Supergravity - Assets - Cambridge University Press

Supergravity

Supergravity, together with string theory, is one of the most significant developments intheoretical physics. Although there are many books on string theory, this is the first-everauthoritative and systematic account of supergravity.

Written by two of the most respected workers in the field, it provides a solid introduc-tion to the fundamentals of supergravity. The book starts by reviewing aspects of relativisticfield theory in Minkowski spacetime. After introducing the relevant ingredients of differ-ential geometry and gravity, some basic supergravity theories (D = 4 and D = 11) and themain gauge theory tools are explained. The second half of the book is more advanced: com-plex geometry and N = 1 and N = 2 supergravity theories are covered. Classical solutionsand a chapter on anti-de Sitter/conformal field theory (AdS/CFT) correspondence completethe text.

Numerous exercises and examples make it ideal for Ph.D. students, and with appli-cations to model building, cosmology, and solutions of supergravity theories, thistext is an invaluable resource for researchers. A website hosted by the authors, fea-turing solutions to some exercises and additional reading material, can be found atwww.cambridge.org/supergravity.

Daniel Z. Freedman is Professor of Applied Mathematics and Physics at the MassachusettsInstitute of Technology. He has made many research contributions to supersymmetry andsupergravity: he was a co-discoverer of the first supergravity theory in 1976. This discoveryhas been recognized by the award of the Dirac Medal and Prize in 1993, and the DannieHeineman Prize of the American Physical Society in 2006.

Antoine Van Proeyen is Head of the Theoretical Physics Section at the KU Leuven, Belgium.Since 1979, he has been involved in the construction of various supergravity theories, theresulting special geometries, and their applications to phenomenology and cosmology.

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Our conventions

The metric is ‘mostly plus’, i.e. −+ . . .+. The curvature is

Rμνρσ = gρρ′ (∂μ�ρ′νσ − ∂ν�

ρ′μσ + �

ρ′μτ�

τνσ − �

ρ′ντ �

τμσ )

= eaρeb

σ

(∂μωνab − ∂νωμab + ωμacων

cb − ωνacωμ

cb).

Ricci tensor and energy–momentum tensors are defined by

Rμν = Rρνρμ , R = gμν Rμν ,

Rμν − 12 gμν R = κ2Tμν .

Covariant derivatives involving the spin connection are, for vectors and spinors,

DμV a = ∂μV a + ωμabVb , Dμλ = ∂μλ+ 1

4ωμabγabλ .

We use (anti)symmetrization of indices with ‘weight 1’, i.e.

A[ab] = 12 (Aab − Aba) and A(ab) = 1

2 (Aab + Aba) .

The Levi-Civita tensor is

ε0123 = 1 , ε0123 = −1 .

The dual, self-dual, and anti-self-dual of antisymmetric tensors are defined by

H̃ab ≡ − 12 iεabcd Hcd , H±

ab = 12 (Hab ± H̃ab) , H±

ab =(

H∓ab

)∗.

Structure constants are defined by [TA, TB

] = f ABC TC .

The Clifford algebra is

γμγν + γνγμ = 2gμν , γμν = γ[μγν] , . . .(γ μ)† = γ 0γμγ 0 ,

γ∗ = (−i)(D/2)+1γ0γ1 . . . γD−1 ;in four dimensions:

γ∗ = iγ0γ1γ2γ3 , εabcdγd = iγ∗γabc .

The Majorana and Dirac conjugates are

λ̄ = λT C , λ̄ = iλ†γ 0 .

We mostly use the former. For Majorana fermions the two are equal.

The main SUSY commutator is [δ(ε1), δ(ε2)

] = 12 ε̄2γ

με1∂μ .

p-form components are defined by

φp = 1

p!φμ1···μp dxμ1 ∧ · · · ∧ dxμp .

The differential acts from the left:

dA = ∂ν Aμ dxν ∧ dxμ , A = Aμdxμ .






Supergravity

DAN IEL Z . FREEDMANMassachusetts Institute of Technology, USA

and

ANTO INE VAN PROEYENKU Leuven, Belgium






C A M B R I D G E U N I V E R S I T Y P R E S S

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, São Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521194013

c© D. Z. Freedman and A. Van Proeyen 2012

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2012

Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication dataFreedman, Daniel Z.

Supergravity / Daniel Z. Freedman and Antoine Van Proeyen.p. cm.

ISBN 978-0-521-19401-3 (hardback)1. Supergravity. I. Van Proeyen, Antoine. II. Title.

