Table Of Content

Arithmetic and Hyperbolic Structures in String Theory Daniel Persson Institut fu¨r Theoretische Physik ETH Zu¨rich CH-8093 Zu¨rich, Switzerland 0 1 [email protected] 0 2 n a Abstract J 9 This monograph is an updated and extended version of the author’s PhD thesis. It consists 1 of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures ] h arising when studying gravity in the vicinity of a spacelike singularity (the “BKL-limit”). t - In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which p e may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic h billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the [ billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic 1 Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional v symmetries of the theory. We develop in detail the relevant mathematics of Lorentzian 4 5 Kac-Moody algebras and hyperbolic Coxeter groups, and explain with many examples how 1 these structures are intimately connected with gravity. We also construct a geodesic sigma 3 model invariant under the hyperbolic Kac-Moody group E , and analyze to what extent its . 10 1 dynamics reproduces the dynamics of type II and eleven-dimensional supergravity. 0 0 Part II of the thesis is devoted to a study of how (U-)dualities in string theory pro- 1 vide powerful constraints on perturbative and non-perturbative quantum corrections. These : v dualities are described by certain arithmetic groups G(Z) which are conjectured to be pre- i X served in the effective action. The exact couplings are generically given by automorphic r forms on the double quotient G(Z)\G/K, where K is the maximal compact subgroup of the a Lie group G. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on the special class of non-holomorphic Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expan- sions. We also show how these techniques can be applied to hypermultiplet moduli spaces in type II Calabi-Yau compactifications, and we provide a detailed analysis for the universal hypermultiplet. In this case, we propose that the quantum theory is invariant under the Picard modular group SU(2,1;Z[i]), and we construct an SU(2,1;Z[i])-invariant Eisenstein series whose non-abelian Fourier expansion exhibits the expected contributions from D2- and NS5-brane instantons. Thesis for the degree of DOCTOR OF PHILOSOPHY in Physics at CHALMERS UNIVERSITY OF TECHNOLOGY AND DOCTOR OF SCIENCE in Theoretical Physics and Mathematics at UNIVERSITE LIBRE DE BRUXELLES Arithmetic and Hyperbolic Structures in String Theory Daniel Persson This doctoral thesis is the result of joint supervision between Chalmers University of Tech- nology and Université Libre de Bruxelles under the framework of the double doctoral degree agreement signed by both universities on the 8th of September 2008. Fundamental Physics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden and Physique Théorique et Mathémathique UNIVERSITE LIBRE DE BRUXELLES & INTERNATIONAL SOLVAY INSTITUTES Bruxelles, Belgium 2009 Arithmetic and Hyperbolic Structures in String Theory Daniel Persson TheresearchpresentedinthisthesishasbeenfundedbyTheInternationalSolvayInstitutes, by a grant from PAI, Belgium, by IISN-Belgium (conventions 4.4505.86, 4.4514.08, 4.4511.06 and 4.4514.08), by the Belgian National Lottery, by the European Commission FP6 RTN programme MRTN-CT-2004-005104 and by the Belgian Federal Science Policy Office through the Interuniversity Attraction Poles P5/27 and P6/11. I am very grateful for all the generous support. This thesis is based on the following eight papers, henceforth referred to as Paper I–VIII: I M. Henneaux, M. Leston, D. Persson and Ph. Spindel, [1] Geometric Configurations, Regular Subalgebras of E and M-Theory Cosmology, 10 JHEP 0610 (2006) 021 [arXiv:hep-th/0606123]. II M. Henneaux, M. Leston, D. Persson and Ph. Spindel, [2] A Special Class of Rank 10 and 11 Coxeter Groups, J. Math. Phys. 48 (2007) 053512 [arXiv:hep-th/0610278]. III M. Henneaux, D. Persson and Ph. Spindel, [3] Spacelike Singularities and Hidden Symmetries of Gravity, Living Rev. Rel. 1 (2008) 1 [arXiv:0710.1818]. IV L. Bao, J. Bielecki, M. Cederwall, B.E.W. Nilsson and D. Persson, [4] U-Duality and the Compactified Gauss-Bonnet Term, JHEP 0807 (2008) 048 [arXiv:0710.4907]. V M. Henneaux, D. Persson and D.H. Wesley, [5] Coxeter Group Structure of Cosmological Billiards on Compact Spatial Manifolds, JHEP 0809 (2008) 052 [arXiv:0805.3793]. VI M. Henneaux, E. Jamsin, A. Kleinschmidt and D. Persson, [6] On the E /Massive Type IIA Supergravity Correspondence, 10 Phys. Rev. D79 (2009) 