Classical Strings and Membranes in the Ads/CFT Correspondence

Classical Strings and Membranes in the AdS/CFT Correspondence GEORGIOS LINARDOPOULOS Faculty of Physics, Department of Nuclear and Particle Physics National and Kapodistrian University of Athens and Institute of Nuclear and Particle Physics National Center for Scientific Research "Demokritos" Dissertation submitted for the Degree of Doctor of Philosophy at the National and Kapodistrian University of Athens June 2015 Doctoral Committee Supervisor Emmanuel Floratos Professor Emer., N.K.U.A. Co-Supervisor Minos Axenides Res. Director, N.C.S.R., "Demokritos" Supervising Committee Member Nikolaos Tetradis Professor, N.K.U.A. Thesis Defense Committee Ioannis Bakas Professor, N.T.U.A. Georgios Diamandis Assoc. Professor, N.K.U.A. Athanasios Lahanas Professor Emer., N.K.U.A. Konstantinos Sfetsos Professor, N.K.U.A. iThis thesis is dedicated to my parents iii Acknowledgements This doctoral dissertation is based on the research that took place during the years 2012–2015 at the Institute of Nuclear & Particle Physics of the National Center for Scientific Research "Demokritos" and the Department of Nuclear & Particle Physics at the Physics Faculty of the National and Kapodistrian University of Athens. I had the privilege to have professors Emmanuel Floratos (principal supervisor), Minos Axenides (co-supervisor) and Nikolaos Tetradis as the 3-member doctoral committee that supervised my PhD. I would like to thank them for the fruitful cooperation we had, their help and their guidance. I feel deeply grateful to my teacher Emmanuel Floratos for everything that he has taught me. It is extremely difficult for me to imagine a better and kinder supervisor. I thank him for his advices, his generosity and his love. I am deeply indebted to Minos Axenides for all his support, encouragement, help, time and advice. I would also like to thank professor Nikolaos Tetradis who was always available to help and advise me. I would also like to express my gratitude to the remaining members of the 7-member doctoral committee, professors Ioannis Bakas, Georgios Diamantis, Athanasios Lahanas and Konstantinos Sfetsos. During the PhD, I had the luck to have my research at N.C.S.R. "Demokritos" financially supported by the General Secretariat for Research and Technology of Greece and from the European Regional Development Fund MIS- 448332-ORASY (NSRF 2007–13 ACTION, KRIPIS). I am very thankful to Dr. George Georgiou who has been my closest collaborator and friend during the PhD and has significantly contributed to its completion. I would like to thank professor Stam Nicolis for many exciting discussions and his encouragement. Finally, I would like to thank all the members of my family and especially my parents to which I dedicate this doctoral dissertation. vProlegomena This doctoral dissertation is based on the publications [1, 2, 3, 4]: • G. Linardopoulos, Large-Spin Expansions of Giant Magnons, [<a href="/goto?url=http://lanl.arxiv.org/abs/1502.01630" target="_blank">arXiv:1502.01630</a>]. To appear in Proceedings of Science, PoS (CORFU2014) 154 • E. Floratos, G. Linardopoulos, Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes, Nucl.Phys. B897 (2015) 229, [<a href="/goto?url=http://lanl.arxiv.org/abs/1406.0796" target="_blank">arXiv:1406.0796</a>] • E. Floratos, G. Georgiou, G. Linardopoulos, Large-Spin Expansions of GKP Strings, JHEP 03 (2014) 018, [<a href="/goto?url=http://lanl.arxiv.org/abs/1311.5800" target="_blank">arXiv:1311.5800</a>] • M. Axenides, E. Floratos, G. Linardopoulos, Stringy Membranes in AdS/CFT, JHEP 08 (2013) 089, [<a href="/goto?url=http://lanl.arxiv.org/abs/1306.0220" target="_blank">arXiv:1306.0220</a>], as well as the following talks: • Large-Spin Expansions of Giant Magnons. Workshop on Quantum Fields and Strings. 14th Hellenic School and Workshops on Elementary Particle Physics & Gravity. Corfu Summer Institute, 17/09/2014, [<a href="/goto?url=http://www.physics.ntua.gr/corfu2014/lectures.html" target="_blank">http://www.physics.ntua.gr/corfu2014/lectures.html</a>] • Dispersion Relation of GKP Strings. Summer School on String Theory and Holography & Mathematica Summer School on Theoretical Physics, 6th Edition. Departamento de Física e Astronomia of Faculdade de Ciências da Universidade do Porto (FCUP), 24/07/2014, [<a href="/goto?url=http://msstp.org/?q=node/294" target="_blank">http://msstp.org/?q= </a><a href="/goto?url=http://msstp.org/?q=node/294" target="_blank">node/294</a>] • Stringy Membranes in AdS/CFT. “Holography 2013: Gauge/Gravity Duality and Strongly Correlated Systems”. Pohang, S. Korea, June 13 – 22. APCTP, POSTECH, 15/06/2013, [<a href="/goto?url=https://www.apctp.org/plan.php/holography2013/660" target="_blank">https: </a><a href="/goto?url=https://www.apctp.org/plan.php/holography2013/660" target="_blank">//www.apctp.org/plan.php/holography2013/660</a>] • Rotating Strings and Membranes in AdS/CFT. “Foundations of Quantum Mechanics and Relativistic Spacetime”. Annual workshop of Working Group 3, “Quantum Theory Meets Relativity”, of the COST Action MP1006, “Fundamental Problems in Quantum Physics”. National & Kapodistrian University of Athens, 26/09/2012, [<a href="/goto?url=http://www.pwallden.gr/foundations.asp" target="_blank">http://www.pwallden.gr/foundations.asp</a>]. The author’s research at N.C.S.R. "Demokritos" is supported by the General Secretariat for Research and Technology of Greece and from the European Regional Development Fund MIS-448332- ORASY (NSRF 2007–13 ACTION, KRIPIS). vii Abstract In its strongest version, the AdS/CFT conjecture states that N = 4, su (Nc) super Yang-Mills (SYM) theory is equal to type IIB string theory on AdS5 × S5. It is by far the most important equation of contemporary theoretical physics, a sort of a "harmonic oscillator" for both the quantum theory of gravity and gauge theories. It goes without saying that it is imperative to fully understand its limits of validity