QC174.17.S9F735 2012530.14′23–dc23

2011053360

ISBN 978-0-521-19401-3 Hardback

Additional resources for this publication at www.cambridge.org/supergravity

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.






Contents

Preface page xvAcknowledgements xvii

Introduction 1

Part I Relativistic field theory in Minkowski spacetime 5

1 Scalar field theory and its symmetries 71.1 The scalar field system 71.2 Symmetries of the system 8

1.2.1 SO(n) internal symmetry 91.2.2 General internal symmetry 101.2.3 Spacetime symmetries – the Lorentz and Poincaré groups 12

1.3 Noether currents and charges 181.4 Symmetries in the canonical formalism 211.5 Quantum operators 221.6 The Lorentz group for D = 4 24

2 The Dirac field 252.1 The homomorphism of SL(2,C)→ SO(3, 1) 252.2 The Dirac equation 282.3 Dirac adjoint and bilinear form 312.4 Dirac action 322.5 The spinors u( p, s) and v( p, s) for D = 4 332.6 Weyl spinor fields in even spacetime dimension 352.7 Conserved currents 36

2.7.1 Conserved U(1) current 362.7.2 Energy–momentum tensors for the Dirac field 37

3 Clifford algebras and spinors 393.1 The Clifford algebra in general dimension 39

3.1.1 The generating γ -matrices 393.1.2 The complete Clifford algebra 403.1.3 Levi-Civita symbol 413.1.4 Practical γ -matrix manipulation 423.1.5 Basis of the algebra for even dimension D = 2m 43

v






vi Contents

3.1.6 The highest rank Clifford algebra element 443.1.7 Odd spacetime dimension D = 2m + 1 463.1.8 Symmetries of γ -matrices 47

3.2 Spinors in general dimensions 493.2.1 Spinors and spinor bilinears 493.2.2 Spinor indices 503.2.3 Fierz rearrangement 523.2.4 Reality 54

3.3 Majorana spinors 553.3.1 Definition and properties 563.3.2 Symplectic Majorana spinors 583.3.3 Dimensions of minimal spinors 58

3.4 Majorana spinors in physical theories 593.4.1 Variation of a Majorana Lagrangian 593.4.2 Relation of Majorana and Weyl spinor theories 603.4.3 U(1) symmetries of a Majorana field 61

Appendix 3A Details of the Clifford algebras for D = 2m 623A.1 Traces and the basis of the Clifford algebra 623A.2 Uniqueness of the γ -matrix representation 633A.3 The Clifford algebra for odd spacetime dimensions 653A.4 Determination of symmetries of γ -matrices 653A.5 Friendly representations 66

4 The Maxwell and Yang–Mills gauge fields 684.1 The abelian gauge field Aμ(x) 69

4.1.1 Gauge invariance and fields with electric charge 694.1.2 The free gauge field 714.1.3 Sources and Green’s function 734.1.4 Quantum electrodynamics 764.1.5 The stress tensor and gauge covariant translations 77

4.2 Electromagnetic duality 774.2.1 Dual tensors 784.2.2 Duality for one free electromagnetic field 784.2.3 Duality for gauge field and complex scalar 804.2.4 Electromagnetic duality for coupled Maxwell fields 83

4.3 Non-abelian gauge symmetry 864.3.1 Global internal symmetry 864.3.2 Gauging the symmetry 884.3.3 Yang–Mills field strength and action 894.3.4 Yang–Mills theory for G = SU(N ) 90

4.4 Internal symmetry for Majorana spinors 93

5 The free Rarita–Schwinger field 955.1 The initial value problem 97






Contents vii

5.2 Sources and Green’s function 995.3 Massive gravitinos from dimensional reduction 102

5.3.1 Dimensional reduction for scalar fields 1025.3.2 Dimensional reduction for spinor fields 1035.3.3 Dimensional reduction for the vector gauge field 1045.3.4 Finally �μ(x, y) 104

6 N = 1 global supersymmetry in D = 4 1076.1 Basic SUSY field theory 109

6.1.1 Conserved supercurrents 1096.1.2 SUSY Yang–Mills theory 1106.1.3 SUSY transformation rules 111

6.2 SUSY field theories of the chiral multiplet 1126.2.1 U(1)R symmetry 1156.2.2 The SUSY algebra 1166.2.3 More chiral multiplets 119

