and is totally antisymmetric in its m SU(N) isospin indices. If the lowest spin s > 0, the superfield strength has the form F(O)a1...a28 and is totally sym- metric in its 2s Weyl spinor indices. To treat both cases together, for s > 0 we write F()alaN al'''a2s - Cal'aNF(o)al...a2S. Then the superfield strength has the form F(o)alamai...a2s and is totally antisymmetric in its isospinor indices and totally symmetric in its spinor indices. It has (mass) dimension s + 1. This superfield contains all the on-shell component field strengths; in particular, at O = 0, it contains the field strength of lowest dimension (and therefore of lowest spin). For s. 0, the suerfield strength describes helicities m-N m-N +1, "m and its 2 'p2 '2' hermitian conjugate describes helicities - m m+1 ... N - m. Since m < N, some 2 2 2 helicities appear in both F(o) and F(o). For s > 0, the superfield strength describes helic- 1 N ities s, s + - ,2--- s + 2 ' and its hermitian conjugate describes helicities N -(s + 2 ). ''''- s. In this case, positive helicities appear only in F(o) and negative 2 helicities only in F(O). For both cases the superfield strength together with its conjugate describe (perhaps multiple) helicities ts , +(s + 1m 2 2  142 3. REPRESENTATIONS OF SUPERSYMMETRY The higher-spin component field strengths occur at 0 = 0 in the superfields F(n) obtained by applying n D's (for n > 0) or -n D's (for n < 0) to F(o). They are totally antisymmetric in their m - n isospinor indices and totally symmetric in their 2s + n spin indices, and satisfy the following Bianchi identities and field equations: n > 0 : D a8..... F(oy" "'"01... 23bi6-a-.-a bn"nF (n)"+1am]ai...a2s1...3, (3.12.4a) n < 0 : U"._- F(o)ala "-"= F "1 " 1ai-ambi.b.n , (3.12.4b) with m - N < n < m; in particular, for s > 0, DF(o) = 0. These equations follow from the requirement that all components of the on-shell superfield strength (defined by pro- jection) are on-shell component field strengths. The 0 = 0 component of the superfield F(0) is the lowest dimension component field strength; this determines the dimension and index structure of the superfield. The higher components of the superfield are either higher dimension component field strengths, or vanish; this determines the superfield equations and Bianchi identities. Note that the difference between maximum and mini- mum helicities in the F is always 1 N. 2 In the special case s = 0, m even, and m = N we have in addition to (3.12.4a,b) 2 the self-conjugacy relation al...aFN 1 a...aN F(o) 2CalaF(o)a .aN .(3.12.4c) (-N)! 2 2 For this case only, F(+n) is related to F(_n); this relation follows from (3.12.4c) for n = 0, and from spinor derivatives of (3.12.4c), using (3.12.4a,b), for n > 0. Eqs. (3.12.4a,b) are U(N) covariant, whereas, because the antisymmetric tensor CalaN is not phase invariant, (3.12.4c) is only SU(N) covariant; thus, self-conjugate multiplets have a smaller symmetry. b. Light-cone formalism When studying only the on-shell properties of a free, massless theory it is simpler to represent the fields in a form where just the physical components appear. As described in sec. 3.9, we use a light-cone formalism, in which an irreducible representa- tion of the Poincard group is given by a single component (complex except for zero  3.12. On-shell representations and superfields 143 helicity). For superfields we make a light-cone decomposition of 0 as well as x. We use the notation (see (3.1.1)): (X+I +' X -+,X-') = (x+, r T _ -) , (8",86") - (8", (") , (3.12.5a) (+ +, - -, _, ) = (0+,or, T, -0-) , ( a+, Oa-) = Oa , (a) . (3.12.5b) (The spinor derivative Oa should not be confused with the spacetime derivative &a). Under the transverse SO(2) part of the Lorentz group the coordinates transform as x± x r, xTi e2i/ XT, a' - e 64, ("' - ae-Z, (",and the corresponding derivatives trans- form in the opposite way. In sec. 3.11 we described the decomposition of general superfields in terms of chiral field strengths, which are irreducible gauge invariant representations of supersymmetry off shell. Although they contain no gauge components, they may contain auxiliary fields that only drop out on shell. To analyze the decomposition of an irreducible off-shell rep- resentation of supersymmetry into irreducible on-shell representations, we perform a nonlocal, nonlinear, nonunitary similarity transformation on the field strengths k and all operators X: XiH X'i- ei -HXe ; H T((" ) .(3.12.6a) 0+ We use this transformation because it makes some of the supersymmetry generators independent of (a in the chiral representation. Dropping primes, we have Qa+ = 2a ,Qa i(a&aO+) OT a aD Qa ( Ya+&a&) , Qa ' j (S a 2 &~ja ). (3.12.6b) 0+ 0+ Thus Q+ and Q+ are local and depend only on &a, 8a, and &+, but not on (a ,(" a _, and &T, whereas Q_ and Q. are nonlocal and depend on all &a and &a. We expand the transformed superfield strength 1 in powers of (" (the external indices of 1 are sup- pressed): m.  144 3. REPRESENTATIONS OF SUPERSYMMETRY where the mth power (', and thus #(m, is totally antisymmetric in isospinor indices. Each is a representation of a subgroup of supersymmetry that we call "on- shell" supersymmetry, and that includes the Q+ transformations, the transverse SO(2) part of Lorentz transformations (and a corresponding conformal boost), SU(N) (or U(N)), and all four translations (as well as scale transformations in the massless case). Although each is a realization of the full supersymmetry group off-shell as well as on-shell, on-shell supersymmetry is the maximal subgroup that can be realized locally (and in the interacting case, linearly). The remaining generators of the full supersymmetry group (including the other Lorentz generators, that mix 8" with (") mix the various 's. In particular, Q_ and Q- allow us to distinguish physical and auxiliary on-shell superfields: Auxiliary fields vanish on-shell, and hence must have transformations proportional to field equations. We go to a Lorentz frame where OT= 0. In this frame, Qa_ = fio and Q* = i(" + i(a D) The Q_ and Q. supersymmetry variation of the highest (a compo- nent of 0. In general, an on- shell representation can be reduced by a reality condition of the form DNS (0)N 5 when the "middle" (82N) component of 0 has helicity 0 (i.e., is invariant under trans- verse SO (2) Lorentz transformations ). (Compare the discussion of reality of off-shell representations in sec. 3.11.) Putting together the results of sec. 3.11 and this section, we have the following reductions: general superfields (4N 0's; physical + auxiliary + gauge) - chiral field strengths (2N 0's; physical + auxiliary = irreducible off-shell representations) - chiral on-shell superfields (N 0's; physical = irreducible on-shell representations). All three types of superfields can satisfy reality conditions; therefore, the smallest type of each has 24N, 22N and 2N components, respectively (when the reality condition is allowed), and is a "real" scalar superfield. All other representations are (real or complex) superfields with (Lorentz or internal) indices, and thus have an integral multiple of this number of components. These counting arguments for off-shell and on-shell components can also be obtained by the usual operator arguments (off-shell, the counting is the same as for on-shell massive theories, since p2 -f 0), but superfields allow an explicit construction, and are thus more useful for applications. Similar arguments apply to higher dimensions: We can use the same numbers there (but taking into account the difference in external indices), if we understand "4N" to mean the number of anticommuting coordinates in the higher dimensional superspace.  146 3. REPRESENTATIONS OF SUPERSYMMETRY For simple supersymmetry in D < 4, because chirality cannot be defined, the counting of states is different.  3.13. Off-shell field strengths and prepotentials 147 3.13. Off-shell field strengths and prepotentials We have shown how superfields can be reduced to irreducible off-shell representa- tions (sec. 3.11), which can be reduced further to on-shell superfield strengths (sec. 3.12). To find a superfield description of a given multiplet of physical states, we need to reverse the procedure: Starting with an on-shell superfield strength F(o) that describes the multiplet, we need to find the off-shell superfield strength W that reduces to F(0) on shell, and then find a superfield prepotential T in terms of which W can be expressed. There is no unambiguous way to do this: The same F(o) is described by different W's, and the same W is described by different l's. However, for a class of theories that includes many of the models that are understood, we impose additional requirements to reduce the ambiguity and find a unique chiral field strength and a family of prepotentials for a given multiplet. The multiplets we consider have on-shell superfield strengths of Lorentz representa- tion type (A, 0) (superspin s = A) and are isoscalars: F(O)a1...a2. From (3.12.4b), this implies that the F(o)'s are chiral and therefore can be generalized to off-shell irreducible (up to bisection) field strengths Wai...a2s, D" Wai...Z2= 0. Physically, the W's correspond to field strengths of conformally invariant models. (They transform in the same way as Ca1''aN: as SU(N) scalars, but not U(N) scalars). In the physical models where these superfields arise, the chirality and bisection conditions on W are linearized Bianchi identities. We can use the projection operator analysis of sec. 3.11 to solve the identities by expressing the W's in terms of appropriate prepotentials. When there is no bisection (s + 1N not an integer), the W's are general chiral 2 superfields: Wai... 2 = 2Ngai...a2s The IV1a...2 's may be expressed in terms of more fun- damental superfields. An interesting class of prepotentials are those that contain the lowest superspins: In that case, the W's have the form N < 2s - 1: Wcv..vD2N( D N!