045008 [arXiv:0811.4358]. VII B. Pioline and D. Persson, [7] The Automorphic NS5-Brane, Commun. Num. Th. Phys. 3, No. 4 (2009), [arXiv:0902.3274]. VIII L. Bao, A Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, [8] Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1), Submitted to Commun. Num. Th. Phys. [arXiv:0909.4299]. Acknowledgments I have been given the great privilege of being guided through my PhD with the aid of two amazing advisors. IamverygratefultoMarcHenneauxandBengtE.W.Nilssonforyourunwaveringsupport and encouragement these last four years. Thank you so much Marc for the many enjoyable and rewarding collaborations, helpful discussions and for always making sure that I am busy. Thank you so much Bengt for very close collaboration during the course of six years now, and for our numerous discussions on theoretical physics in general and string theory in particular, during which I have learned so many things. In the second half of my PhD, I have also been fortunate enough to work on several projects in close collaboration with Axel Kleinschmidt, who has become much like an extra advisor. I am very grateful for our extensive discussions on everything from E to number theory, and for many helpful 10 remarks which greatly improved the contents of this thesis. Thank you so much Axel! Part II of this thesis had probably not been very coherent if it had not been for the generous help I have received from Boris Pioline during the last two years. I am enormously grateful to Boris for all your explanations and clarifications regarding automorphic forms and their role in string theory, as well as for very enjoyable collaborations, and for numerous useful comments and suggestions on a previous draft of this thesis. I want to extend a very special thanks to my two companions Christoffer Petersson and Jakob Palmkvistforyourclosefriendshipandsupportduringalltheseyears. ToChristoffer,Iamparticularly grateful for our endless discussions on string theory, and in particular on the topics presented in Chapters 8 and 11 of this thesis. I also want to thank Jakob especially for being the only one who has actually read through this entire manuscript and given numerous helpful suggestions. IamgratefultoLingBao,JohanBielecki,MartinCederwall,LaurentHouart,EllaJamsin,Mauri- cio Leston, Josef Lindman Hörnlund, Philippe Spindel, Nassiba Tabti and Daniel H. Wesley for very enjoyable and rewarding collaborations. A big thanks goes out to the entire theoretical physics groups at Chalmers and ULB, for creating extremelyfriendlyandstimulatingresearchatmospheres. Ialsowishtothankeveryoneatthephysics department of Ećole Normale Supérieure in Paris for kind hospitality during the fall of 2008. Finally, I owe thanks to many people with whom I have benefitted from stimulating discussions and correspondence over the years, which has greatly improved my understanding of the contents of this thesis, including Riccardo Argurio, Costas Bachas, Raphael Benichou, Sophie de Buyl, Stephane Detournay,MirandaCheng,JarahEvslin,LucaForte,MatthiasGaberdiel,GastonGiribet,UlfGran, Stefan Hohenegger, Nick Halmagyi, Bernard Julia, Arjan Keurentjes, Chethan Krishnan, Stanislav Kuperstein, Carlo Maccaferri, Yan Soibelman, Dieter Van den Bleeken, Stefan Vandoren, Pierre Van- hove, Alexander Wijns, Niclas Wyllard and Genkai Zhang. “Man cannot survive except by gaining knowledge, and reason is his only means to gain it. Reason is the faculty that perceives, identifies and integrates the material provided by his senses. The task of his senses is to give him the evidence of existence, but the task of identifying it belongs to his reason, his senses tell him only that something is, but what it is must be learned by his mind.” -Ayn Rand Till Johanna Contents 1 Introduction – A Tour Through the Dualities of String Theory 15 1.1 The Principle of Beauty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 The Duality Web of Type II String Theory . . . . . . . . . . . . . . . . . . . 17 1.2.1 Duality and Maximal Supersymmetry . . . . . . . . . . . . . . . . . . 17 1.2.2 Duality in N = 2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Instanton Effects and String Dualities . . . . . . . . . . . . . . . . . . 21 1.3 Hyperbolic Billiards and Infinite-Dimensional Dualities . . . . . . . . . . . . . 22 1.3.1 Infinite-Dimensional U-Duality? . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Hyperbolic Weyl Groups and Cosmological Billiards . . . . . . . . . . 24 1.3.3 Geodesic Motion on Infinite-Dimensional Coset Spaces . . . . . . . . . 