Supergravity. Cambridge University Press Supergravity Daniel Z. Freedman and Antoine Van Proeyen Frontmatter More information

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1 Supergravity Supergravity, together with string theory, is one of the most significant developments in theoretical physics. Although there are many books on string theory, this is the first-ever authoritative and systematic account of supergravity. Written by two of the most respected workers in the field, it provides a solid introduction to the fundamentals of supergravity. The book starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories (D = 4 and D = 11) and the main gauge theory tools are explained. The second half of the book is more advanced: complex geometry and N = 1 and N = 2 supergravity theories are covered. Classical solutions and a chapter on anti-de Sitter/conformal field theory (AdS/CFT) correspondence complete the text. Numerous exercises and examples make it ideal for Ph.D. students, and with applications to model building, cosmology, and solutions of supergravity theories, this text is an invaluable resource for researchers. A website hosted by the authors, featuring solutions to some exercises and additional reading material, can be found at /supergravity. Daniel Z. Freedman is Professor of Applied Mathematics and Physics at the Massachusetts Institute of Technology. He has made many research contributions to supersymmetry and supergravity: he was a co-discoverer of the first supergravity theory in This discovery has been recognized by the award of the Dirac Medal and Prize in 1993, and the Dannie Heineman Prize of the American Physical Society in 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, the resulting special geometries, and their applications to phenomenology and cosmology.

2 Our conventions The metric is mostly plus, i.e The curvature is R μνρσ = g ρρ ( μ Ɣνσ ρ ν Ɣμσ ρ + Ɣμτ ρ Ɣνσ τ Ɣρ ντɣμσ τ ) = eρ a ( eb σ μ ω νab ν ω μab + ω μac ω c ν b ω νac ω c ) μ b. Ricci tensor and energy momentum tensors are defined by R μν = R ρ νρμ, R = g μν R μν, R μν 1 2 g μν R = κ 2 T μν. Covariant derivatives involving the spin connection are, for vectors and spinors, D μ V a = μ V a + ω ab μ Vb, D μ λ = μ λ ω μ ab γ ab λ. We use (anti)symmetrization of indices with weight 1, i.e. A [ab] = 1 2 (A ab A ba ) and A (ab) = 1 2 (A ab + A ba ). The Levi-Civita tensor is ε 0123 = 1, ε 0123 = 1. The dual, self-dual, and anti-self-dual of antisymmetric tensors are defined by H ab 1 2 iεabcd H cd, H ± ab = 1 2 (H ab ± H ab ), H ± ab = ( H ab). Structure constants are defined by [ TA, T B ] = f AB C TC. The Clifford algebra is γ μ γ ν + γ ν γ μ = 2g μν, γ μν = γ [μ γ ν],... (γ μ ) = γ 0 γ μ γ 0, γ = ( i) (D/2)+1 γ 0 γ 1...γ D 1 ; in four dimensions: γ = iγ 0 γ 1 γ 2 γ 3, ε abcd γ d = iγ γ abc. The Majorana and Dirac conjugates are λ = λ T C, λ = iλ γ 0. We mostly use the former. For Majorana fermions the two are equal. The main SUSY commutator is [ δ(ɛ1 ), δ(ɛ 2 ) ] = 1 2 ɛ 2γ μ ɛ 1 μ. p-form components are defined by φ p = 1 p! φ μ 1 μ p dx μ 1 dx μp. The differential acts from the left: da = ν A μ dx ν dx μ, A = A μ dx μ.

3 Supergravity DANIEL Z. FREEDMAN Massachusetts Institute of Technology, USA and ANTOINE VAN PROEYEN KU Leuven, Belgium

4 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York Information on this title: / c D. Z. Freedman and A. Van Proeyen 2012 This publication is in copyright. Subject to statutory exception and 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 data Freedman, Daniel Z. Supergravity / Daniel Z. Freedman and Antoine Van Proeyen. p. cm. ISBN (hardback) 1. Supergravity. I. Van Proeyen, Antoine. II. Title. QC S9F dc ISBN Hardback Additional resources for this publication at /supergravity Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Preface Acknowledgements page xv xvii Introduction 1 Part I Relativistic field theory in Minkowski spacetime 5 1 Scalar field theory and its symmetries The scalar field system Symmetries of the system SO(n) internal symmetry General internal symmetry Spacetime symmetries the Lorentz and Poincaré groups Noether currents and charges Symmetries in the canonical formalism Quantum operators The Lorentz group for D = The Dirac field The homomorphism of SL(2, C) SO(3, 1) The Dirac equation Dirac adjoint and bilinear form Dirac action The spinors u( p, s) and v( p, s) for D = Weyl spinor fields in even spacetime dimension Conserved currents Conserved U(1) current Energy momentum tensors for the Dirac field 37 3 Clifford algebras and spinors The Clifford algebra in general dimension The generating γ -matrices The complete Clifford algebra Levi-Civita symbol Practical γ -matrix manipulation Basis of the algebra for even dimension D = 2m 43 v