(a...aN~ &cN+1 N+ M iMgGNM1''~jl' N;> 2s -1: W1=I52NDN [b..b-M 1(f "l''"M 1 where 'I is an arbitrary (complex) superfield and M= s - -(N + 1).  148 3. REPRESENTATIONS OF SUPERSYMMETRY If W is bisected (s +91 N integer, (1 - K)W = 0), then T must be expressed in 2 terms of a real prepotential V that has maximum superspin s. W has a form similar to (3.13.1): N < 2s: W. 1 2NDN & 1 .2).. iM +1/ 0103 (2s)! (al''aN aN+1 aN+~M aN+M+1'''a2s)f31''' M N > 2s: W = U...a2 1I2NDN [bl...bMlb- l-M(3.13.2) (-M)! [a1...aM] 1..2s 1l -M where M = s - 1N. 2 Whether or not W is bisected, ambiguity remains in the prepotentials T , V, since they may still be expressed as derivatives of more fundamental superfields: This leads to "variant off-shell multiplets" (see sec. 4.5.c). Our expression (3.13.1) for T in terms of IT is an example of such an ambiguity: There is no a priori reason why T must take the special form, unless it is obtained as a submultiplet of a bisected higher-N multiplet (as, 3 for example, in the case of the N = 1 spin , 1 m tiplet of the N = 2 supergravity multiplet). Modulo such ambiguities, the expressions for W in terms of T and V are the most general local solutions to the Bianchi identities constraining W (i.e., chirality, and if possible, bisection).  Contents of 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS 4.1. The scalar multiplet 149 a. Renormalizable models 149 a.1. Actions 149 a.2. Auxiliary fields 151 a.3. R-invariance 153 a.4. Superfield equations 153 b. Nonlinear o-models 154 4.2. Yang-Mills gauge theories 159 a. Prepotentials 159 a.1. Linear case 159 a.2. Nonlinear case 162 a.3. Covariant derivatives 165 a.4. Field strengths 167 a.5. Covariant variations 168 b. Covariant approach 170 b.1. Conventional constraints 171 b.2. Representation-preserving constraints 172 b.3. Gauge chiral representation 174 c. Bianchi identities 174 4.3. Gauge-invariant models 178 a. Renormalizable models 178 b. CP(n) models 179 4.4. Superforms 181 a. General 181 b. Vector multiplet 185 c. Tensor multiplet 186 c.1. Geometric formulation 186 c.2. Duality transformation to chiral multiplet 190 d. Gauge 3-form multiplets 193 d.1. Real 3-form 193 d.2. Complex 3-form 195 e. 4-form multiplet 197 4.5. Other gauge multiplets 198 a. Gauge Wess-Zumino model 198  b. The nonminimal scalar multiplet 199 c. More variant multiplets 201 c.1. Vector multiplet 201 c.2. Tensor multiplet 203 d. Superfield Lagrange multipliers 203 e. The gravitino matter multiplet 206 e.1. Off-shell field strength and prepotential 206 e.2. Compensators 208 e.3. Duality 211 e.4. Geometric formulations 212 4.6. N-extended multiplets 216 a. N=2 multiplets 216 a.1. Vector multiplet 216 a.2. Hypermultiplet 218 a.2.i. Free theory 218 a.2.ii. Interactions 219 a.3. Tensor multiplet 223 a.4. Duality 224 a.5. N=2 superfield Lagrange multiplier 227 b. N=4 Yang-Mills 228 b.1. Minimal formulation 228 b.2. Lagrange multiplier formulation 229  4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS In this chapter we discuss interacting field theories that can be built out of the superfields of global N = 1 Poincard supersymmetry. This restricts us to theories describing particles with spins no higher than 1. The simplest description of such theo- ries is in terms of chiral scalar superfields for particles of the scalar multiplet (spins 0 1 1 and -), and real scalar gauge superfields for particles of the vector multiplet (spins - and 1). However, other descriptions are possible; we treat some of these in a general framework provided by superforms. We describe N = 1 theories and also extended N < 4 theories in terms of N = 1 superfields. Our primary goal is to explain the struc- ture of these theories in superspace. We do not discuss phenomenological models. 4.1. The scalar multiplet a. Renormalizable models The lowest superspin representation of the N = 1 supersymmetry algebra is car- ried by a chiral scalar superfield. In sec. 3.6 we described its components and transfor- mations. In the chiral representation we have + m(b2 4 ) - -2mA(AJF + AA2) - ±4A2A2F + A(A@2 + A47)] , (4.1.8) and @ =- (- i&"A - ca(ml + ~AAE) . (4.1.9) Therefore the Wess-Zumino action gives equal masses to the scalars and the spinor, cubic and quartic self-interactions for the scalars, and Yukawa couplings between the  152 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS scalars and the spinor, all governed by a common coupling constant. After eliminating F, the supersymmetry transformations of the spinor are nonlin- ear; this makes an analysis of the supersymmetry Ward identities without the auxiliary fields difficult. This is not the only problem caused by eliminating auxiliary component fields: The transformations are not only nonlinear, but also dependent on parameters in the Lagrangian, and it is difficult to discover further supersymmetric terms that could be added to the component Lagrangian (e.g., gauge couplings). Furthermore, equation (4.1.7) is not itself supersymmetric unless the equation of motion of the spinor is satis- fied; only then is bF = - E - Ocivbo(4.1.10a) the same as bF(A) 6= (-mA - 1 AA2) - (m + A A)E( . (4.1.lOb) 2 For this reason, formulations of supersymmetric theories that lack the component auxil- iary fields are often called "on-shell supersymmetric". Indeed, if we calculate the com- mutator of two supersymmetry transformations acting on the spinor, we find that the fields A, b, form a representation of the algebra (i.e., the algebra closes) only if the spinor equation of motion is satisfied. The Wess-Zumino model can be generalized to include several chiral superfields. The most general action that leads to a conventional renormalizable theory is S = fd4xd4O i +fd4x d2OP( 1 , (4.1.27b) Ki r , (4.1.27c) with T1,3 (1, 1, ... -1, -1, ...) depending on the signature of the manifold. If the V describe physical matter multiplets, rIZ 5 = . In a normal gauge, all the connections vanish at the point Go, the Riemann curvature tensor has the form: RZ3 k1 = Kiri , (4.1.28a) with all other components related by the usual symmetries of the Riemann tensor or zero, and hence the Ricci tensor is simply: Rh3 = Kielf . (4. 1.28b) In a general gauge, the connection is where (K-1);k is the inverse of the metric 1Kk1; all other components are related by complex conjugation or are zero. The contracted connection is, as always,  4.1. The scalar multiplet 157 rz = zj 3= [ln det Kkl ] Z. (4.1.29b) The Riemann tensor in a general gauge is RZsI -=ZKI - (K-1)m" ]Kkm ]K 31 (4.1.30a) and the Ricci tensor has the simple form Rk3 - Rik1 ]Ki [ in det ]KZI ]k3 . (4.1.30b) Manifolds can have symmetries, or isometries. On a Kshler manifold, an isometry of the metric is, in general, an invariance of the Kshler potential 1K up to a Kshler gauge transformation (4.1.26). One can require the isometry to be an invariance of the potential. (Actually, this is only true if there is a point on the manifold where the isom- etry group is unbroken, i.e., the transformations do not shift the point.) This (partially) fixes the Kshler gauge invariance: It is no longer possible to go to a normal gauge (4.1.27). However, holomorphic coordinate transformations still make it possible to choose normal coordinates, where the metric ]KZ' satisfies (4.1.27c), and its holomorphic derivatives (]K13)2... a i... satisfy (4.1.27b) (likewise for the antiholomorphic derivatives) but the conditions (4.1.27a) are not satisfied. In arbitrary coordinate systems, the isometries act on the coordinates as -4b = AA kz , Aog = X kAi (4.1.31) where the A's are infinitesimal parameters (A A=X are constant unless we introduce gauge fields and gauge the isometry group; supersymmetric gauge theories are discussed in the remainder of this chapter), and the k(<, 4)'s are Killing vectors. These satisfy Killing's equations: kA z; + kA3; = kA ;j + kA ;Z = 0 (4.1.32a) kAs;j + (]K-1)kAk]Kl= 0 . (4.1.32b) where k - a - (4. 