String Theory: exact solutions, marginal deformations and ... Jean Iliopoulos, Bernard Julia, Volodia

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    String Theory: exact solutions, marginal deformations and hyperbolic

    spaces

    Domenico Orlando

    http://arxiv.org/abs/hep-th/0610284v1

  • A Lei

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  • Contents

    1 Introduction 1

    2 Wess-Zumino-Witten Models 5

    Wess-Zumino-Witten models constitute a large class of the exact string theory solutions which we will use as starting points for most of the analysis in the following. In this chapter we see how they can be studied from differ- ent perspectives and with different motivations both from a target space and world-sheet point of view.

    2.1 The two-dimensional point of view . . . . . . . . . . . . . . . . . 5 2.2 The target space point of view . . . . . . . . . . . . . . . . . . . . 14

    3 Deformations 17

    In this rather technical chapter we describe marginal deformations of Wess-Zumino-Witten models. The main purpose for these constructions is to reduce the symmetry of the system while keeping the integrability proper- ties intact, trying to preserve as many nice geometric properties as possible.

    3.1 Deformed WZW models: various perspectives . . . . . . . . . . . 18 3.2 Background fields for the asymmetric deformation . . . . . . . . 22 3.3 Geometry of squashed groups . . . . . . . . . . . . . . . . . . . . 26 3.4 A no-renormalization theorem . . . . . . . . . . . . . . . . . . . 29 3.5 Partition functions . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.6 The deformation as a gauging . . . . . . . . . . . . . . . . . . . . 37

    4 Applications 43

    In this chapter we present some of the applications for the construction outlined above. After an analysis of the most simple (compact and non- compact) examples, we describe the near-horizon geometry for the Bertotti- Robinson black hole, show some new compactifications and see how Horne and Horowitz’s black string can be described in this framework and gener- alized via the introduction of an electric field.

    4.1 The two-sphere CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 SL(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Near horizon geometry for the Bertotti-Robinson black hole . . 59 4.4 The three-dimensional black string revisited . . . . . . . . . . . 62 4.5 New compactifications . . . . . . . . . . . . . . . . . . . . . . . . 76

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  • 5 Squashed groups in type II 85

    In this chapter we start deviating from the preceding ones because we will no longer deal with WZW models but with configurations in which the group manifold geometry is sustained by RR fields. In particular, then, we see how the squashed geometries can be obtained in type II theories by T- dualizing black brane configurations.

    5.1 SL(2, R)× SU(2) as a D-brane solution . . . . . . . . . . . . . . 85 5.2 T duality with RR fields . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 The squashed sphere . . . . . . . . . . . . . . . . . . . . . . . . . 88

    6 Out of the conformal point: Renormalization Group Flows 91

    This chapter is devoted to the study of the relaxation of squashed WZW models further deformed by the insertion of non-marginal operators. The calculation is carried from both the target space and world-sheet points of view, once more highlighting the interplay between the two complementary descriptions. In the last part such techniques are used to outline the con- nection between the time evolution and the RG-flow which is seen as a large friction limit description; we are hence naturally led to a FRW-type cosmo- logical model.

    6.1 The target space point of view . . . . . . . . . . . . . . . . . . . . 92 6.2 The CFT approach . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 RG flow and friction . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.4 Cosmological interpretation . . . . . . . . . . . . . . . . . . . . . 110

    7 Hyperbolic Spaces 115

    In this chapter we investigate type II and M-theory geometries written as direct products of constant-curvature spaces. We find in particular a class of backgrounds with hyperbolic components and we study their stabil- ity with respect to small fluctuations.

    7.1 M-theory solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.3 Type IIB backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 128

    8 Conclusions and further perspectives 133

    A Table of conventions 135

    B Explict parametrizations for some Lie groups 137

    B.1 The three-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 B.2 AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 B.3 SU (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.4 USp (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

    C Symmetric deformations of SL(2, R) 147

    D Spectrum of the SL(2, R) super-WZW model 149

    Bibliography 153

  • Acknowledgements

    Never had nay man a wit so presently excellent, as that it could raise itselfe; but there must come both matter, occasion, commenders, and favourers to it.

