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SUPERGRAVITY P. Van NIEUWENHUIZEN to Joel Scherk 0370—1573/81/0000--0000/$42.OO © North-Holland Publishing Company PHYSICS REPORTS (Review Section of Physics Letters) 68, No. 4 (1981) 189—398. North-Holland Publishing Company SUPERGRAVITY P. van NIEUWENHUIZEN Institute for Theoretical Physics, State University of New York at Stony Brook, LI., New York 11794, U.S.A. Received July 1980 To Joel Scherk Contents: Preface 192 2.10. Explicit calculations 256 1. Simple supergravity in ordinary space-time 192 2.10.1. Green’s functions are not finite 256 1.1. Introduction and outlook 192 2.10.2. Matter couplings are not finite 257 1.2. Physical arguments leading to supergravity 198 2.10.3. Calculations of finite processes 259 1.2.1. Supersymmetry 198 2.11. Higher-order invariants, multiplets and auxiliary 1.2.2. Supergravity 200 fields 260 1.3. The gauge action of simple supergravity 205 2.12. On-shell counterterms and non-linear invariances 262 1.4. Palatini formalism 207 2.13. Canonical decomposition of the action 263 1.5. Flat supergravity with torsion 209 2.14. Dirac quantization of the free gravitino 264 1.6. The 1.5 order formalism 210 2.15. Anomalies 267 1.7. Explicit proof of gauge invariance 211 2.16. Chirality, self-duality and counterterms 270 1.8. Auxiliary fields in global supersymmetry 214 2.17. Super index theorems 271 1.9. Auxiliary fields for the gauge algebra 217 3. Group theory 272 1.10. Field equations and consistency 221 3.1. The superalgebras 272 1.11. Matter coupling: the supersymmetric Maxwell—Em- 3.2. The gauging of (super) algebras 280 stein system 223 3.3. The Haag—Lopuszanski—Sohnius theorem 283 1.12. The scalar multiplet coupling 226 3.4. Supergravity as a gauge theory on the group manifold 285 1.13. Free spin-2 and spin-3/2 field theory 229 3.5. Supergravity and spinning space 289 1.14. Higher spin theory 234 4. Conformal simple supergravity 293 2. Quantum supergeavity 235 4.1. Conformal supersymmetry 293 2.1. Introduction 235 4.2. The action of conformal simple supergravity 297 2.2. General covariant quantization of gauge theories 236 4.3. Constraints and gauge algebra 299 2.3. Covariant quantization of simple supergravity 239 4.4. Conformal tensor calculus 302 2.4. Path-integral quantization of gauge theories 242 4.5. The origins of the auxiliary fields 5, P, A,,, 305 2.5. Gauge independenceof supergravity 244 4.6. Tensor calculus for ordinary supergravity 308 2.6. Unitarity of supergravity 245 5. Superspace 310 2.7. Supersymmetric regularization 247 5.1. Introduction to superspace 310 2.8. BRST invariance and open algebras 249 5.2. Superspace from ordinary space 313 2.9. Finiteness of quantum supergravity? 254 5.3. Flat superspace 318 Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 68, No. 4 (1981) 189—398. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 95.00, postage induded. 0370—1573/81/0000--0000/$42.OO © North-Holland Publishing Company SUPERGRAVITY P. van NIEUWENHUIZEN Institute for Theoretical Physics, State University of New York at Stony Brook, L.I., New York 11794, U.S.A. 1931 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM In memoriam Joel Scherk Joel Scherk died on May 16, 1980. Although 33 years old, he had made many very important discoveries in dual string theory and supergravity, as will be manifest in this review. It is striking to note how logical and sustained the development of his thinking was. His first important contribution was the renormalization of the one-loop open dual string amplitude. Then he developed the idea of the zero-slope limit when all masses except the lightest tend to infinity, and found that this leads to quantum field theories. In particular the closed string sector yielded general relativity, and from then on he was fascinated by gravitation. The one-loop contributions of the open string model in d = 26 dimensions were shown to correspond to a new ghost-free set of states, corresponding to closed Strings and interpreted as Pomeron states. One of these states was described by an antisymmetric tensor gauge field, which is nowadays of interest. Then he started wondering what happens with the extra dimensions in our four-dimensional world. This led him to the idea of spontaneous compactification of space-time, extending ideas of Kalusza and Klein. In supergravity he started by solving the matter coupling problem and worked out the super-Higgs effect. Then he constructed the N = 3 supergravity, and both the SO(4) and the SU(4) versions of N = 4 extended supergravity. The latter model was suggested by taking the zero slope limit of the d 10 dimensional closed string dual model with fermions. The d = 10 open string model yielded in the zero slope limit N = 4 supersymmetric Yang—Mills theory. In