1.32c) iV - eYv(V e-2V + e-V[VeV e-V = 0 , (4.2.26b) and hence 2 sinh( {L,)(SV ) =e-{v ( ev e-{v L [e-ViAe2V - e2Vie-V]  4.2. Yang-Mills gauge theories 165 = iLv[cosh(} L~)(A1-A) - sinh(}2L~)(XA+A)] , (4.2.27) from which it follows 8V =-{ iL~[X+ A + coth({L )(A - A1)] i (A-A) - }-i[V ,A1 + A] + 0(V2) . (4.2.28) From the transformations (4.2.28) and the parameter (4.2.11-12) we find the non- abelian WZ gauge-preserving "supersymmetry" transformations: 8Aa _ ( EE Aa + EaAc ) 6Aa = - E3f a + EieD' -D' = } V (-aA - Aa) , (4.2.29) where now f1 is the self-dual part of the nonabelian field strength and Va; = Bag - iA g. The nonlinearity comes from the gauge-covariantization of the linear transformations (4.2.13). The components of the nonabelian vector multiplet are covari- ant generalizations of the abelian components; in the WZ gauge, they are the same as (4.2.4a) (see also (4.3.5)). a.3. Covariant derivatives The gauge field V can be used to construct derivatives, gauge covariant with respect to A transformations VA = DA - FA = a , V3 , Va&) , (4.2.30) defined by the requirement (VA1)' =eiA (VAIb) , (4.2.31) i.e., V'A eiAVAe-iA , (4.2.32a) or  166 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS VA = i[A,VAI . (4.2.32b) Since A is chiral, V= D is covariant without further modification: V' =iAeiZA=V . (4.2.33) The undotted spinor derivative Da is covariant with respect to A transformations. We can use eV to convert it into a derivative covariant with respect to A (see (4.2.21)); V a e-VDaeV transforms correctly: V'a_ (eiA e-ve -A)Da(eiAeve-iA) _ eiA e-VDaeVe-iA = eiA Vae-1 . (4.2.34) Finally, we construct Va by analogy with (3.4.9): Va=V - i{V , V}. Its covari- ance follows from that of Va and Vg. We summarize: VA = (e-VDaeV, D , - i{Va , V }) . (4.2.35) These derivatives are not hermitian. Their conjugates VA are covariant with respect to A transformations: VA =(Da , JDeDe- zV V, Vc}) V'A = eA V Ae-21 . (4.2.36) The derivatives VA (VA) are called gauge chiral (antichiral) representation covariant derivatives. They are related by a nonunitary similarity transformation V A= eVVAe-V . (4.2.37) This is analogous to the relation between global supersymmetry chiral and antichiral representations DN=eUDA(+e-U (4.2.38) of (3.4.8).  4.2. Yang-Mills gauge theories 167 The gauge covariant derivatives are usually defined in terms of vector representa- tion DA's; if we express these in terms of ordinary derivatives, (4.2.35) becomes VA= (e-V e-U eU eV e2Uj"g-jU , - i{V , V}) . (4.2.39) By a further similarity transformation VA - e-2U VA e2U, we go to a new representation that is chiral with respect to both global supersymmetry and gauge transformations: VA =e-2 e-V e2U ejU ev eVj ,,-i{V,,V}) (4.2.40) We define V by e2UeV e2jU eU+f .(4.2.41) In this form, it is clear that V gauge covariantizes U: iOa'(Oa( - - - + i6jOz(O&" - iA ") + -"- -. This combination transforms as (eU+Vf eiAeU+Ve-iA , & =0 A=0 . (4.2.42) There also exists a symmetric gauge vector representation that treats chiral and antichi- ral fields on the same footing. Such a representation uses a complex scalar gauge field Q, and requires a larger gauge group. We discuss the vector representation in subsec. 4.2.b, where the covariant derivatives are defined abstractly, and where it enters natu- rally. a.4. Field strengths The covariant derivatives define field strengths by commutation: [VA , VB} TABCV C - iFAB , (4.2.43) with V = VA TA, and TA in the adjoint representation. From the explicit form of the covariant derivatives (4.2.35) we find that the torsion TABC is the same one as in flat global superspace (3.4.19), and some field strengths vanish: Fay =F . =F . =0 . (4.2.44) The remaining field strengths are F. .=C..D2(e-VD ev)=iC..W ,  168 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS F ..- --(C.V Wn +C V"W Wa = iD2(e-VDaeV) W( e-v W eV e-V (-Wa)teV .(4.2.45) (Recall that W (Wa)t implies W _= (-Wa)f (3.1.20).) Thus all the field strengths of the theory are expressed in terms of a single spinor W( that is the nonlinear version of (4.2.2). It satisfies Bianchi identities analogous to (4.2.1): VWa= - VaW3 . (4.2.46) 3 It is chiral, has dimension -, and can be used to construct a gauge invariant action S = 1trfd4xd2O W2 = - 2tr f d4x d40 (evDaev)D2(evDaev) V = VATA , trTTB AB*(4.2.47) As in the abelian case, this action is hermitian up to a surface term (see discussion fol- lowing (4.2.15)). a.5. Covariant variations To derive the field equations from the action (4.2.47), we need to vary the action with respect to V. However, since V is not a covariant object, this results in noncovari- ant field equations (although multiplication by a suitable (but complicated) invertible operator covariantizes them). In addition, variation with respect to V is complicated because V appears in eV factors. We therefore define a covariant variation of V by 1- e-Lv AV= e-VoeV = 5V = 6V +... . (4.2.48) AV satisfies the chiral representation hermiticity condition as in (4.2.37). In practice, we always vary an action with respect to V by expressing its variation in terms of SeV, and then rewriting that in terms of AV. We thus define a covariant functional deriva- AF[EV tive by (cf. (3.8.3))  4.2. Yang-Mills gauge theories 169 F [V +6V ] - F [V ] =( AV AVI) + O((AV)2) . (4.2.49) F[V+ ~1 F[] AV We now obtain the equations of motion from: g26S = i tr f d4x d40 6(e-VDaev)Wa = i tr f d4x d40 [evDaev , AV]Wa - i trfd4xd4OAVVaWa , (4.2.50) which gives 2AS g AV - iVWa= 0 . (4.2.51) * * * At the end of sec. 3.6 we expressed supersymmetry transformations in terms of the spinor derivatives Da. Using the covariant derivatives that we have constructed, we can write manifestly gauge covariant supersymmetry transformations by using the form (3.6.13) (for w = 0) and adding the gauge transformation A = iD2("7Da() , (4.2.52a) where FA is defined in (4.2.30). We then find e-Vog eV = (WVa + WaV )( = (Wa'e-vDaev + e-VWeViDa)( (4.2.52b) (where ( is a real x-independent superfield that commutes with the group generators, e.g., Va( Da(). Since (4.2.52b) is manifestly gauge covariant, it preserves the Wess- Zumino gauge (but it is not a symmetry of the action after gauge-fixing). The corre- sponding supersymmetry transformations for covarianitly chiral superfields @, V$5 0 with arbitrary R-weight w are - iV2[(V(a)Va + w(V2()]I .5 (4.2.52c)  170 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS b. Covariant approach In this subsection we discuss another approach to supersymmetric Yang-Mills theory that reverses the direction of the previous section. We postulate derivatives transforming covariantly under a gauge group, impose constraints on them, and discover that they can be expressed in terms of prepotentials. This procedure will prove espe- cially useful in studying supergravity and extended super-Yang-Mills, so we give a detailed analysis for the simpler case of N = 1 super-Yang-Mills. We start with the ordinary superspace derivatives DA satisfying [DA, DB} TABcDc, where TABc is the torsion and has only one nonzero component T ." (see(3.4.19)). For a Lie algebra with generators TA we covariantize the derivatives by introducing connection fields VA = DA -iFA , (4.2.53) where FA = FABTB is hermitian and VA - - (-)AVA. At the component level we have Fa = va +±Ov +.. F,a Wa+ - --,(4.2.54a) and hence Va Oa -vca + - C,(&a&z va&).. V2-iQ O(a j- t ) + Va=a-iWa+--- , (4.2.54b) so that the component derivatives are covariantized. Under gauge transformations the covariant derivatives are postulated to transform as V'A eiKV -iK ,(4.2.55) where the parameter K =KA TA is a real superfield. K =w(x) + O"K~la(x) + OaK~l)(x) + --- . (4.2.56) This is very different from what emerged in the previous section: Instead of chiral  4.2. Yang-Mills gauge theories 171 representation derivatives transforming with the chiral parameter A, we have vector rep- resentation hermitian derivatives, transforming with the hermitian parameter K. The asymmetric form of the previous section will emerge when we make a similarity transfor- mation to go to the chiral representation. For infinitesimal K, we find the component transformations: (v - a& - v , wl - iw ow -1=[8,3- iw,3 , w] , (4.2.57) where w= KI, owa = [D , Dc]KI = Jg. The component gauge parameter w can be used to gauge away Im va algebraically; however, the component fields Re vc, and was both remain as two a priori independent gauge fields for the same component gauge transformation. To avoid this we impose constraints on the covariant derivatives. b.1. Conventional constraints Field strengths FAB are defined by (4.2.43). Substituting (4.2.53) we find FAB = D[AIFB) - i[FA ,FB} - TABCITC . (4.2.58) In particular, Fac = D4I + D47 - i{F, 1,7} - iFe; . (4.2.59) If we impose the constraint F 3 = 0 , (4.2.60) (4.2.59) defines the vector connection 1,3a in terms of the spinor connections. (In com- ponents, this expresses w ; in terms of v , and va.) In any theory one can add covariant terms to the connections (e.g., (3.10.22)) without changing the transformation of the covariant derivatives. If we did not impose the constraint (4.2.60) on the connections 'A, we could define equally satisfactory new connections 17'A (17a,17s, F,3 - iFa3) that identically satisfy the constraints. For this reason (4.2.60) is called a conventional constraint. It implies VA = Va,,{ia , V }) .26 (4.2.61)  172 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS The theory now is expressed entirely in terms of the connection Fa. However, it contains spin s > 1 gauge covariant component fields, for example " - =F "| = i[b, D a]F/3)| + - . (4.2.62) It also contains a superfield strength F(1 whose 0-independent component fa/ = Fajp|= D(al) + -... , (4.2.63) is a dimension one symmetric spinor (equivalent to an antisymmetric second rank ten- sor). Because of its dimension, it cannot be the Yang-Mills field strength. Although in principle the theory might contain such fields (as auxiliary, not physical, components), in the covariant approach there are generally further types of constraints that eliminate (many) such components. b.2. Representation-preserving constraints To couple scalar multiplets described by chiral scalar superfields to super-Yang- Mills