    Il Volpone BEN JOHNSON

    While writing these acknowledgements I realized how much I’m indebted to so many people. I learned a lot from them all – not just physics – and with- out them this work simply wouldn’t have been possible. I’m pretty sure I’m forgetting someone. I plead guilty. But even if you don’t find your name rest assured that you’ll always have my personal gratitude.

    In any case, I simply can’t start without mentioning my parents for – al- though far – they’ve always been present in their peculiar way.

    I must thank the members of my thesis committee: Luis Alvarez-Gaumé, Edouard Brezin, Costas Kounnas, Marios Petropoulos, Alex Sevrin, Philippe Spindel, Arkady Tseytlin for they extremely precious feedback. They turned my defence into a valuable learning experience.

    I wish to thank Patrick Mora for having welcomed me as a PhD student at the Centre de physique théorique of the École polytechnique. Also I thank Edouard Brezin at École normale supèrieure, Dieter Lüst at Max Planck Institut (Munich) and Alex Sevrin at Vrije Universiteit Brussel. During the last year they hosted me for several months at their respective institutions. Each and all of them have made of my stays invaluable experiences from both the professional and human points of view.

    Then I thank my advisor Marios Petropoulos for all what he taught me and for his continuous support. In these years I learned to think of him as a friend. Costas Kounnas for letting me camp in his office at ENS where he would devote me hours and hours of his time, sharing his experience and his amazing phys- ical intuition. Costas Bachas, who was among the first professors I had when I arrived in Paris and (unknowingly) pushed me towards string theory. Since then he helped me in many an occasion and I’m almost embarrassed if I think how much I learned from him and in general I owe him.

    Doing one’s PhD in Paris is a great opportunity for there are not so many places in the world with such a physics community. And it’s definitely surpris-

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  • ing to a PhD student, like I was, how everybody is always willing to discuss and makes you feel one of them. I learned a lot from many people and I hope that those I don’t remember explicitly won’t take it too bad. On the other hand I couldn’t not mention Adel Bilal, Eugène Cremmer, Pierre Fayet, Mariana Graña, Jean Iliopoulos, Bernard Julia, Volodia Kazakov, Ruben Minassian, Hervé Partouche, Michela Petrini, Boris Pioline, Ian Troost, Pierre Vanhove.

    Sometimes discussions would lead to a publication and I’m honored to count among my collaborators Costas Bachas, Ioannis Bakas, Davide Cassani, Stéphane Detournay, Dan Israël, Marios Petropoulos, Susanne Reffert, Kostas Sfet- sos, Philippe Spindel. I really wish to thank you for what I learned and keep learning from you all.

    I must remember my office mates Luciano Abreu, Stéphane Afchain, Yacine Dolivet, Stéphane Fidanza, Pascal Grange, Claudia Isola, Liguori Jego, Jerome Levie, Xavier Lacroze, Alexey Lokhov, Liuba Mazzanti, Hervé Moutarde, Chloé Papineau, Susanne Reffert, Sylvain Ribault, Barbara Tumpach and especially Claude de Calan. With all of you I shared more than an office and our never-ending discussions have really been among the most precious moments of my PhD. I hope none of you will get mad at me if in particular I thank Sylvain for all the advice he gave me, even before starting my PhD. Not to speak of my brother Pascal.

    A final thanks (last but not least) goes to the computer people and secre- taries at CPHT and ENS, without whom life couldn’t be that easy.

    Thanks, all of you.

  • CHAPTER 1

    Introduction

    There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable

    The Restaurant at the End of the Universe DOUGLAS ADAMS

    IN THE LAST FIFTY YEARS theoretical physics has been dominated by two ap-parently incompatible models: the microscopic world being described by quantum field theory and the macroscopic word by general relativity. QFT is by far the most successful theory ever made, allowing to reach an almost incredible level of accuracy in its measurable predictions. But gravity is d