the SU(4) model he found a noncompact SU(1, 1) global symmetry, and in the N = 2, 3, 4 models he gave a complete treatment of the duality and chiral symmetries. He then formulated N = 1 d = 11 supergravity, which yields the N = 8 d = 4 model upon dimensional reduction. He also developed the idea of spontaneous symmetry breaking by means of dimensional reduction, and worked out the particle spectrum. There he found a massless vector boson and proposed the idea that it yields the (testable) force of antigravity. Despite this impressive record, Joel was a relaxed and very friendly person. He was among the best and most pedagogical speakers and his articles were crystal clear. He was always open to discussion and had a very developed sense of humour. He also contributed actively to the popularization of theoretical physics by radio talks, lectures for lay-men and articles in popular scientific magazines. This short review was written by his closest friends. We will miss him deeply. 192 P. van Nieuwenhuizen, Supergravity 5.4. Construction of multiplets using the closed gauge al- B. Majorana spinors 362 gebra 321 C. Symmetries of covariants built from Majorana spinors 364 5.5. The Wess—Zumino approach to supergravity 324 D. Fierz rearrangements 364 5.6. Chiral superspace 329 E. Two-component formalism 365 5.7. Gauge supersymmetry 335 F. Supermatrix algebra 369 6. Extended supergravities 343 G. Superintegration 371 6.1. The N = 2, 3, 4 models 343 H. The super Jacobian is the superdeterminant 374 6.2. The N = 8 model in II dimensions 349 I. Elementa;y general relativity 376 6.3. The N = 8 model in 4 dimensions by dimensional J. Duffin—Kemmer—Petiau formalism for supergravity 377 reduction 353 K. Glossary 379 6.4. Spontaneoussymmetry breaking in the N = 8 model by L. Conventions 380 dimensional reduction 355 References 381 6.5. Auxiliary fields for N = 2 supergravity 359 Addendum 7. Appendices 361 6.6. Supergravity and phenomenology 395 A. Gamma matrices 361 Preface Supergravity was discovered four years ago [234,120, 235] and interest in this new theory has been great from the beginning. However, the rapid development of the field and various technical barriers (such as properties of Majorana spinors) have prevented many interested physicists from starting active research. This report hopes to provide an introduction which will allow the reader to go from no knowledge at all about supergravity up to the point where present research begins. It is written as a course for graduate students and thus the focus is more on explaining than on reviewing. Most importantly: it is self-contained. We have tried to discuss all major areas of supergravity, including the advanced aspects, but we have placed the emphasis on supergravity as a gauge theory. In this way, this report might be useful as the basis for a graduate course in gauge fields and quantum field theory. Rather than giving some general discussions or refer to the literature for details, we have tried to be explicit. Thus we have not shied away from working out details, and this should be helpful to the reader. Actually, large parts of this report were tried out on students at Stony Brook, and in this way we discovered where more explicit work was required. We must confess that as a result this report is rather long, and we have had nightmares in which the publisher provided it together with a wheelbarrow. In ~rder to describe the work of our colleagues in detail, we have visited many of them for long discussions. Thus the preparation of this report took more than two years. It is a pleasure to thank them for their cooperation. In addition, we have asked all of our colleagues to send us a list of their publications as a basis for the references in this report. This report is dedicated to our superfriends, who have created light (and heat) by their enthusiasm. Let us hope that nature is equally enthusiastic and has reserved an important role for this most elegant of all gauge theories. 1. Simple supergravity in ordinary space-time 1.1. Introduction and outlook The gravitational force is the oldest force known to man and the least understood. Some reasons for this curious fact are that gravitational experiments are difficult to perform, due to the smallness of the gravitational constant K, while at the theoretical level the dimensionful character of the gravitational constant has prevented the construction of a predictive, i.e. renormalizable, quantum theory of gravity, P. van Nieuwenhuizen, Supergravity 193 as we will discuss. All experimental data known at present confirm Einstein’s theory of gravity and it seems that any future quantum theory of gravitation should be an extension rather than a replacement of general relativity.
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