theory, we must define covariantly chiral superfields 1: The covariant derivatives transform with the hermitian parameter K, and all fields must either be neutral or transform with the same parameter. However, K is not chiral, and gauge transforma- tions will not preserve chirality defined with D3. Instead we define a covariantly chiral superfield by p 3b = 0 , '=eK Vab= 4'4eiZK . (4.2.64) This implies 0 ={V3,V} I-iF 4T. (4.2.65) Consistency requires that we impose the representation-preserving constraint F Fp .0 . (4.2.66 ) This can be written as {Va , V} =0 . (4.2.67) The most general solution is  4.2. Yang-Mills gauge theories 173 Va = e-Dae , = QATA , (4.2.68) where QA is an arbitrary complex superfield. Eq. (4.2.67) states that Va satisfies the same algebra as Da, and the solution expresses the fact that they are equivalent up to a complex gauge transformation. Hermitian conjugation yields V = eQDge-Q . (4.2.69) Thus VA is completely expressed in terms of the unconstrained prepotential Q by the solutions (4.2.61,68,69) to the constraints (4.2.60,66). The K gauge transformations are realized by (e) = eQe-iK . (4.2.70) However, the solution to the constraint (4.2.67) has introduced an additional gauge invariance: The covariant derivatives (4.2.68) are invariant under the transformation (e )' = ede , DMA = 0 . (4.2.71) Therefore, the gauge group of Q is larger than that of IA We define the K-invariant hermitian part of Q by eV QCCn . (4.2.72) The K gauge transformations can be used to gauge away the antihermitian part of Q. 1 In this gauge, Q = - V, and A transformations must be accompanied by gauge- 2 restoring K transformations: e)'=eAeQeiK(A) eiK(A) - e-sAe-iX ie2Qe-iA In any gauge, the transformation of V is (eC- eA2 eiA(4.2.74) We have defined covariantly chiral superfields b by (4.2.64). We can use Q (see (4.2.69)) to express them in terms of ordinary chiral superfields , is a (in general reducible) matrix representa- tion of the generators TA of some group. However, in contrast to (4.3.9), when we vary (4.3.11) with respect to V, we get an equation that in general does not have an explicit solution: 4DeVTAD -ctrTA =0 (4.3.12) (To derive (4.3.12), we use the covariant variation (4.2.48) AV = AVATA tr AV = troV .) e-VbeV, and  4.4. Superforms 181 4.4. Superforms a. General In ordinary spacetime, there is a family of gauge theories that can be constructed systematically; these theories are expressed in terms of p-forms I, = do" /\ dom/\ - - -"/\dz... where the differentials satisfy dxmA dxo = -dxA dxr. The "tower" of theories based on forms is: F07= scalar, 1 = vector gauge field, F2 = tensor gauge field, F3 = auxiliary field, and F4 = "nothing" field. Their gauge transformations, field strengths, and Bianchi identities are given by gauge transformation: of, = dK,_1 field strength: Fp+1 = d1, Bianchi identity : dFp+1 = 0 . (4.4.1) Here K,, Fp, FP are p-form gauge parameters, gauge fields, and field strengths respec- tively, and d= dX mOn. By definition, -1-forms vanish, and 5-forms (or (D+1)-forms in D dimensions) vanish by antisymmetry. The Bianchi identities and the gauge invariance of the field strengths are automatic consequences of the Poincard lemma dd = 0. In superspace the same construction is possible, using super p-forms: F,= (-P(P-1 dzM1A ... A dzMp..M (note the ordering of the indices), where now dzMA dzN - ()MNdzNA dzM , (4.4.2b) the coefficients of the form are superfields, and d = dZM&M. The same tower of gauge parameters, gauge fields, field strengths, and Bianchi identities can be built up (now using the sup erPoincard lemma dd =0). An advantage of this description of flat super- space theories is that it generalizes immediately to curved superspace and determines the coupling of these global multiplets to supergravity. However, superforms do not describe irreducible representations of supersymmetry unless we impose constraints. To maintain gauge invariance, these constraints should be  182 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS imposed on the coefficients of the field strength form; when the constraints are solved, the coefficients of the (gauge) potential form are expressed in terms of prepotentials. In table 4.4.1 the prepotentials A correspond to the constrained super p-form A, and the expressions dA, correspond to dAn. 0o i( - @) 1 V iD2D"V 1 2 2 (D$4V + D ,a) 3 V 1217 4 4D0 Table 4.4.1. Simple superfields (prepotentials) corresponding to superforms In this Table k and 4V are chiral and V is real. The relation A = A4_, corresponds to Hodge duality of the component forms. The constrained super p-forms correspond to particular prepotentials A whether A, is a gauge parameter K,, a potential FP, a field strength FP, or a Bianchi identity (dF),. The explicit expressions for A, in terms of A take the same form whether A is K,F,F,or dF. Thus the prepotentials give rise to a tower of theories that mimics (4.4.1): The gauge field strength and Bianchi identities at one level are the gauge parameter and field strength at the next level. If A,_1, A,, and Ap+1 are the gauge parameter K,_1, the gauge field FP, and the field strength Fp+1 superforms, respectively, then the gauge transformation, field strength, and Bianchi identities of the prepotentials are gauge transformation: field strength: Bianchi identity : oF, = dK,_1 Fp+1 = dF, dF p+1 = 0 (4.4.3) The Lagrangians for all p-form theories are quadratic in the field strengths, without extra derivatives. We discuss details in the subsections that follow.  4.4. Superforms 183 Under a supersymmetry transformation the superforms are defined to transform as I'(z' , dz') = I'(z , dz) , (4.4.4) where (cf. (3.3.15)) dz' (d' ,dO', dx'")= (d9", dO8, dx"" - ± (-dOl" + E= 1DF Fa 3 =D(aF 3 + D;Fa+ O F aOc = D (aF/)c + &cFa,/ a c =DF"+ D Fac + &cF - C 17- i Ca-YF(0.) Fabc =C .(DaFr(13) -&I O F ) + C(D F(".) +10&F) 2 Fab c F F. 6abcdF (OiN) (4.4.19) where we have used (3.1.22). We can impose two conventional constraints. The first, Fa /M 0 F Z ~ [D far "> + D F ]~ (4.4.20) gives (4.4.21) which implies F31.=i CI b+ijD F .+ ±D~17r for an arbitrary spinor 4b y. The second conventional constraint, (aF . 3= (4.4.22) (4.4.23) gives  188 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS F = - i ( [D(C' 3+ UF /p-0 F ) , (4.4.24) which implies F ) = lD( ) - 2D2F - z/F . (4.4.25) The potential I is pure gauge: It can be gauged to zero using (4.4.18). To eliminate the remaining unwanted physical states we choose two additional constraints F a,, =F O0 . (4.4.26) The first implies F, is pure gauge, and the second imposes Dae) = 0 , UIbp = 0 . (4.4.27) In the gauge F,= =F = 0, all of FAB is expressed in terms of 4Ica; thus the superfield 4b, is the chiral spinor prepotential that describes the tensor gauge multiplet. The constraints also imply that all the nonvanishing field strengths can be expressed in terms of a single independent field strength G = - 2 (D"Go + D 3) . (4.4.28) For example, F . "= i SC?6 jG = T ."G . (4.4.29) G is a linear superfield: D2G = 0. It is invariant under gauge transformations of the pre- potential oa = iD2DaL , L=L . (4.4.30) Projecting the components of Ga we have: The components of the gauge parameter that enter o~a are: La = iD2DaL|  4.4. Superforms 189 L -) = DaD2Da L = L() L D = (a &(2D L| = [Da Dv1, ] L| P- - (La)(,La(=Lay 3.(4.4.32) The components xa and B can be algebraically gauged away by La and L(1) respectively, whereas La& is the parameter of the usual gauge transformation for the tensor gauge field t,/. The spinor be is the physical spinor of the theory (up to terms that vanish in the WZ gauge). The gauge invariant components are found by projecting from the field strength G: A = G|I f=|DaG = [ ,ba]GiOtaac-t) fa =Fa I=[D , Da1 G I-= Z(&/ta/3 -&a'7 ") D2G = D2G= 0 . (4.4.33) Since there is only one physical spinor in the multiplet, G has dimension one. This determines the kinetic action uniquely: Sk-7-f d'x d4O G2 . (4.4.34) The corresponding component action is Sk -fd.x [ ADA + (f")2 + y 8ga . (4.4.35) Note that none of the fields is auxiliary. The physical degrees of freedom are those of the scalar multiplet. On shell, the only difference is the replacement of the physical pseudoscalar by the field strength of the antisymmetric tensor.  190 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS c.2. Duality transformation to chiral multiplet We can write two first order actions that are equivalent to Sk. Introducing an auxiliary superfield X, we define S'k-fd'x d4[ X2 - GX . (4.4.36a) Varying X and substituting the result back into S'k, we reobtain Sk. We also see that the tensor multiplet is classically equivalent to a chiral scalar multiplet: Varying W + -1(} D2Wc + D D2) 2 U gb = 0 . (4.5.35) They induce the following changes in the actions (1) S(i) - S() - fd6z W2 + fd'z [- (WWc, + W% 3) + G2 - G(D"'Wc, + Da()], (2) S(2) - S(2) - f d6z W 2 + f d'z [ - (W"W c + W% ) - 24I~I - (D"Wc, + NU )] . (4.5.36) Although the redefinitions are nonlocal, the actions remain local. (Actually, in case (1) we can also use simply W + i 'DV + iD. In the above field redefinitions we introduced a vector multiplet V and either a tensor multiplet Ic, or a chiral scalar multiplet 1. These choices are a reflection of the representations that were introduced by the additional projections: A vector multiplet H1! _ 1T, a tensor multiplet I2 + ,, and a chiral scalar multiplet II1,0Pc,. In the pres- '22 ence of the compensating multiplets the gauge variation of Wc, is given by (4.5.27). The compensating multiplets transform as follows: (1) oV = i(c - Q) , 64OZ -AOz + ig2DOK3, (2) bV = i(SZ - U) ,64D= - 2U ,(4.5.37) Since they are compensators, they can be algebraically gauged to zero. In the resulting gauge, the transformations (4.5.27) of 'Pc are restricted back to (4.5.34). The two inequivalent formulations of the gravitino multiplet, one using a tensor multiplet compensator and the other using a chiral scalar compensator, lead to different auxiliary field structures at the component level. In the Wess-Zumino gauge for case (1) the components of the gravitino multiplet are  210 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS AaDADa'T + D24' + 1±Da(DI Tj+ D/ i + Wa - DaG, P -4jD aD2W a - DazD2IP, + DavWac) A'a =(D2DzW'Pa D2Da4'&~) Xa D2Da[G - 2 (D/3Ti + D Ti3) Ba 1[DD~]V+(DaTj+D4a) =t- ~[Da DJ4' + b Da'D"W,+ D2TL W" + [ ] .