6.3 SUSY gauge theories 1206.3.1 SUSY Yang–Mills vector multiplet 1216.3.2 Chiral multiplets in SUSY gauge theories 122

6.4 Massless representations of N -extended supersymmetry 1256.4.1 Particle representations of N -extended supersymmetry 1256.4.2 Structure of massless representations 127

Appendix 6A Extended supersymmetry and Weyl spinors 129Appendix 6B On- and off-shell multiplets and degrees of freedom 130

Part II Differential geometry and gravity 133

7 Differential geometry 1357.1 Manifolds 1357.2 Scalars, vectors, tensors, etc. 1377.3 The algebra and calculus of differential forms 1407.4 The metric and frame field on a manifold 142

7.4.1 The metric 1427.4.2 The frame field 1437.4.3 Induced metrics 145

7.5 Volume forms and integration 1467.6 Hodge duality of forms 1497.7 Stokes’ theorem and electromagnetic charges 1517.8 p-form gauge fields 1527.9 Connections and covariant derivatives 154

7.9.1 The first structure equation and the spin connection ωμab 1557.9.2 The affine connection �ρ

μν 1587.9.3 Partial integration 160

7.10 The second structure equation and the curvature tensor 161






viii Contents

7.11 The nonlinear σ -model 1637.12 Symmetries and Killing vectors 166

7.12.1 σ -model symmetries 1667.12.2 Symmetries of the Poincaré plane 169

8 The first and second order formulations of general relativity 1718.1 Second order formalism for gravity and bosonic matter 1728.2 Gravitational fluctuations of flat spacetime 174

8.2.1 The graviton Green’s function 1778.3 Second order formalism for gravity and fermions 1788.4 First order formalism for gravity and fermions 182

Part III Basic supergravity 185

9 N = 1 pure supergravity in four dimensions 1879.1 The universal part of supergravity 1889.2 Supergravity in the first order formalism 1919.3 The 1.5 order formalism 1939.4 Local supersymmetry of N = 1, D = 4 supergravity 1949.5 The algebra of local supersymmetry 1979.6 Anti-de Sitter supergravity 199

10 D = 11 supergravity 20110.1 D ≤ 11 from dimensional reduction 20110.2 The field content of D = 11 supergravity 20310.3 Construction of the action and transformation rules 20310.4 The algebra of D = 11 supergravity 210

11 General gauge theory 21211.1 Symmetries 212

11.1.1 Global symmetries 21311.1.2 Local symmetries and gauge fields 21711.1.3 Modified symmetry algebras 219

11.2 Covariant quantities 22111.2.1 Covariant derivatives 22211.2.2 Curvatures 223

11.3 Gauged spacetime translations 22511.3.1 Gauge transformations for the Poincaré group 22511.3.2 Covariant derivatives and general coordinate transformations 22711.3.3 Covariant derivatives and curvatures in a gravity theory 23011.3.4 Calculating transformations of covariant quantities 231

Appendix 11A Manipulating covariant derivatives 23311A.1 Proof of the main lemma 23311A.2 Examples in supergravity 234






Contents ix

12 Survey of supergravities 23612.1 The minimal superalgebras 236

12.1.1 Four dimensions 23612.1.2 Minimal superalgebras in higher dimensions 237

12.2 The R-symmetry group 23812.3 Multiplets 240

12.3.1 Multiplets in four dimensions 24012.3.2 Multiplets in more than four dimensions 242

12.4 Supergravity theories: towards a catalogue 24412.4.1 The basic theories and kinetic terms 24412.4.2 Deformations and gauged supergravities 246

12.5 Scalars and geometry 24712.6 Solutions and preserved supersymmetries 249

12.6.1 Anti-de Sitter superalgebras 25112.6.2 Central charges in four dimensions 25212.6.3 ‘Central charges’ in higher dimensions 253

Part IV Complex geometry and global SUSY 255

13 Complex manifolds 25713.1 The local description of complex and Kähler manifolds 25713.2 Mathematical structure of Kähler manifolds 26113.3 The Kähler manifolds CPn 26313.4 Symmetries of Kähler metrics 266

13.4.1 Holomorphic Killing vectors and moment maps 26613.4.2 Algebra of holomorphic Killing vectors 26813.4.3 The Killing vectors of CP1 269

14 General actions withN = 1 supersymmetry 27114.1 Multiplets 271

14.1.1 Chiral multiplets 27214.1.2 Real multiplets 274

14.2 Generalized actions by multiplet calculus 27514.2.1 The superpotential 27514.2.2 Kinetic terms for chiral multiplets 27614.2.3 Kinetic terms for gauge multiplets 277