and thoroughly investigate its implications. In particular, it would be desirable to solve the theory, i.e. to be able to compute all of its observables. One of the most important observables of AdS/CFT is its spectrum. According to the AdS/CFT "dictionary", the spectrum of the theory comprises the energies of its string states, each of which must be equal to the scaling dimensions of its dual gauge theory operator. The full spectral problem of AdS/CFT is solved by integrability, in the sense that integrability provides the full set of algebraic equations that determine it. Integrability methods are however severely limited in the regime of long, strongly coupled operators, such as those that are dual to the Gubser-Klebanov-Polyakov (GKP) strings, giant magnons and single spike strings. In this thesis we study classical strings and branes in the context of the AdS/CFT correspondence. Our goal is twofold: (1) develop methods for computing the AdS5/CFT4 spectrum in the case of long, strongly coupled operators, by using classical strings and (2) understand the role of classical membranes in AdS/CFT by investigating their stringy limits. With regard to the first objective, we compute the classical spectra of long rotating GKP strings, giant magnons and single spikes. The conserved linear and angular momenta of these string configurations, that live either in AdS3 or R×S2, are known in parametric form in terms of the strings’ linear and angular velocities. We eliminate the linear and angular velocities from the expressions that give the energy of the strings, in favor of the strings’ conserved charges of linear and angular momenta. This way, we find all the leading, subleading and next-to-next-to-leading terms in the dispersion relations of the aforementioned string configurations. Our results are expressed in closed forms with Lambert’s W-function. For the second objective we introduce and study "stringy membranes", a new class of membranes that live in AdS4/7 × S7/4 or AdS4 × S7/Zk and have the same equations of motion, constraints and conserved charges with strings that live in an appropriate subset of AdS5. Stringy membranes can be constructed whenever the target spacetime contains a compact submanifold, by identifying one of the submanifold’s compact coordinates with one of the membrane worldvolume coordinates. For the stringy membranes that reproduce the pulsating and rotating GKP strings in AdS, we find that the spectrum of their transverse quadratic fluctuations displays a multiple band/gap structure governed by the Lamé equation. Conversely, string excitations are represented by a single-band/single-gap Lamé pattern. These findings confirm the picture that we have of membranes as collective excitations of some stringy counterparts. ix Contents 1 Introduction 61.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 <ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">Introduction to AdS/CFT </li></ul>11 2 Gauge/Gravity Duality 11 2.1 Large-Nc Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Holographic Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 AdS/CFT Correspondence 18 3.1 Open String Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Closed String Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 N = 4 Super Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 IIB String Theory on AdS5×S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Parameter Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6 The BMN Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.7 Maldacena Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.8 ABJM Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.9 Field/Operator Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.10 Testing the AdS5/CFT4 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.10.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.10.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.10.3 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.10.4 Anomalies, Moduli Spaces, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 AdS/CFT Integrability 31 4.1 Classical & Quantum Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.1 Classical Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.2 Quantum Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Integrability in AdS5/CFT4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Integrability in the su (2) Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3.1 Coordinate Bethe Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.2 Asymptotic Bethe Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 II Spinning Strings in AdS5 × S5 39 5 Introduction and Motivation 39 5.1 Classical Bosonic Strings in AdS5 × S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2 Pohlmeyer Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 Neumann-Rosochatius Reduction in R × S5 . . . . . . . . . . . . . . . . . . . . . . . . 43 6 The Gubser-Klebanov-Polyakov (GKP) String 45 6.1 Rotating AdS3 GKP String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.1.1 Short Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1.2 Long Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1 6.1.3 Short-Long Strings Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Rotating R × S2 GKP String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2.1 Short Folded Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.2 Long Folded Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2.3 Slow Circular Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2.4 Fast Circular Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2.5 Short-Long Strings Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 Pulsating AdS3 GKP String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.3.1 Semiclassical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 Dispersion Relations of GKP Strings 66 7.1 Rotating R × S2 GKP String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.1.1 Inverse