(4.5.38) For case (2) we have instead Aa =UfD"+ D2'Pa + W a-l9$ P - (DavD2'p - DavD2IP, + DavWa)  4.5. Other gauge multiplets 211 Ba[D}[D IV+i(Da&+D&34'a) - [D " , ]4 + ioj[ D-D75Tf + D2TJ + W ] .(4.5.39) In case (1), the gauge field t'c, of the tensor multiplet replaces the complex scalar com- ponent field J, which corresponds to the auxiliary field of the chiral scalar multiplet of case (2). In the component action t'c, only appears as t', &oaA'/" - P'.67 A' . This term is invariant under separate gauge transformations of t', and A'1". Also it should be noted that the field A'a is real. In case (2) A~ # (A .) has no gauge transforma- tions because the physical scalars of the chiral scalar multiplet have become the longitu- dinal parts of A and cancel the transformation in (4.5.30). In both cases, the physical vector of the compensating vector multiplet has become the physical vector of the grav- .1 itino multiplet, while the physical spinor of the vector multiplet becomes the spin - part of the gravitino and cancels the spinor translation in (4.5.30). e.3. Duality Since the two formulations above differ in that (1) has a tensor compensator where (2) has a chiral compensator, using the approach of sec. 4.4.c.2, we can write first order actions that demonstrate the duality between the two formulations. For example, we can start with (1) and write S'(i)= S(i) [T , V] + fdz [X2 - X(D"WTy + DT&3) - 2X(4 + 4)] . (4.5.40a) Varying the chiral field k leads to X = G and formulation (1), whereas eliminating X results in formulation (2). Similarly, we can start with (2) and write S'2 (2)[( ,V ] + fdoz [-X2 - X(D"ay +U DNg) + 2XG ] . (4.5.40b) Varying the linear superfield C we find X = @+ and formulation (2), whereas elimi- nating X leads directly to (1). There are other inequivalent formulations where we replace V by the variant vec- tor multiplet and/or replace 1 by either the real or complex three-form multiplets. This  212 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS simply replaces some of the scalar auxiliary fields with gauge three-form field strengths. e.4. Geometric formulations Finally, we give geometrical formulations of the theories. To describe a multiplet that gauges a symmetry with a spinorial gauge parameter, we introduce a super 1-form FA/ with an additional spinor "group" index. The analysis is simplified if the irreducible multiplet is considered first. The irreducible theory was described by W,/ in (4.5.26). To describe this multiplet geometrically, we introduce more gauge fields (in particular a complex super 1-form FA (# PA)) and enlarge the gauge group. When we get to super- gravity we will find that this process can also be carried out. There the irreducible mul- tiplet is the Weyl multiplet and the enlarged group is the conformal group. The final form of the (4,1) multiplet with more irreducible multiplets and compensators is analo- gous to Poincard supergravity. We will use the words "Poincard" and "Weyl" for the (4, 1) multiplet to emphasize this analogy. The complete set of gauge fields and gauge transformations that describe the Weyl 3 (- , 1) multiplet is: 6FA / = DAK - oAL , L= DAK b - bA6L oFA = DAL , 5Al=T DAL. (4.5.41) The K-terms are the usual gauge transformations associated with a superform and the L-terms are the "conformal" transformation. Recall that we found that the gauge trans- formations of the irreducible multiplet contain an S-supersymmetry term. L is the superfield parameter that contains these component parameters. The vector component of the complex super 1-form FA is the component gauge field whose field strength appears in (4.5.28). The field strengths for FA are those for an ordinary (complex) vec- tor multiplet, but those for 03 and its conjugate I 3 must be L-covariantized: F AB? D[AFB)~ - TAB DpD +[ASB) , F ABi D[AFB)4 - I ABD D [ASB) . (4.5.42) We can now impose the constraints :  4.5. Other gauge multiplets 213 Fa = F .?'= F =Fa, = 0, Fa'" + Fa = 0 , F = F . =0 , (F.0 --0). (4.5.43) Even with the L invariance the geometrical description here does not quite reduce to the irreducible multiplet W(/. However, these constraints reduce the super 1-forms to the irreducible multiplet plus the compensating vector multiplet, which are the two irre- ducible multiplets common to both forms of the Poincard (4 ,1) multiplet, and thus are sufficient for their general analysis. The explicit solution of these constraints is in terms of prepotentials C, '1<, 'T(complex), and V(real): F) - - i[ DDQTY' + DaD jTJO + 6aci(F ~- DT) Fa = DaT ,F3 = - DUg(No- a) - D 2 (T( - 0 ) + U - W3 , Fa = - i D2D( Sa - Na) + OaW ; (4.5.44) where W D = iD2DaV. The prepotentials transform under K, and L, as well as under new parameters Q(complex) and Aa(complex) under which the F's are invariant (this is analogous to the A-group parameters in super-Yang-Mills): 6TP=Ka + DaQ , Sea = Ka - Aa, D Aa = 0 S = L -D20Z , SV = i( 0- G) . (4.5.45) As with the vector multiplet, we can go to a chiral representation where $Ta and NTo only appear as the combination WTe = Na -N'T', with a - a) =Aa + DaQ. (4.5.46)  214 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS At this point, by comparison with (4.5.27), we can identify T, and V with the cor- responding quantities there. To recover the full Poincard theory, we must break the L invariance. To break the L invariance, we introduce a "tensor" compensator 0 or $, to obtain the tensor-multiplet or scalar-multiplet, respectively. These "tensor" (scalars) are not prepotentials, and transform covariantly under all of the gauge transformations defined thus far. By covariant, we mean that these transform without derivatives DA. 60 = - ( L + ) , (4.5.47) 64D= -LE. (4.5.48) We now impose the L-covariantized form of the usual constraints (D2G = 0 and Ug b = 0) which describe tensor and chiral scalar multiplets; U []UG O + (F& + I" )] + h.c.= 0, (4.5.49) DUge+I, = 0 . (4.5.50) The invariance of these constraints follows directly (4.5.27,37,47,48). (The hermitian conjugate term above is necessary to avoid constraining T itself.) These constraints can be solved in terms of prepotentials: O=G-( W + - (DW , (4.5.51) 2 $D=_+4 . (4.5.53) (In case (2), using (4.5.50), these simplify to DI + F = 0.)  216 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS 4.6. N-extended multiplets So far in this chapter we have described the multiplets of N = 1 global supersym- metry. For interacting theories there are two such multiplets, with spins (-,0), and (1, -), although their superfield description may take many forms. For N-extended supersymmetry, global multiplets exist for N <4. They are naturally described in terms of extended superfields. It is possible, however, to discuss these multiplets, and their interactions, in terms of N = 1 superfields describing their N = 1 submultiplets. In many cases of interest this is the most complete description that we have at the present time. a. N=2 multiplets As discussed in sec. 3.3, there exist two global N = 2 multiplets: a vector multi- 1 1 1 1 plet with spins (1, 2 ' 20,0), and a scalar multiplet with spins ( 0,0,0,0). There exists only one global N = 4 multiplet: the N = 4 vector multiplet, with SU(4) 1 representation®spins (101,40®- ,600). (The only N = 3 multiplet is the same as that of N = 4.) We begin by discussing the N = 2 situation. a.1. Vector multiplet The N = 2 vector multiplet consists of an N = 1 Yang-Mills multiplet coupled to a scalar multiplet in the same (adjoint) representation of the internal symmetry group. The action is 1 4 4 4 2P S tr (I dxd4O Ib + d'xd2 )(4.6.1) in the vector representation. In addition to the usual gauge invariance, it is invariant under the following global transformations with parameters x, (: o@=- WaVaX - i[V2(Va()Va + (Vo()ijWa]I e-Q oeQ ix4 + WV(~V (4.6.2)  4.6. N-extended multiplets 217 (For the ( transformation we use (4.2.52), and also a gauge transformation with K = i(ID2 _ ag2Da() + paa [D3, Da](.) Due to the identity SVa = [Va, (e-QoeQ)] (4.2.77), the second transformation can be written as Spa =- -Va(ixb - WN~() = iVax + W [v3 , Va]( .