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Gauge/Gravity Duality and Field Theories at Strong Coupling

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Gauge/Gravity Duality and Field Theories at Strong Coupling Th`ese de Doctorat Pr´esent´ee`al'Universit´eParis{Sud XI Sp´ecialit´e:Physique Th´eorique P´eriode acad´emique 2008{2011 Gauge/Gravity Duality and Field Theories at Strong Coupling Gregory C. Giecold Institut de Physique Th´eorique,CNRS{URA 2306 CEA/Saclay, F{91191 Gif{sur{Yvette, France Th`esesoutenue le 17 juin 2011 devant le jury compos´ede: Iosif Bena Directeur de Th`ese Emilian Dudas Membre du Jury Edmond Iancu Directeur de Th`ese Henning Samtleben Rapporteur Kostas Skenderis Rapporteur Acknowledgements I would like to thank my advisors Iosif Bena and Edmond Iancu for giving much leeway for exploring alleyways of my own interest in the maze of string theory and field theory. I am beholden for fast{paced, illuminating discussions with them and their careful guidance with non{scientific matters. I am grateful to Mariana Gra~naand Al Mueller for being sharp yet very kind collaborators. Special credit is due to Nick Halmagyi from whom I have learnt so much, Aussie vernacular included. Many thanks to Ruben Minasian and Robi Peschanski for their interest in my research orientations and for tips on relocating to Stony Brook. I am grateful to Emilian Dudas and Henning Samtleben for agreeing to be part of the jury of my PhD defense. Special thanks to Kostas Skenderis whose research has been very important to my work. Discussions with Michael Bon, Gaetan Borot, Roberto Bondesan, J´eromeDubail, Hadi Go- dazgar, Mahdi Godazgar, Andrea Puhm, Bruno Sciolla and Piotr Tourkine were appreciated. As for those conversations that have been of direct relevance to my field of research interests, I am grateful to Guillaume Beuf, Paul Chesler, Sheer El{Showk, Yoshitaka Hatta, Akikazu Hashimoto, Jan Manschot, Diego Marqu`es, Carlos Nu~nez,Hagen Triendl, Pierre Vanhove and Bert Vercnocke. Special thanks to my excellent collaborator Stefano Massai and to Tae{Joon Cho, Cl´ement Gombeaud, Enrico Goi, Francesco Orsi and Cl´ement Ruef. The generous financial funding from the CEA/Saclay over those three years is greatly ap- preciated, as is the help with administrative matters from Catherine Cataldi, Laure Sauboy and Sylvie Zaffanella. Finally, I am beholden to my family for their support. R´esum´e L'objet de cette th`eseest l'´etudede certaines propri´et´esde th´eoriesdes champs `afort couplage via la dualit´eavec la th´eoriedes cordes, dans la limite de supergravit´e.L'analyse exp´erimentale du plasma de quarks et de gluons produit au RHIC et au LHC tend en effet `aindiquer que cet ´etat de la mati`erese comporte comme un fluide quasiment parfait. Les m´ethodes perturbatives de la QCD sont impuissantes `ad´ecrireses propri´et´eset la chromodynamique quantique sur r´eseaufait face `ades probl`emestant techniques que conceptuels pour calculer les observables dynamiques d'un tel syst`eme.La correspondance AdS/CFT offre par cons´equent un outil unique permettant d'´etudieren premi`ereapproximation cette phase de la QCD. L'un des aspects de cette th`ese consiste en la description par une ´equationstochastique de Langevin d'un parton massif se propageant dans le plasma d'une th´eoriede Yang{Mills maximalement supersym´etrique. Bien que cette th´eoriesemble d´ecrirede mani`eresatisfaisante la phase d´econfin´eede la QCD, il est toutefois d´esirablede chercher un dual en th´eoriedes cordes rendant compte des aspects de la QCD `abasse ´energie. L'autre axe directeur de cette th`esepropose ainsi de rendre compte de solutions de moindre supersym´etrie,sans invariance conforme, et avec confinement. On obtient le dual gravitationnel d'´etatsmetastables de telles th´eories. En particulier, on d´erive une contribution au potentiel inflationnaire dans le cadre d'un mod`ele cosmologique g´en´erique de la th´eoriedes cordes. Abstract In this thesis, we apply the gauge/string duality in its supergravity limit to infer some properties of field theories at strong coupling. Experiments at RHIC and at the LHC indeed suggest that the quark{gluon plasma behaves as one of the most perfect fluid ever achieved in any controlled experimental setup. Perturbative approaches fail at accounting for its properties, whereas lattice QCD methods face technical as well as conceptual difficulties in computing dynamical aspects of this new state of matter. As a result, the AdS/CFT correspondence currently is the best tool at our disposal for analytically modelling this phase of QCD. One of the contributions of this thesis amounts to deriving a stochastic Langevin equation for a heavy quark moving across a maximally supersymmetric Yang{Mills plasma at strong coupling. Even though this theory seems to describe in a surprisingly satisfactory way the high{energy, deconfined phase of QCD, it is also of much interest to try and search for a string theory dual making closer contact with QCD at lower energies. As such, the other main focus of this thesis deals with supergravity solutions of lesser supersymmetry, without conformal invariance and exhibiting confinement. We build for the first time the gravity dual to metastable states of such theories. In particular, we find the contribution from anti{branes to the inflation potential in some general scenario of string cosmology. Contents 1 Introduction 4 2 Fermionic Schwinger{Keldysh Propagators from AdS/CFT 22 2.1 Introduction....................................... 