Spin Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.1.2 Anomalous Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.1.3 Leading Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.1.4 Next-to-Leading Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.1.5 NNL Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.1.6 Fast Circular Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.2 Rotating AdS3 GKP String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.2.1 Inverse Spin Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.2.2 Anomalous Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2.3 Leading Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.2.4 Next-to-Leading Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2.5 NNL Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8 Infinite-Size Giant Magnons and Single Spikes 91 8.1 The Hofman-Maldacena (HM) Giant Magnon . . . . . . . . . . . . . . . . . . . . . . . 94 8.2 Infinite-Momentum Single Spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.3 Bound States & Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3.1 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3.2 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9 Finite-Size Giant Magnons and Single Spikes 102 9.1 Giant Magnon: Elementary Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2 Giant Magnon: Doubled Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.3 Single Spike: Elementary Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.4 Single Spike: Doubled Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10 Dispersion Relations of Giant Magnons and Single Spikes 111 10.1 Giant Magnon: Elementary Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 10.1.1 Inverse Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.1.2 Inverse Spin Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 10.1.3 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 10.2 Giant Magnon: Doubled Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3 Single Spike: Elementary Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.4 Single Spike: Doubled Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 211 Part II Summary and Discussion 124 11.1 GKP Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 11.2 Giant Magnons & Single Spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 III Rudiments of p-Branes & M-Theory 128 12 Generalities 128 12.1 Uses of Extended Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12.1.1 Description of Elementary Particles . . . . . . . . . . . . . . . . . . . . . . . . . 129 12.1.2 Study of the Strong Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 12.1.3 Generalization of Superstrings and Superparticles . . . . . . . . . . . . . . . . . 129 12.1.4 The Membrane Paradigm of Black Holes . . . . . . . . . . . . . . . . . . . . . . 130 12.1.5 M-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 12.1.6 AdS/CFT Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 12.1.7 Brane World Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12.2 Towards M(embrane) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12.2.1 Membrane Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12.2.2 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 12.2.3 Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 12.2.4 Non-Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 12.2.5 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 12.2.6 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 12.2.7 Membrane Interactions & Perturbation Theory. . . . . . . . . . . . . . . . . . . 134 13 Introduction to Membranes 135 13.1 Bosonic Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.1.1 Dirac-Nambu-Goto Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.1.2 Polyakov Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.1.3 Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 13.1.4 Membranes in Flat Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.2 Bosonic p-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.2.1 Volume-Preserving Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 140 13.3 Supermembranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13.3.1 11-Dimensional Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 13.4 M(atrix) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 13.4.1 Matrix-Regularized Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 13.4.2 The Matrix Theory Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 13.4.3 Matrix Theory in Curved Backgrounds . . . . . . . . . . . . . . . . . . . . . . . 145 IV Rotating Membranes 147 147 149 150 14 Introduction and Motivation 15 Spinning Membranes in AdS7 × S4 16 Spinning Membranes as Spinning Strings 16.1 Stringy Membranes in AdS7 × S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 16.2 Stringy Membranes in AdS4 × S7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 16.3 Stringy Membranes in AdS4 × S7/Zk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 317 Membrane Fluctuations 155 17.1 Rotating Stringy Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 17.2 Pulsating Stringy Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 18 Part IV Summary and Discussion 164 <ul style="display: flex;"><li style="flex:1">V</li><li style="flex:1">Appendixes </li></ul>167 A Anti-de Sitter Space 167 A.1 Global Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.2 "Sausage" Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.3 Horospheric/Poincaré Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.4 Stereographic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 A.5 "Static" Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 A.6 AdS as a Ruled Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 A.6.1 Light-Cone Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.7 AdS Coordinate Systems Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Classical Strings and Membranes in the Ads/CFT Correspondence

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