(4.6.3) Both parameters are x-independent superfields and commute with the group generators (e.g., Vax = Day). The parameter x is chiral and mixes the two N = 1 multiplets, whereas ( is the real parameter of the N = 1 supersymmetry transformations (3.6.13). Since ( has the (x-independent) gauge invariance 8( = i(A - A), the global superparame- ters themselves form an abelian N = 2 vector multiplet. Referring to the components of this parameter multiplet (x , () by the names of the corresponding components in the field multiplet (4, V), we find the following: The "physical bosonic fields" give transla- tions (from the vector ( 2 2[Da, Da](I) and central charges (from the scalars z x|); the "physical fermionic fields" give supersymmetry transformations ( 1a iD2Da(, 2 a DaX|); and the "auxiliary fields" give internal symmetry U(2)/SO(2) transforma- 1- tions (r - DaD2DaQ(, q -D2X). (The full U(2) symmetry has, in addition to (r, q, q) transformations, phase rotations D= iu, SV = 0). The algebra of the N = 2 global transformations closes off shell; e.g., the commuta- tor of two X transformations gives a ( transformation: [o1 ,X2 <= o52, (12 = X[lX2] = ( i1X2 - X2X1) . (4.6.4) The transformations take a somewhat different form in the chiral representation: 64D=- WaVaX - iT2(Vag)gag eVoeV i( - x ) + (W/V/ + W )( , (4.6.5a) and hence oVa =Va[i(f@ - x3) + (W4V4 + WIVg)( . (4.6.5b) Now the i (A- A) part of the ( transformation does contribute, but only as a field-depen- dent gauge transformation A =WaVa A.  218 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS We can add an N = 2 Fayet-Iliopoulos term (parametrized by constants v =vi + iv2,21/3 = /3) d4xd4O v3V + (-if d4xd2O v1 + h. c.) (4.6.6) to the above action in the abelian (free) case. This is invariant under (4.6.5) if we restrict the global parameters by v2= vD2 , 3D2 _ iv(2D2D(+ u) , (4.6.7) where u is the real constant parameter of the phase SO(2) part of U(2). The constraint on the parameters implies that the U(2) is broken down to SO(2)®U(1). This model has some interesting quantum properties. It has gauge invariant diver- gences at one-loop, but explicit calculations show their absence at the two- and three- loop level. In sec. 7.7 we present an argument to establish their absence at all higher loops. a.2. Hypermultiplet a.2.i. Free theory The N = 2 scalar multiplet can be described by a chiral scalar isospinor superfield V" (the "" hypermultiplet") with the free action S f d4xd4O 8aa + (f d4xd2O amavbG + h. c.) , (4.6.8) where the symmetric matrix m satisfies the condition mac Ccb = ac acb (4.6.9) (The explicit form is mab = iMCaTVc, M M=, with Ta = 0 and Tb a -=Tab. Without loss of generality, mab can be chosen proportional to oa.) The free action is invariant under the global symmetries o"=- (D27C"bI6b - XZ@") - iD2[(Da()Da@" + (D2()Ia] , (4.6.10) where Z is a central charge: Z a = Cab mc , ZIa = Cab"bcD . (4.6.111  4.6. N-extended multiplets 219 On shell, we also have Za - CabD2b . (4.6.12) We can use either of the forms (4.6.11,12) in the transformation (4.6.10), because of the local invariance SS 4" =_TlCabSb , Sb = b -g ,(4.6.13) for arbitrary x-dependent chiral r1. (This is an invariance because the variation of the action is proportional to SS SaCabSb = 0.) If we use the form (4.6.12), the variations do not depend on the parameters mab. An interesting feature of the algebra (4.6.10) is that it does not close off-shell if we use realization (4.6.11) for Z. On the other hand, if we use realization (4.6.12) instead, the symmetries (4.6.10) contain part of the field equations, and hence become nonlinear and coupling-dependent when interactions are introduced. These effects are a signal that in the decomposition of the N = 2 superfield that describes the theory into N = 1 superfields, some auxiliary N = 1 superfields have been discarded. We discuss further aspects of this problem below. Without the mass term, the internal symmetries of the free scalar multiplet are the explicit SU(2) that acts on the isospinor index of @a and the U(2) made up of the r and q transformations in ( and x, and of the uniform phase rotations aD"= in". The mass term breaks the explicit SU(2) to the U(1) subgroup that commutes with mac. a.2.ii. Interactions The N = 2 scalar multiplet can interact with an N = 2 vector multiplet, and it can have self-interactions describing a nonlinear o model. A class of supersymmetric o--mod- els can be found by coupling an abelian N = 2 vector multiplet (with no kinetic term but with a Fayet-Iliopoulos term) to n N = 2 scalar multiplets described by the n-vector ~. The supersymmetry transformations of the vector multiplet are the same as those given above in (4.6.2) or (4.6.5) for the abelian case. (They are independent of the fields in the scalar multiplets.) However, the transformations of the scalar multiplets (each of which is described by a pair of chiral superfields @") are gauge covariantized: o"=- 52L cryT)cb Cab] - iD2[(Da()Va@" + (D2g>I] -_ _a.  220 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS (4.6.14) The matrix T is an SU(2) generator that breaks the explicit SU(2) of the scalar multi- plet down to U(1). Because SU(2) preserves the alternating tensor Cab, (eTv)ab (eTV)cd Cac = Cbd. The action that is left invariant by these transformations is: S f d4x do0 [4a (evab. .1b + V3V] +fd4x d2 I4baCabTbc.'c-v] + h.c. (4.6.15) provided (4.6.7) are satisfied. The theory is also invariant under local abelian gauge transformations: Sp" = i AT 74b 4 , SV = i(11-1A) ,64D = 0; (4.6.16) as well as global SU(n) rotations of b4. For explicit computation, it is useful to choose a specific T: We choose T = T3. We write b"a_ (b+ , 4-) ( + , 4-Z) where i = 1.-.-.- n is the SU(n) index, +, - are the SU(2) isospin indices, and I+ transforms under the SU(n) representation conjugate to 4_. The transformations (4.6.14) and the action (4.6.15) become (using (4.6.7)) S + D2( erV) - 273 (D2 _I giD2(Da()V$a (4.6.17a) S = d4x d40 + eve+i + _e-v 4i + v3V] + fJd4x d20 - i+ - v] + h. c. (4.6.17b) We now proceed as we did in the case of the CP(n) models (see (4.3.9)): We eliminate the vector multiplet by its (algebraic) equations of motion. In this case, 1 acts as a Lagrange multiplier to impose the constraint: @_G~ =v. (4.6.18) Choosing a gauge (e.g., @+ =Ia-), we can easily solve this constraint; for example, we can parametrize the solution as:  4.6. N-extended multiplets 221 @+ 1+ U+ - U_)- V2(1,U+ _ = (1+ u+ -u_)Zv (1, u_) . (4.6.19) The V equation of motion gives: ve+ i e-v_ + v3 = 0 (4.6.20a) or Mse±V =2[(v32 + 4M+M_) -Tv3] ; (4.6.20b) where M s = 4b+ ' + + 2= II||1 + u+ ' U-| (1 + lui l2) .(4.6.20c) Substituting, we find the action S = d4x d40 {(v32 + 4M+M_) + v3 lm[(v32 + 4M+M_) - Iv3]} . (4.6.21) In terms of the unconstrained chiral superfields u±, the transformations (4.6.17a) become u j2 [ZeTrV(1 2(+ u+. u>(1 + u+* -<2(ur- u±)] - ji2[(Da()Daus] , (4.6.22a) where the auxiliary gauge field V is expressed in terms of u± by (4.6.20). The super- symmetry transformations (4.6.22a) include a compensating gauge transformation with parameter iA =-D2[(cosh V)(-)>(1 + u+ '"u-)2(1 + u+ -f_)-2] (4.6.22b) that must be added to (4.6.17a) to maintain the gauge choice we made in (4.6.19). As for the free N =2 scalar multiplet, we can add an invariant mass term (which introduces a nonvanishing central charge). The mass term necessarily breaks SU(n) and has the form I,=i ~ dgg Cb cM@+ h.c. ,(4.6.23) where M is any traceless n x n matrix (M's differing by SU(n) transformations are  222 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS equivalent). The supersymmetry transformations that leave this term invariant are the same as before, including the Z term of (4.6.10). The realization of Z given in (4.6.11) is preferable, since it is linear, whereas the realization (4.6.12) must be gauge covari- antized. These nonlinear u-models live on Kshler manifolds with three independent com- plex coordinate systems related by nonholomorphic coordinate transformations (they have three independent complex structures (see the end of sec. 4.1); the constants v3 , v , i- parametrize the linear combination of complex structures chosen by the particu- lar coordinate system). Thus these manifolds are hyperKshler. Just as we found that for every Kshler manifold there is an N = 1 nonlinear u-model (and conversely), one can show that for every hyperKshler manifold there is an N = 2 nonlinear u-model, and con- versely, N = 2 nonlinear u-models are defined only on hyperKshler manifolds. An immediate consequence of this relation is a strong restriction on possible off-shell formu- lations of the N = 2 scalar multiplet: No formulation can exist that contains as physical submultiplets two N = 1 scalar multiplets (e.g., such as we have considered), that can be used to describe N = 2 nonlinear u-models, and that has supersymmetry transformations independent of the form of the action. If such a formulation existed, then the sum of two N = 2 invariant actions would neces- sarily be invariant; however, the sum of the Kshler potentials of two hyperKshler mani- folds is not in general the Kshler potential of a hyperKshler manifold. We will see below that we can give an off-shell formulation of the N = 2 scalar multiplet that avoids this problem. We can generalize the action (4.6.15) in the same way that we generalized the CP(n) models (see (4.3.11)): S f d'x d40 [+eVTA4D+ - 5_T_ eVI_ + v3trTA= 0 . (4.6.24b) As in the N = 1 case, these do not, in general, have an explicit solution. a.3. Tensor multiplet Just as the N = 