6 vi The highest rank Clifford algebra element Odd spacetime dimension D = 2m Symmetries of γ -matrices Spinors in general dimensions Spinors and spinor bilinears Spinor indices Fierz rearrangement Reality Majorana spinors Definition and properties Symplectic Majorana spinors Dimensions of minimal spinors Majorana spinors in physical theories Variation of a Majorana Lagrangian Relation of Majorana and Weyl spinor theories U(1) symmetries of a Majorana field 61 Appendix 3A Details of the Clifford algebras for D = 2m 62 3A.1 Traces and the basis of the Clifford algebra 62 3A.2 Uniqueness of the γ -matrix representation 63 3A.3 The Clifford algebra for odd spacetime dimensions 65 3A.4 Determination of symmetries of γ -matrices 65 3A.5 Friendly representations 66 4 The Maxwell and Yang Mills gauge fields The abelian gauge field A μ (x) Gauge invariance and fields with electric charge The free gauge field Sources and Green s function Quantum electrodynamics The stress tensor and gauge covariant translations Electromagnetic duality Dual tensors Duality for one free electromagnetic field Duality for gauge field and complex scalar Electromagnetic duality for coupled Maxwell fields Non-abelian gauge symmetry Global internal symmetry Gauging the symmetry Yang Mills field strength and action Yang Mills theory for G = SU(N) Internal symmetry for Majorana spinors 93 5 The free Rarita Schwinger field The initial value problem 97

7 vii 5.2 Sources and Green s function Massive gravitinos from dimensional reduction Dimensional reduction for scalar fields Dimensional reduction for spinor fields Dimensional reduction for the vector gauge field Finally μ (x, y) N = 1 global supersymmetry in D = Basic SUSY field theory Conserved supercurrents SUSY Yang Mills theory SUSY transformation rules SUSY field theories of the chiral multiplet U(1) R symmetry The SUSY algebra More chiral multiplets SUSY gauge theories SUSY Yang Mills vector multiplet Chiral multiplets in SUSY gauge theories Massless representations of N -extended supersymmetry Particle representations of N -extended supersymmetry Structure of massless representations 127 Appendix 6A Extended supersymmetry and Weyl spinors 129 Appendix 6B On- and off-shell multiplets and degrees of freedom 130 Part II Differential geometry and gravity Differential geometry Manifolds Scalars, vectors, tensors, etc The algebra and calculus of differential forms The metric and frame field on a manifold The metric The frame field Induced metrics Volume forms and integration Hodge duality of forms Stokes theorem and electromagnetic charges p-form gauge fields Connections and covariant derivatives The first structure equation and the spin connection ω μab The affine connection Ɣμν ρ Partial integration The second structure equation and the curvature tensor 161

8 viii 7.11 The nonlinear σ -model Symmetries and Killing vectors σ -model symmetries Symmetries of the Poincaré plane The first and second order formulations of general relativity Second order formalism for gravity and bosonic matter Gravitational fluctuations of flat spacetime The graviton Green s function Second order formalism for gravity and fermions First order formalism for gravity and fermions 182 Part III Basic supergravity N = 1 pure supergravity in four dimensions The universal part of supergravity Supergravity in the first order formalism The 1.5 order formalism Local supersymmetry of N = 1, D = 4 supergravity The algebra of local supersymmetry Anti-de Sitter supergravity D = 11 supergravity D 11 from dimensional reduction The field content of D = 11 supergravity Construction of the action and transformation rules The algebra of D = 11 supergravity General gauge theory Symmetries Global symmetries Local symmetries and gauge fields Modified symmetry algebras Covariant quantities Covariant derivatives Curvatures Gauged spacetime translations Gauge transformations for the Poincaré group Covariant derivatives and general coordinate transformations Covariant derivatives and curvatures in a gravity theory Calculating transformations of covariant quantities 231 Appendix 11A Manipulating covariant derivatives A.1 Proof of the main lemma A.2 Examples in supergravity 234