14.3 Kähler geometry from chiral multiplets 27814.4 General couplings of chiral multiplets and gauge multiplets 280

14.4.1 Global symmetries of the SUSY σ -model 28114.4.2 Gauge and SUSY transformations for chiral multiplets 28214.4.3 Actions of chiral multiplets in a gauge theory 28314.4.4 General kinetic action of the gauge multiplet 28614.4.5 Requirements for an N = 1 SUSY gauge theory 286

14.5 The physical theory 288






x Contents

14.5.1 Elimination of auxiliary fields 28814.5.2 The scalar potential 28914.5.3 The vacuum state and SUSY breaking 29114.5.4 Supersymmetry breaking and the Goldstone fermion 29314.5.5 Mass spectra and the supertrace sum rule 29614.5.6 Coda 298

Appendix 14A Superspace 298Appendix 14B Appendix: Covariant supersymmetry transformations 302

Part V Superconformal construction of supergravity theories 305

15 Gravity as a conformal gauge theory 30715.1 The strategy 30815.2 The conformal algebra 30915.3 Conformal transformations on fields 31015.4 The gauge fields and constraints 31315.5 The action 31515.6 Recapitulation 31715.7 Homothetic Killing vectors 317

16 The conformal approach to pureN = 1 supergravity 32116.1 Ingredients 321

16.1.1 Superconformal algebra 32116.1.2 Gauge fields, transformations, and curvatures 32316.1.3 Constraints 32516.1.4 Superconformal transformation rules of a chiral multiplet 328

16.2 The action 33116.2.1 Superconformal action of the chiral multiplet 33116.2.2 Gauge fixing 33316.2.3 The result 334

17 Construction of the matter-coupledN = 1 supergravity 33717.1 Superconformal tensor calculus 338

17.1.1 The superconformal gauge multiplet 33817.1.2 The superconformal real multiplet 33917.1.3 Gauge transformations of superconformal chiral multiplets 34017.1.4 Invariant actions 342

17.2 Construction of the action 34317.2.1 Conformal weights 34317.2.2 Superconformal invariant action (ungauged) 34317.2.3 Gauged superconformal supergravity 34517.2.4 Elimination of auxiliary fields 34717.2.5 Partial gauge fixing 351

17.3 Projective Kähler manifolds 351






Contents xi

17.3.1 The example of CPn 35217.3.2 Dilatations and holomorphic homothetic Killing vectors 35317.3.3 The projective parametrization 35417.3.4 The Kähler cone 35717.3.5 The projection 35817.3.6 Kähler transformations 35917.3.7 Physical fermions 36317.3.8 Symmetries of projective Kähler manifolds 36417.3.9 T -gauge and decomposition laws 36517.3.10 An explicit example: SU(1, 1)/U(1) model 368

17.4 From conformal to Poincaré supergravity 36917.4.1 The superpotential 37017.4.2 The potential 37117.4.3 Fermion terms 371

17.5 Review and preview 37317.5.1 Projective and Kähler–Hodge manifolds 37417.5.2 Compact manifolds 375

Appendix 17A Kähler–Hodge manifolds 37617A.1 Dirac quantization condition 37717A.2 Kähler–Hodge manifolds 378

Appendix 17B Steps in the derivation of (17.7) 380

Part VI N = 1 supergravity actions and applications 383

18 The physicalN = 1matter-coupled supergravity 38518.1 The physical action 38618.2 Transformation rules 38918.3 Further remarks 390

18.3.1 Engineering dimensions 39018.3.2 Rigid or global limit 39018.3.3 Quantum effects and global symmetries 391

19 Applications ofN = 1 supergravity 39219.1 Supersymmetry breaking and the super-BEH effect 392

19.1.1 Goldstino and the super-BEH effect 39219.1.2 Extension to cosmological solutions 39519.1.3 Mass sum rules in supergravity 396

19.2 The gravity mediation scenario 39719.2.1 The Polónyi model of the hidden sector 39819.2.2 Soft SUSY breaking in the observable sector 399

19.3 No-scale models 40119.4 Supersymmetry and anti-de Sitter space 40319.5 R-symmetry and Fayet–Iliopoulos terms 404

19.5.1 The R-gauge field and transformations 405






xii Contents

19.5.2 Fayet–Iliopoulos terms 40619.5.3 An example with non-minimal Kähler potential 406

Part VII ExtendedN = 2 supergravity 409

20 Construction of the matter-coupledN = 2 supergravity 41120.1 Global supersymmetry 412

20.1.1 Gauge multiplets for D = 6 41220.1.2 Gauge multiplets for D = 5 41320.1.3 Gauge multiplets for D = 4 41520.1.4 Hypermultiplets 41820.1.5 Gauged hypermultiplets 422