1 scalar multiplet can be described by different superfields, we can describe the N = 2 scalar multiplet by superfields other than the chiral isodoublet V". We now discuss the N = 2 tensor formulation of the scalar multiplet. This is dual to the previous description in the same way that the N = 1 tensor and scalar multiplets are dual (see sec. 4.4.c). We write the tensor form of the scalar multiplet in terms of one chiral scalar field r, and a chiral spinor gauge field #a with linear field strength G = (Da#a + Dg#), D2G D= 2G = 0. The N = 2 supersymmetry transformations of this theory are 6#a = - 2/Daty - iD2[(D()D1#a + (D2 1 =- D21(G) - iD2[(D()Der1 + 2(D2()1] . (4.6.25) In contrast to the V" hypermultiplet realization of the N = 2 scalar multiplet, these transformations close off-shell; they have the same algebra as the transformations of the N = 2 vector multiplet (4.6.4) (up to a gauge transformation of #a). However, although the superfields describe a scalar multiplet, the central charge transformations z = x| leave the fields #,1/ inert; this gives one guide to understanding the duality to the hyper- multiplet. The simplest action invariant under the transformations (4.6.25) is the sum of the usual free chiral and tensor actions ((4.1.1) and (4.4.34)): S i-fd'xdO0[ -G+ 1] . (4.6.26 ) To find other actions, we consider a general ansatz, and require invariance under the  224 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS transformations (4.6.25). Actually, we can consider a slightly more general case that still has full off-shell N = 2 invariance by restricting the chiral parameter x by D2x = 0, and the real parameter ( by D2D2( = 0. This means that we do not impose SU(2) invariance. An action S = d4x d4O f (G , r ,) (4.6.27) is invariant under (4.6.25) (with D2)(= 0) if f satisfies + = f GG + fg 0-- (4.6.28) (f contains no derivatives of G, r1, .) It describes a general N = 2 tensor multiplet interacting model. We also can consider more than one multiplet G , - G ln(G + (G2 + 4 T',>)] (4.6.35) This complicated nonlinear action corresponds to a free hypermultiplet! It is, however, an off-shell formulation, invariant under the transformations (4.6.25). It generalizes directly to give an off-shell formulation of the nonlinear o-models we discussed above. An alternative derivation of the improved tensor multiplet does not require an N = 2 vector multiplet, but uses the central charge invariance of the tensor multiplet. We begin with the free hypermultiplet action (4.6.8) (without loss of generality, we take mab = imCab(T3)bc). We wish to Legendre transform one of the chiral fields 4" =_ (4+,_), and keep the other field as the chiral field r, of the tensor multiplet. However, though r, is inert under central charge transformations, 4± are not; we there- fore define the invariant combination r,= iI+4_, and in terms of it write the first order action S' =fd4x d4O [7Ye-V + eV - GV] + m[f d4x d2Or+ h. c.] . (4.6.36) Varying C, we recover the hypermultiplet action (4.6.8) with V =ln(&+ +); varying V, we recover the improved tensor multiplet action (4.6.35) with a linear r, term that acts as a mass term. The algebra of transformations that act on the massive scalar multiplet has a central charge; however, the description of the multiplet given by the N =2 tensor multiplet only involves fields that are inert under the central charge.  4.6. N-extended multiplets 227 Finally, we note that the gauge interactions of the N = 2 tensor multiplet are anal- ogous to the N = 1 case (see sec. 4.4.c). a.5. N=2 superfield Lagrange multiplier Another formulation of the N = 2 scalar multiplet with off-shell N = 2 supersym- metry is the N = 2 Lagrange multiplier multiplet. It is the N = 2 generalization of the multiplet discussed in sec. 4.5.d, and contains that N = 1 multiplet as a submultiplet. Unlike the off-shell N = 2 supersymmetric scalar multiplet discussed above (the N = 2 tensor multiplet of secs. 4.6.a.3,4), this multiplet can be coupled to the (N = 2) non- abelian vector multiplet, though only in real representations. By using the adjoint rep- resentation, this allows construction of N = 4 Yang-Mills with off-shell N = 2 supersym- metry, as discussed below in sec. 4.6.b.2. The N = 2 Lagrange multiplier multiplet is described by the following N = 1 superfields: (1) TI and Y, describing an N = 1 Lagrange multiplier multiplet as in (4.5.18), with the gauge invariance of (4.5.19), and field strength E '= UTg (for which F and G of (4.5.18) are the real and imaginary parts); (2) a second spinor W2", with the same dimension and gauge invariance, but which is auxiliary; (3) a complex Lagrange multiplier E, which constrains all of E2 to vanish (instead of just the imaginary part, as does Y for E1), and has a field strength DE with gauge invariance 6BE= A (for A chi- ral); (4) a minimal scalar multiplet, described by a complex gauge field T1 with chiral field strength 1 (see sec. 4.5.a); and (5) two more minimal scalar multiplets T2 and W3, but auxiliary. We thus have an N = 1 Lagrange multiplier multiplet, a minimal scalar multiplet, and assorted auxiliary superfields. The action is S =- fd4x d40 [. (E1 + Z1 )2 + 2Y( 1 - Z1 )] + fd4x d40 [I$@ + (EZE2 + 5Z2 ) + ( '2 3 + 5253 ) ] . (4.6.37) The most interesting properties of this theory appear when it is coupled to N =2 super- Yang-Mills. We do this by N =2 gauge covariantizing the N =2 Lagrange multiplier multiplet field strengths. (In the absence of Yang-Mills coupling, the Vs can be  228 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS considered as ordinary scalar multiplets, rather than field strengths.) This coupling is discussed in sec. 4.6.b.2. b. N=4 Yang-Mills In many respects, the N = 2 nonlinear u-models, when studied in two dimen- sions, are analogs of N = 4 Yang-Mills theory in four dimensions. Despite power-count- ing arguments, they are completely finite on shell, and they are the maximally super- symmetric models containing only scalar multiplets (the vector is auxiliary and can be eliminated). The N = 4 Yang-Mills theory is the first and best-studied 4-dimensional theory that is ultraviolet finite to all orders of perturbation theory, and thus scale invari- ant at the quantum as well as the classical level. (Its f-function has been calculated to vanish through three loops; arguments for total finiteness are given in sec. 7.7. Inde- pendent arguments using light-cone superfields have been given elsewhere.) It is self- conjugate and is the maximally extended globally supersymmetric theory. Two super- field formulations of the theory have been given: One uses an N = 2 vector multiplet coupled to a V" hypermultiplet and has only N = 1 supersymmetry off shell, and the other uses an N = 2 vector multiplet coupled to an N = 2 Lagrange multiplier multiplet and has N = 2 supersymmetry off shell (however, it has a large number of auxiliary superfields). b.1. Minimal formulation 1 At the component level the theory contains a gauge vector particle, four spin 2 Weyl spinors, and six spin 0 particles, all in the adjoint representation of the internal symmetry group. It can be described by one real scalar gauge superfield V and three chiral scalar superfields V , and is the same as an N = 2 vector multiplet coupled to an N = 2 scalar multiplet. If we use a matrix representation for the V , the (chiral repre- sentation) conjugate can be written as Ie = eV . The N =1 supersymmetric action (in the chiral representation) is given by S= tr(I d'xd4O e-~iv v +f d'xd 2O + + fd'xd2O iC~j [I,@ ] +- d'xd2O iCisk4i [, ,,] ) . (4.6.38)  4.6. N-extended multiplets 229 In addition to the manifest SU(3) symmetry on the i, j, k indices of < and <, it has the following global symmetries: 64 - (WaVcvX( + CiikV2 Ik)- 22,7 + 2 (V)2 >V1 bVa =_Va[i(xiV - xzIi) + (W V + WIV )(] ; (4.6.39) in the chiral representation, and in the vector representation 64D-Z(WaVcXv +CiikV 2N jI4k - i[yjI4D3 , i] ) - j~2(a~)~~ + (Va(9jW + 2V2(V2 Zi 3 bVa - Va(ixIi4 + W V() . (4.6.40) The x2 are the generalization of those given for the N = 2 multiplets above, but now they form an SU(3) isospinor, as does V. The identification of the components of x and ( is the same: The "physical bosonic fields" are the translations and the central charge parameters (3 complex = 6 real, as follows from dimensional reduction from D=10: see sec. 10.6), the spinors are the supersymmetry parameters, and the "auxiliary fields" are internal symmetry parameters of SU(4)/SU(3). The algebra does not close off-shell. Upon reduction to its N = 2 submultiplets, (4.6.39) (or (4.6.40)) reduces to (4.6.5) (or (4.6.2)) and (4.6.14) (but with different R-weights). The corresponding component action has a conventional appearance, with gauge, Yukawa, and quartic scalar couplings all governed by the same coupling constant. In sec. 6.4 we discuss some of the quantum properties of this theory. b.2. Lagrange multiplier formulation We now briefly describe another N 1 superfield formulation of N 4 super- Yang-Mills; it employs the (unimproved) type of N =1 scalar multiplet of sec. 4.5.d. Although even less of the SU(4) symmetry is manifest, this formulation is off-shell N =2 supersymmetric: It follows from the N =2 superfield formulation of the theory, as described by the coupling of N =2 super Yang-Mills to an N =2 (Lagrange multi- plier) scalar