Gauge/Gravity Duality Foundations and Applications Gauge/gravity duality creates new links between quantum theory and gravity. It has led to new concepts in mathematics and physics, and provides new tools for solving problems in many areas of theoretical physics. This book is the first comprehensive textbook on this important topic, enabling graduate students and researchers in string theory and particle, nuclear and condensed matter physics to become acquainted with the subject. Focusing on the fundamental aspects as well as on applications, this textbook guides readers through a thorough explanation of the central concepts of gauge/gravity duality. For the AdS/CFT correspondence, it explains in detail how string theory provides the conjectured map. Generalisations to less symmetric cases of gauge/gravity duality and their applications are then presented, in particular to finite temperature and density, hydrodynamics, QCD-like theories, the quark–gluon plasma and condensed matter sys- tems. The textbook features a large number of exercises, with solutions available online at www.cambridge.org/9781107010345. Johanna Erdmenger is a Research Group Leader at the Max Planck Institute for Physics (Werner Heisenberg Institute), Munich, Germany, and Honorary Professor at Ludwig Maximilian University, Munich. She is one of the pioneers of applying gauge/gravity duality to elementary particle, nuclear and condensed matter physics. Martin Ammon is a Junior Professor at Friedrich Schiller University, Jena, Germany, leading a research group on gauge/gravity duality. He was awarded the prestigious Otto Hahn Medal of the Max Planck Society for his Ph.D. thesis on applying gauge/gravity duality to condensed matter physics. Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:52:14 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:52:15 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 Gauge/Gravity Duality Foundations and Applications MARTIN AMMON Friedrich Schiller University, Jena JOHANNA ERDMENGER Max Planck Institute for Physics, Munich Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:52:15 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107010345 © M. Ammon and J. Erdmenger 2015 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 2015 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library Ammon, Martin, 1981- Gauge-gravity duality : foundations and applications / Martin Ammon, Friedrich Schiller University, Jena, Johanna Erdmenger, Max Planck Institute for Physics, Munich. pages cm. Includes bibliographical references and index. ISBN 978-1-107-01034-5 (hardback : alk. paper) 1. Gauge fields (Physics). 2. Gravity. 3. Holography. 4. Mathematical physics. I. Erdmenger, Johanna. II. Title. QC793.3.G38A56 2015 530.1435–dc23 2015003328 ISBN 978-1-107-01034-5 Hardback 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. Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:52:15 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 Contents Preface page ix Acknowledgements xiii Part I Prerequisites 1 1 Elements of field theory 3 1.1 Classical scalar field theory 3 1.2 Symmetries and conserved currents 6 1.3 Quantisation 9 1.4 Wick rotation and statistical mechanics 18 1.5 Regularisation and renormalisation 19 1.6 Dirac fermions 28 1.7 Gauge theory 30 1.8 Symmetries, Ward identities and anomalies 45 1.9 Further reading 48 References 48 2 Elements of gravity 50 2.1 Differential geometry 50 2.2 Einstein’s field equations 65 2.3 Maximally symmetric spacetimes 66 2.4 Black holes 76 2.5 Energy conditions 88 2.6 Further reading 89 References 89 3 Symmetries in quantum field theory 91 3.1 Lorentz and Poincaré symmetry 91 3.2 Conformal symmetry 102 3.3 Supersymmetry 118 3.4 Superconformal symmetry 139 3.5 Further reading 143 References 143 v Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:53:25 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 vi Contents 4 Introduction to superstring theory 145 4.1 Bosonic string theory 145 4.2 Superstring theory 156 4.3 Web of dualities 164 4.4 D-branes and other non-perturbative objects 168 4.5 Further reading 174 References 174 Part II Gauge/Gravity Duality 177 5 The AdS/CFT correspondence 179 5.1 The AdS/CFT correspondence: a first glance 180 5.2 D3-branes and their two faces 182 5.3 Field-operator map 189 5.4 Correlation functions 196 5.5 Holographic renormalisation 204 5.6 Wilson loops in N = 4 Super Yang–Mills theory 211 5.7 Further reading 216 References 216 6 Tests of the AdS/CFT correspondence 219 6.1 Correlation function of 1/2 BPS operators 220 6.2 Four-point functions 229 6.3 The conformal anomaly 233 6.4 Further reading 237 References 238 7 Integrability and scattering amplitudes 240 7.1 Integrable structures on the gauge theory side 241 7.2 Integrability on the gravity (string theory) side 248 7.3 BMN limit and classical string configurations 254 7.4 Dual superconformal symmetry 258 7.5 Further reading 271 References 271 8 Further examples of the AdS/CFT correspondence 273 8.1 D3-branes at singularities 273 8.2 M2-branes: AdS4/CFT3 278 8.3 Gravity duals of conformal field theories: further examples 287 8.4 Towards non-conformal field theories 289 8.5 Further reading 294 References 294 Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:53:26 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 vii Contents 9 Holographic renormalisation group flows 296 9.1 Renormalisation group flows in quantum field theory 296 9.2 Holographic renormalisation group flows 302 9.3 ∗Supersymmetric flows within IIB Supergravity in D = 10 308 9.4 Further reading 323 References 324 10 Duality with D-branes in supergravity 326 10.1 Branes as flavour degrees of freedom 326 10.2 AdS/CFT correspondence with probe branes 331 10.3 D7-brane fluctuations and mesons in N = 2 theory 335 10.4 ∗D3/D5-brane system 340 10.5 Further reading 342 References 343 11 Finite temperature and density 344 11.1 Finite temperature field theory 344 11.2 Gravity dual thermodynamics 352 11.3 Finite density and chemical potential 362 11.4 Further reading 365 References 365 Part III Applications 367 12 Linear response and hydrodynamics 369 12.1 Linear response 370 12.2 Hydrodynamics 380 12.3 Transport coefficients from linear response 384 12.4 Fluid/gravity correspondence 389 12.5 Further reading 396 References 396 13 QCD and holography: confinement and chiral symmetry breaking 399 13.1 Review of QCD 399 13.2 Gauge/gravity duality description of confinement 405 13.3 Chiral symmetry breaking from D7-brane probes 413 13.4 Non-Abelian chiral symmetries: the Sakai–Sugimoto model 421 13.5 AdS/QCD correspondence 426 13.6 Further reading 430 References 431 Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:53:26 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 viii Contents 14 QCD and holography: finite temperature and density 435 14.1 QCD at finite temperature and density 435 14.2 Gauge/gravity approach to the quark–gluon plasma 438 14.3 Holographic flavour at finite temperature and density 445 14.4 Sakai–Sugimoto model at finite temperature 454 14.5 Holographic predictions for the quark–gluon plasma 455 14.6 Further reading 456 References 457 15 Strongly coupled condensed matter systems 460 15.1 Quantum phase transitions 461 15.2 Charges and finite density 463 15.3 Holographic superfluids and superconductors 477 15.4 Fermions 485 15.5 Towards non-relativistic systems and hyperscaling violation 489 15.6 Entanglement entropy 496 15.7 Further reading 501 References 502 Appendix A Grassmann numbers 505 Appendix B Lie algebras and superalgebras 508 Appendix C Conventions 526 Index 527 Downloaded from Cambridge Books Online by IP 150.244.109.203 on Sat Jun 20 19:53:27 BST 2015. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511846373 Cambridge Books Online © Cambridge University Press, 2015 Preface Gauge/gravity duality is a major new development within theoretical physics. It brings together string theory, quantum field theory and general relativity, and has applications to elementary particle, nuclear and condensed matter physics. Gauge/gravity duality is of fundamental importance since it provides new links between quantum theory and gravity which are based on string theory.