20.2 N = 2 superconformal calculus 42520.2.1 The superconformal algebra 42520.2.2 Gauging of the superconformal algebra 42720.2.3 Conformal matter multiplets 43020.2.4 Superconformal actions 43220.2.5 Partial gauge fixing 43420.2.6 Elimination of auxiliary fields 43620.2.7 Complete action 43920.2.8 D = 5 and D = 6, N = 2 supergravities 440

20.3 Special geometry 44020.3.1 The family of special manifolds 44020.3.2 Very special real geometry 44220.3.3 Special Kähler geometry 44320.3.4 Hyper-Kähler and quaternionic-Kähler manifolds 452

20.4 From conformal to Poincaré supergravity 45920.4.1 Kinetic terms of the bosons 45920.4.2 Identities of special Kähler geometry 45920.4.3 The potential 46020.4.4 Physical fermions and other terms 46020.4.5 Supersymmetry and gauge transformations 461

Appendix 20A SU(2) conventions and triplets 463Appendix 20B Dimensional reduction 6 → 5 → 4 464

20B.1 Reducing from D = 6 → D = 5 46420B.2 Reducing from D = 5 → D = 4 464

Appendix 20C Definition of rigid special Kähler geometry 465

21 The physicalN = 2matter-coupled supergravity 46921.1 The bosonic sector 469

21.1.1 The basic (ungauged) N = 2, D = 4 matter-coupledsupergravity 469

21.1.2 The gauged supergravities 47121.2 The symplectic formulation 472






Contents xiii

21.2.1 Symplectic definition 47221.2.2 Comparison of symplectic and prepotential formulation 47421.2.3 Gauge transformations and symplectic vectors 47421.2.4 Physical fermions and duality 475

21.3 Action and transformation laws 47621.3.1 Final action 47621.3.2 Supersymmetry transformations 477

21.4 Applications 47921.4.1 Partial supersymmetry breaking 47921.4.2 Field strengths and central charges 48021.4.3 Moduli spaces of Calabi–Yau manifolds 480

21.5 Remarks 48221.5.1 Fayet–Iliopoulos terms 48221.5.2 σ -model symmetries 48221.5.3 Engineering dimensions 482

Part VIII Classical solutions and the AdS/CFT correspondence 485

22 Classical solutions of gravity and supergravity 48722.1 Some solutions of the field equations 487

22.1.1 Prelude: frames and connections on spheres 48722.1.2 Anti-de Sitter space 48922.1.3 AdSD obtained from its embedding in RD+1 49022.1.4 Spacetime metrics with spherical symmetry 49622.1.5 AdS–Schwarzschild spacetime 49822.1.6 The Reissner–Nordström metric 49922.1.7 A more general Reissner–Nordström solution 501

22.2 Killing spinors and BPS solutions 50322.2.1 The integrability condition for Killing spinors 50522.2.2 Commuting and anti-commuting Killing spinors 505

22.3 Killing spinors for anti-de Sitter space 50622.4 Extremal Reissner–Nordström spacetimes as BPS solutions 50822.5 The black hole attractor mechanism 510

22.5.1 Example of a black hole attractor 51122.5.2 The attractor mechanism – real slow and simple 513

22.6 Supersymmetry of the black holes 51722.6.1 Killing spinors 51722.6.2 The central charge 51922.6.3 The black hole potential 521

22.7 First order gradient flow equations 52222.8 The attractor mechanism – fast and furious 523Appendix 22A Killing spinors for pp-waves 525






xiv Contents

23 The AdS/CFT correspondence 52723.1 The N = 4 SYM theory 52923.2 Type IIB string theory and D3-branes 53223.3 The D3-brane solution of Type IIB supergravity 53323.4 Kaluza–Klein analysis on AdS5 ⊗ S5 53423.5 Euclidean AdS and its inversion symmetry 53623.6 Inversion and CFT correlation functions 53823.7 The free massive scalar field in Euclidean AdSd+1 53923.8 AdS/CFT correlators in a toy model 54123.9 Three-point correlation functions 54323.10 Two-point correlation functions 54523.11 Holographic renormalization 550