multiplet. This formulation has a number of other novel features: (1)  230 4. CLASSICAL, GLOBAL, SIMPLE (N=1) SUPERFIELDS renormalizable couplings between nonminimal and other scalar multiplets, (2) the neces- sary appearance (in interaction terms) of the minimal scalar multiplet in the form of a gauge multiplet (sec. 4.5.a), and (3) loss of (super)conformal invariance off shell (this occurs because the model includes an unimproved Lagrange multiplier multiplet). The action can be written as (in the super-Yang-Mills vector representation) the sum of (4.6.1) and (4.6.37). However, the definitions of the field strengths EZ and 4I are now modified: i- = 2TJ (i = 0, 1, 2) , 3 _V2 3 - [o, E] ; (4.6.41) where Io (with prepotential T0) is the chiral superfield of the N = 2 Yang-Mills multi- plet. The VA in these definitions is the Yang-Mills covariant derivative. In addition to the usual (adjoint, vector representation) Yang-Mills gauge transformations, we have many new local symmetries of the action: b2 = V Kj (i = 0, 1, 2) , 63 = VK3 + i [40 , A] ; (4.6.42a) b 2" = VOK " + i[10 , Kjo] (i = 1, 2) ; _ = A ; bY = 6Q = 0 ; (4.6.42b) where A is covariantly chiral (VA = 0), and Q is the Yang-Mills vector-representation prepotential. Under these transformations the field strengths 4I (i = 0, ... , 3), EZ (i = 1, 2), V3E, Y, and Wc are invariant. In the abelian (or linearized) case, the sum of (4.6.1) and (4.6.37) as modified by (4.6.42) describes an N = 2 vector multiplet (WC and o) plus an N = 2 scalar multiplet consisting of the N = 1 Lagrange multiplier multiplet of (4.5.18) (4' and Y), a minimal N = 1 scalar multiplet (i), and some auxiliary superfields (24j, E, 2, and W3). However, in the interacting case the formulation is somewhat unusual in that k3 is not just N = 1 covariantly chiral (0(13 # 0) nor are EZ N =1 covariantly linear (V2E, # 0), but they satisfy the N =2 covariant Bianchi iden- tities V2E, = - i[10 , G]. (4.6.43) The interaction terms of the auxiliary superfields (introduced through the nonlinearities  4.6. N-extended multiplets 231 of the field strengths E2 and 43) cancel among themselves: Their terms in the action can be rewritten as, in the chiral representation, f d4xd0( ED T2c + h. c.) + [ d4x d2 2(D2DTs3)+h.c.] . (4.6.44) By combining the Bianchi identities (4.6.43), the usual constraint V35, = 0 (for i = 0, 1, 2), and V&W, = 0, V"Wa + VW3I= 0 with the field equations which follow from the action, we obtain the on-shell equations for all of the superfields DUg ElE1 l 5 = E2 = 42 =43 = 0, iV We = -iY17 =[4D0 , 4o] + [1, 51] + [( E1l+2Y ), ( E1+iY )] , 4 V o=0 V2 0 + i[ 1, (E1 + iY)] 0, V4b1 =VY2b1 +i[ 1(E1+iY),o]=0, V3(E1 + iY) =_V2(E1 + iY) + 2i[40 ,#1] = 0 . (4.6.45) We can thus identify this formulation on shell with that given above in subsec. 4.6.b.1. by the correspondences W jWa ,(@o,1,(ZE+iY))Yi . (4.6.46) 2  Contents of 5. CLASSICAL N=1 SUPERGRAVITY 5.1. Review of gravity 232 a. Potentials 232 b. Covariant derivatives 235 c. Actions 238 d. Conformal compensator 240 5.2. Prepotentials 244 a. Conformal 244 a.1. Linearized theory 244 a.2. Nonlinear theory 247 a.3. Covariant derivatives 249 a.4. Covariant actions 254 b. Poincard 255 c. Density compensators 259 d. Gauge choices 261 e. Summary 263 f. Torsions and curvatures 264 5.3. Covariant approach to supergravity 267 a. Choice of constraints 267 a.1. Compensators 267 a.2. Conformal supergravity constraints 270 a.3. Contortion 273 a.4. Poincard supergravity constraints 274 b. Solution to constraints 276 b.1. Conventional constraints 276 b.2. Representation preserving constraints 278 b.3. The A gauge group 279 b.4. Evaluation of 1. and R 281 b.5. Chiral representation 284 b.6. Density compensators 286 1 b.6.i. Minimal (n = 3) supergravity 287 1 b.6.ii. Nonminimal (n # 3 ) supergravity 287 b.6.iii. Axial (nm 0) supergravity 288 b.7. Degauging 289  5.4. Solution to Bianchi identities 292 5.5. Actions 299 a. Review of vector and chiral representations 299 b. The general measure 300 c. Tensor compensators 300 d. The chiral measure 301 e. Representation independent form of the chiral measure 301 f. Scalar multiplet 302 f.1. Superconformal interactions 303 f.2. Conformally noninvariant actions 304 f.3. Chiral self-interactions 305 g. Vector multiplet 306 h. General matter models 307 i. Supergravity actions 309 i.1. Poincard 309 i.2. Cosmological term 312 i.3. Conformal supergravity 312 j. Field equations 313 5.6. From superspace to components 315 a. General considerations 315 b. Wess-Zumino gauge for supergravity 317 c. Commutator algebra 320 d. Local supersymmetry and component gauge fields 321 e. Superspace field strengths 323 f. Supercovariant supergravity field strengths 325 g. Tensor calculus 326 h. Component actions 331 5.7. DeSitter supersymmetry 335  5. CLASSICAL N=1 SUPERGRAVITY 5.1. Review of gravity a. Potentials Our review is intended to describe the approach to gravity that is most useful in understanding supergravity. We treat gravity as the theory of a massless spin-2 particle described by a gauge field with an additional vector index as a group index (so that it contains spin 2). By analogy with the theory of a massless spin 1 particle its linearized transformation law is oham = MaA . (5.1.1) Since the only global symmetry of the S-matrix with a vector generator is translations, we choose partial spacetime derivatives (momentum) as the generators appearing con- tracted with the gauge field's group index in the covariant derivative e a =& - iham(ipm) (Sam + har)m a=ejmI . (5.1.2a) Thus, in contrast with Yang-Mills theory, we are able to combine the derivative and "group" terms into a single term. The gauge field earn is the vierbein, which reduces to a Kronecker delta in flat space. It is invertible: Its inverse em" is defined by emaan = om" , eamem = oG . (5.1.2b) Finite gauge transformations are also defined by analogy with Yang-Mills theory: e'_ = e2 e_ e-, A Ami0m . (5.1.3) The linearized transformation takes the form of (5.1.1), whereas the full infinitesimal form takes the form of a Lie derivative: (og)m=i[A, eq] =- [A_ , ejgh8ml, (5.1.4a) or, in more conventional notation, (eam =ea"51Ab - A)e, (5.1.4b)  5.1. Review of gravity 233 The gauge transformation of a scalar matter field is, again by analogy with Yang-Mills theory, eil eZ9ea , (5.1.5) and in infinitesimal form S= i[A, @] =- A' . (5.1.6) Equation (5.1.5) can also be written as the more common general coordinate transforma- tion (X') (x) , X' = e-axell . (5.1.7) (This can be verified by a Taylor expansion.) For the case of constant A it takes the familiar form of global translations. Orbital (global) Lorentz transformations are obtained by choosing A = Q[,x"[ + Q[ix " (which just equals 6x' in the infinitesimal case); Q is traceless. Scale transformations are obtained by choosing A =coxm. We could at this point define field strengths in terms of the covariant derivatives (5.1.2), but the invariance group we have defined is too small for two reasons: (1) The vierbein is a reducible representation of the (global) Lorentz group, so more of it should be gauged away; and (2) there are difficulties in realizing (global) Lorentz transforma- tions on general representations, as we now discuss. Since under global Lorentz transformations (x) transforms as a scalar field, its gradient &,9 will transform as a covariant vector. In general, we define a covariant vec- tor to be any object that transforms like Om9. We can define a contravariant vector to belong to the "adjoint" representation of our gauge group. Indeed, if we define V - V iz, and require that [V, L'] = Vmi mI transform as a scalar, i.e., V' - V'mio m = eZAVe-ZA , (5.1.8) then VrI transforms contravariantly under global Lorentz transformations. However, this procedure does not allow us to define objects which transform as spinors under global Lorentz transformations, and in fact it is impossible to define a field, transforming linearly under the A group, which also transforms as a spinor when the A's are restricted to represent global Lorentz transformations. It is possible to get around this difficulty by realizing the A transformations nonlinearly, but this is not a convenient solution.  234 5. CLASSICAL N=1 SUPERGRAVITY Comparing (5.1.3) to (5.1.8), we see ear transforms as four independent contravariant vectors under the global Lorentz group: The A transformations do not act on the a indices. To solve these problems we enlarge the gauge group by adjoining to the A transfor- mations a group of local Lorentz transformations, and define spinors with respect to this group. This is a procedure familiar in treatments of nonlinear o models. Nonlinear real- izations of a group are replaced by linear representations of an enlarged (gauge) group. The nonlinearities reappear only when a definite gauge choice is made. Similarly here, by enlarging the gauge group, we obtain linear spinor representations. The nonlinear spinor representations of the general coordinate group reappear only if we fix a gauge for the local Lorentz transformations. It will thus turn out that our final gauge group for gravity can be interpreted physically as the direct product of the translation (general coordinate) group with the spin (internal) angular momentum group. We define the action of the local Lorentz group on the vierbein to be (eam = - Aje