22 2.2 Review of Schwinger{Keldysh formalism for fermions................ 23 2.3 Review of fermionic retarded correlators in AdS/CFT............... 25 2.4 Spinors in complexified space{time.......................... 28 2.5 Real{time correlators from gravity.......................... 30 3 Stochastic Trailing String and Langevin Dynamics from AdS/CFT 34 3.1 Introduction....................................... 34 3.2 Boundary picture of the stochastic motion...................... 37 3.2.1 The trailing string and its small fluctuations................ 37 3.2.2 Keldysh Green function in AdS/CFT.................... 38 3.2.3 A Langevin equation for the heavy quark.................. 42 3.3 Bulk picture of the stochastic motion......................... 44 3.4 Discussion and physical picture............................ 49 3.4.1 Momentum distributions from the Langevin equations........... 49 3.4.2 Physical picture: Medium{induced radiation................ 50 4 Heavy Quark in an Expanding Plasma in AdS/CFT 54 4.1 Introduction....................................... 54 4.2 Transverse and rapidity fluctuations in an expanding plasma and the Langevin description....................................... 55 4.3 Trailing string in the BF background......................... 59 4.3.1 Dispersion relations and drag force...................... 61 4.3.2 Fluctuating trailing string and momentum diffusion............ 63 5 Finite{Temperature Fractional D2{Branes and the Deconfinement Transition in 2+1 Dimensions 66 5.1 Introduction....................................... 66 5.2 Non{extremal generalization of the fractional D2{brane Ansatz.......... 69 5.3 Derivation of the equations of motion........................ 71 5.4 Three simple solutions................................. 75 5.4.1 Singular non{extremal fractional D2{brane................. 75 5.4.2 The extremal Herzog{Klebanov fractional D2{brane solution....... 77 5.4.3 The non{extremal ordinary black D2{brane................. 77 5.5 Asymptotics of the regular non{extremal fractional D2{branes.......... 78 5.6 Perturbation theory in P................................ 79 5.6.1 Leading{order solution for K......................... 81 5.6.2 Solutions for other fields............................ 82 5.7 Conclusion....................................... 85 2 6 The Backreaction of Anti{M2 Branes on a Warped Stenzel Space 87 6.1 Introduction and discussion.............................. 87 6.2 Perturbations around a supersymmetric solution.................. 90 6.2.1 The first–order Borokhov{Gubser formalism................. 90 6.2.2 The perturbation Ansatz........................... 91 6.2.3 The supersymmetric background....................... 93 6.2.4 Explicit equations............................... 93 6.3 The force on a probe M2................................ 94 6.4 The space of solutions................................. 96 6.4.1 Analytic solutions for the ξ~'s......................... 96 6.4.2 Solving the φi equations............................ 99 6.5 Boundary conditions for M2 branes.......................... 104 6.5.1 BPS M2 branes................................. 104 6.6 Constructing the anti{M2 brane solution....................... 106 7 On The Inflaton Potential From Antibranes in Warped Throats 108 7.1 Introduction and motivation.............................. 108 7.2 Computing the force using the action of probe antibranes............. 110 7.2.1 Backreacted D3 branes in the warped deformed conifold.......... 110 7.2.2 M{Theory on a warped Stenzel space.................... 111 7.3 Computing the force on probe branes......................... 112 7.3.1 Warped deformed conifold........................... 112 7.3.2 M{Theory on a warped Stenzel space.................... 113 Bibliography 113 3 Chapter 1 Introduction This introduction aims at presenting the different topics included in the bulk of this thesis in a more qualitative and informal way than found in the later chapters, which mostly consist in a collection of my published work. This is primarily an opportunity to motivate work done over a two years and a half period on loosely{connected subjects, with as few equations and technicalities as possible. Gauge/gravity duality Let us start with introducing heuristically gauge/gravity
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