9 ix 12 Survey of supergravities The minimal superalgebras Four dimensions Minimal superalgebras in higher dimensions The R-symmetry group Multiplets Multiplets in four dimensions Multiplets in more than four dimensions Supergravity theories: towards a catalogue The basic theories and kinetic terms Deformations and gauged supergravities Scalars and geometry Solutions and preserved supersymmetries Anti-de Sitter superalgebras Central charges in four dimensions Central charges in higher dimensions 253 Part IV Complex geometry and global SUSY Complex manifolds The local description of complex and Kähler manifolds Mathematical structure of Kähler manifolds The Kähler manifolds CP n Symmetries of Kähler metrics Holomorphic Killing vectors and moment maps Algebra of holomorphic Killing vectors The Killing vectors of CP General actions with N = 1 supersymmetry Multiplets Chiral multiplets Real multiplets Generalized actions by multiplet calculus The superpotential Kinetic terms for chiral multiplets Kinetic terms for gauge multiplets Kähler geometry from chiral multiplets General couplings of chiral multiplets and gauge multiplets Global symmetries of the SUSY σ -model Gauge and SUSY transformations for chiral multiplets Actions of chiral multiplets in a gauge theory General kinetic action of the gauge multiplet Requirements for an N = 1 SUSY gauge theory The physical theory 288

10 x Elimination of auxiliary fields The scalar potential The vacuum state and SUSY breaking Supersymmetry breaking and the Goldstone fermion Mass spectra and the supertrace sum rule Coda 298 Appendix 14A Superspace 298 Appendix 14B Appendix: Covariant supersymmetry transformations 302 Part V Superconformal construction of supergravity theories Gravity as a conformal gauge theory The strategy The conformal algebra Conformal transformations on fields The gauge fields and constraints The action Recapitulation Homothetic Killing vectors The conformal approach to pure N = 1 supergravity Ingredients Superconformal algebra Gauge fields, transformations, and curvatures Constraints Superconformal transformation rules of a chiral multiplet The action Superconformal action of the chiral multiplet Gauge fixing The result Construction of the matter-coupled N = 1 supergravity Superconformal tensor calculus The superconformal gauge multiplet The superconformal real multiplet Gauge transformations of superconformal chiral multiplets Invariant actions Construction of the action Conformal weights Superconformal invariant action (ungauged) Gauged superconformal supergravity Elimination of auxiliary fields Partial gauge fixing Projective Kähler manifolds 351

11 xi The example of CP n Dilatations and holomorphic homothetic Killing vectors The projective parametrization The Kähler cone The projection Kähler transformations Physical fermions Symmetries of projective Kähler manifolds T -gauge and decomposition laws An explicit example: SU(1, 1)/ U(1) model From conformal to Poincaré supergravity The superpotential The potential Fermion terms Review and preview Projective and Kähler Hodge manifolds Compact manifolds 375 Appendix 17A Kähler Hodge manifolds A.1 Dirac quantization condition A.2 Kähler Hodge manifolds 378 Appendix 17B Steps in the derivation of (17.7) 380 Part VI N = 1 supergravity actions and applications The physicaln = 1 matter-coupled supergravity The physical action Transformation rules Further remarks Engineering dimensions Rigid or global limit Quantum effects and global symmetries Applications of N = 1 supergravity Supersymmetry breaking and the super-beh effect Goldstino and the super-beh effect Extension to cosmological solutions Mass sum rules in supergravity The gravity mediation scenario The Polónyi model of the hidden sector Soft SUSY breaking in the observable sector No-scale models Supersymmetry and anti-de Sitter space R-symmetry and Fayet Iliopoulos terms The R-gauge field and transformations 405

12 xii Fayet Iliopoulos terms An example with non-minimal Kähler potential 406 Part VII Extended N = 2 supergravity Construction of the matter-coupled N = 2 supergravity Global supersymmetry Gauge multiplets for D = Gauge multiplets for D = Gauge multiplets for D = Hypermultiplets Gauged hypermultiplets N = 2 superconformal calculus The superconformal algebra Gauging of the superconformal algebra Conformal matter multiplets Superconformal actions Partial gauge fixing Elimination of auxiliary fields Complete action D = 5 and D = 6, N = 2 supergravities Special geometry The family of special manifolds Very special real geometry Special Kähler geometry Hyper-Kähler and quaternionic-kähler manifolds From conformal to Poincaré supergravity Kinetic terms of the bosons Identities of special Kähler geometry The potential Physical fermions and other terms Supersymmetry and gauge transformations 461 Appendix 20A SU(2) conventions and triplets 463 Appendix 20B Dimensional reduction B.1 Reducing from D = 6 D = B.2 Reducing from D = 5 D = Appendix 20C Definition of rigid special Kähler geometry The physicaln = 2 matter-coupled supergravity The bosonic sector The basic (ungauged) N = 2, D = 4 matter-coupled supergravity The gauged supergravities The symplectic formulation 472