26 I Spacelike Singularities and Hyperbolic Structures in (Super-)Gravity 29 2 Kac-Moody Algebras 31 2.1 Preliminary Example: A – The Fundamental Building Block . . . . . . . . . 31 1 2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 The Chevalley-Serre Presentation . . . . . . . . . . . . . . . . . . . . . 34 2.2.2 The Cartan Matrix and Dynkin Diagrams . . . . . . . . . . . . . . . . 36 2.2.3 The Root System and the Root Lattice . . . . . . . . . . . . . . . . . 37 2.2.4 The Invariant Bilinear Form . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.5 Example: A versus A+ . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 1 2.2.6 The Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.7 Regular Subalgebras of Kac-Moody Algebras . . . . . . . . . . . . . . 51 2.3 Affine Kac-Moody Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3.1 The Center of a Kac-Moody Algebra . . . . . . . . . . . . . . . . . . . 52 2.3.2 The Derived Algebra and the Derivation . . . . . . . . . . . . . . . . . 53 2.3.3 The Affine Root System . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3.4 The Affine Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 Lorentzian Kac-Moody Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.1 Extensions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.2 Hyperbolic Kac-Moody Algebras . . . . . . . . . . . . . . . . . . . . . 61 2.4.3 Extended Example: The Weyl Group of A++ . . . . . . . . . . . . . . 62 1 2.5 Level Decomposition in Terms of Finite Regular Subalgebras . . . . . . . . . 68 9 2.5.1 A Finite-Dimensional Example: sl(3,R) . . . . . . . . . . . . . . . . . 69 2.5.2 Some Formal Considerations . . . . . . . . . . . . . . . . . . . . . . . 72 2.5.3 Level Decomposition of A++ . . . . . . . . . . . . . . . . . . . . . . . 75 1 2.5.4 Level Decomposition of E . . . . . . . . . . . . . . . . . . . . . . . . 84 10 3 Chaotic Cosmology and Hyperbolic Coxeter Groups 91 3.1 Cosmological Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.1.2 Hamiltonian Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.1.3 Iwasawa Decomposition of the Metric . . . . . . . . . . . . . . . . . . 94 3.1.4 Decoupling of Spatial Points . . . . . . . . . . . . . . . . . . . . . . . 97 3.1.5 The Sharp Wall Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1.6 Dynamics as a Billiard in Hyperbolic Space . . . . . . . . . . . . . . . 101 3.1.7 Recovering the Kasner Solution . . . . . . . . . . . . . . . . . . . . . . 104 3.2 Kac-Moody Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2.1 The Kac-Moody Billiard of Pure Gravity . . . . . . . . . . . . . . . . 105 3.2.2 The Kac-Moody Billiard for the Coupled Gravity-3-Form System . . . 106 3.2.3 Dynamics in the Cartan Subalgebra . . . . . . . . . . . . . . . . . . . 109 3.2.4 Hyperbolicity Implies Chaos. . . . . . . . . . . . . . . . . . . . . . . . 111 4 Compactification, Cohomology and Coxeter Groups 113 4.1 Intermediate Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 The “Uncompactified Billiard” . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.3 Compactification and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3.1 Selection Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.4 Gram Matrices and Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . 116 4.4.1 Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.4.2 Relative Positions of Walls in Hyperbolic Space . . . . . . . . . . . . . 117 4.4.3 Acute-Angled Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.4.4 Coxeter Polyhedra and the Billiard Group . . . . . . . . . . . . . . . . 118 4.5 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5.1 Rules of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.5.2 Describing the Billiard Group After Compactification . . . . . . . . . 120 4.5.3 Fundamental Domains, Chamber Complexes and Galleries . . . . . . . 123 4.5.4 Determining the Chaotic Properties After Compactification . . . . . . 127 5 Beyond the Weyl Group – Infinite Symmetries Made Manifest 131 5.1 Nonlinear Sigma Models on Finite-Dimensional Coset Spaces . . . . . . . . . 132 5.1.1 The Cartan Involution and Symmetric Spaces . . . . . . . . . . . . . . 132 5.1.2 Nonlinear Realisations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1.3 Three Ways of Writing the Quadratic K ×G-Invariant Action . . . . 134 5.1.4 Equations of Motion and Conserved Currents . . . . . . . . . . . . . . 136 5.1.5 Parametrization of K\G . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2 Geodesic Sigma Models on Infinite-Dimensional Coset Spaces . . . . . . . . . 139 5.2.1 Formal Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140