23.11.1 The scalar two-point function in a CFTd 55423.11.2 The holographic trace anomaly 555

23.12 Holographic RG flows 55823.12.1 AAdS domain wall solutions 55923.12.2 The holographic c-theorem 56223.12.3 First order flow equations 563

23.13 AdS/CFT and hydrodynamics 564

Appendix A Comparison of notation 573A.1 Spacetime and gravity 573A.2 Spinor conventions 575A.3 Components of differential forms 576A.4 Covariant derivatives 576

Appendix B Lie algebras and superalgebras 577B.1 Groups and representations 577B.2 Lie algebras 578B.3 Superalgebras 581

References 583Index 602






Preface

The main purpose of this book is to explore the structure of supergravity theories at theclassical level. Where appropriate we take a general D-dimensional viewpoint, usuallywith special emphasis on D = 4. Readers can consult the Contents for a detailed list ofthe topics treated, so we limit ourselves here to a few comments to guide readers. Wehave tried to organize the material so that readers of varying educational backgroundscan begin to read at a point appropriate to their background. Part I should be accessibleto readers who have studied relativistic field theory enough to appreciate the importanceof Lagrangians, actions, and their symmetries. Part II describes the differential geometricbackground and some basic physics of the general theory of relativity. The basic super-gravity theories are presented in Part III using techniques developed in earlier chapters. InPart IV we discuss complex geometry and apply it to matter couplings in global N = 1supersymmetry. In Part V we begin a systematic derivation of N = 1 matter-coupledsupergravity using the conformal compensator method. The going can get tough on thissubject. For this reason we present the final physical action and transformation rules andsome basic applications in two separate short chapters in Part VI. Part VII is devotedto a systematic discussion of N = 2 supergravity, including a short chapter with theresults needed for applications. Two major applications of supergravity, classical solu-tions and the AdS/CFT correspondence, are discussed in Part VIII in considerable detail.It should be possible to understand these chapters without full study of earlier parts of thebook.

Many interesting aspects of supergravity, some of them subjects of current research,could not be covered in this book. These include theories in spacetime dimensions D < 4,higher derivative actions, embedding tensors, infinite Lie algebra symmetries, and the pos-itive energy theorem.

Like many other subjects in theoretical physics, supersymmetry and supergravity arebest learned by readers who are willing to ‘get their hands dirty’. This means activelyworking out problems that reinforce the material under discussion. To facilitate this aspectof the learning process, many exercises for readers appear within each chapter. We give arough indication of the level of each exercise as follows:

xv






xvi Preface

Level 1. The result of this exercise will be used later in the book.Level 2. This exercise is intended to illuminate the subject under discussion, but it is not

needed in the rest of the book.Level 3. This exercise is meant to challenge readers, but is not essential.

These levels are indicated respectively by single, double or triple gray bars in the outsidemargin.

A website featuring solutions to some exercises, errata and additional reading material,can be found at www.cambridge.org/supergravity.

Dan FreedmanToine Van Proeyen

October 2011






Acknowledgements

We thank Eric Bergshoeff, Paul Chesler, Bernard de Wit, Eric D’Hoker, Henriette Elvang,John Estes, Gary Gibbons, Joaquim Gomis, Renata Kallosh, Hong Liu, Marián Lledó,Samir Mathur, John McGreevy, Michael Peskin, Leonardo Rastelli, Kostas Skenderis,Stefan Vandoren, Bert Vercnocke and Giovanni Villadoro. We thank the students in var-ious courses (Leuven advanced field theory course, Doctoral schools in Paris, Barcelona,Hamburg), and also Frederik Coomans, Serge Dendas, Daniel Harlow, Andrew Larkoski,Jonathan Maltz, Thomas Rube, Walter Van Herck and Bert Van Pol for their input in thepreparation of this text and their critical remarks.

Our home institutions have supported the writing of this book over a period of years, andwe are grateful. We also thank the Galileo Galilei Institute in Florence and the Departmentof Applied Mathematics and Theoretical Physics in Cambridge for support during extendedvisits, and the Stanford Institute for Theoretical Physics for support and hospitality, indeeda home away from home, during multiple visits when we worked closely together.

A.V.P. wil in het bijzonder zijn moeder bedanken voor de sterkte en voortdurende steundie hij van haar gekregen heeft. He also thanks Marleen and Laura for the strong supportduring the work on this book. D.Z.F. thanks his wife Miriam for her encouragement to startthis project and continuous support as it evolved.

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Supergravity - Assets - Cambridge University Press