13 xiii Symplectic definition Comparison of symplectic and prepotential formulation Gauge transformations and symplectic vectors Physical fermions and duality Action and transformation laws Final action Supersymmetry transformations Applications Partial supersymmetry breaking Field strengths and central charges Moduli spaces of Calabi Yau manifolds Remarks Fayet Iliopoulos terms σ -model symmetries Engineering dimensions 482 Part VIII Classical solutions and the AdS/CFT correspondence Classical solutions of gravity and supergravity Some solutions of the field equations Prelude: frames and connections on spheres Anti-de Sitter space AdS D obtained from its embedding in R D Spacetime metrics with spherical symmetry AdS Schwarzschild spacetime The Reissner Nordström metric A more general Reissner Nordström solution Killing spinors and BPS solutions The integrability condition for Killing spinors Commuting and anti-commuting Killing spinors Killing spinors for anti-de Sitter space Extremal Reissner Nordström spacetimes as BPS solutions The black hole attractor mechanism Example of a black hole attractor The attractor mechanism real slow and simple Supersymmetry of the black holes Killing spinors The central charge The black hole potential First order gradient flow equations The attractor mechanism fast and furious 523 Appendix 22A Killing spinors for pp-waves 525

14 xiv 23 The AdS/CFT correspondence The N = 4 SYM theory Type IIB string theory and D3-branes The D3-brane solution of Type IIB supergravity Kaluza Klein analysis on AdS 5 S Euclidean AdS and its inversion symmetry Inversion and CFT correlation functions The free massive scalar field in Euclidean AdS d AdS/CFT correlators in a toy model Three-point correlation functions Two-point correlation functions Holographic renormalization The scalar two-point function in a CFT d The holographic trace anomaly Holographic RG flows AAdS domain wall solutions The holographic c-theorem First order flow equations AdS/CFT and hydrodynamics 564 Appendix A Comparison of notation 573 A.1 Spacetime and gravity 573 A.2 Spinor conventions 575 A.3 Components of differential forms 576 A.4 Covariant derivatives 576 Appendix B Lie algebras and superalgebras 577 B.1 Groups and representations 577 B.2 Lie algebras 578 B.3 Superalgebras 581 References 583 Index 602

15 Preface The main purpose of this book is to explore the structure of supergravity theories at the classical level. Where appropriate we take a general D-dimensional viewpoint, usually with special emphasis on D = 4. Readers can consult the for a detailed list of the topics treated, so we limit ourselves here to a few comments to guide readers. We have tried to organize the material so that readers of varying educational backgrounds can begin to read at a point appropriate to their background. Part I should be accessible to readers who have studied relativistic field theory enough to appreciate the importance of Lagrangians, actions, and their symmetries. Part II describes the differential geometric background and some basic physics of the general theory of relativity. The basic supergravity theories are presented in Part III using techniques developed in earlier chapters. In Part IV we discuss complex geometry and apply it to matter couplings in global N = 1 supersymmetry. In Part V we begin a systematic derivation of N = 1 matter-coupled supergravity using the conformal compensator method. The going can get tough on this subject. For this reason we present the final physical action and transformation rules and some basic applications in two separate short chapters in Part VI. Part VII is devoted to a systematic discussion of N = 2 supergravity, including a short chapter with the results needed for applications. Two major applications of supergravity, classical solutions 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 the book. 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 positive energy theorem. Like many other subjects in theoretical physics, supersymmetry and supergravity are best learned by readers who are willing to get their hands dirty. This means actively working out problems that reinforce the material under discussion. To facilitate this aspect of the learning process, many exercises for readers appear within each chapter. We give a rough indication of the level of each exercise as follows: xv

16 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 outside margin. A website featuring solutions to some exercises, errata and additional reading material, can be found at /supergravity. Dan Freedman Toine Van Proeyen October 2011

17 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 various 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 the preparation of this text and their critical remarks. Our home institutions have supported the writing of this book over a period of years, and we are grateful. We also thank the Galileo Galilei Institute in Florence and the Department of Applied Mathematics and Theoretical Physics in Cambridge for support during extended visits, and the Stanford Institute for Theoretical Physics for support and hospitality, indeed a 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 steun die hij van haar gekregen heeft. He also thanks Marleen and Laura for the strong support during the work on this book. D.Z.F. thanks his wife Miriam for her encouragement to start this project and continuous support as it evolved. xvii