Spacelike Singularities and Hidden Symmetries of Gravity Marc Henneaux Physique Th´eoriqueet Math´ematique Universit´eLibre de Bruxelles & International Solvay Institutes Boulevard du Triomphe, ULB { C.P. 231 B-1050 Bruxelles, Belgium email: [email protected] Daniel Persson Physique Th´eoriqueet Math´ematique Universit´eLibre de Bruxelles & International Solvay Institutes Boulevard du Triomphe, ULB { C.P. 231 B-1050 Bruxelles, Belgium email: [email protected] Philippe Spindel Service de M´ecaniqueet Gravitation Universit´ede Mons-Hainaut, Acad´emieWallonie-Bruxelles Avenue du Champ de Mars 6, B-7000 Mons, Belgium email: [email protected] Abstract We review the intimate connection between (super-)gravity close to a spacelike singularity (the \BKL-limit") and the theory of Lorentzian Kac{Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter arXiv:0710.1818v2 [hep-th] 28 Apr 2008 groups are the Weyl groups of infinite-dimensional Kac{Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self- contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac{Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity. 1 Contents 1 Introduction 7 1.1 Cosmological billiards and hidden symmetries of gravity . 7 1.2 Outline of the paper . 8 2 The BKL Phenomenon 10 2.1 The general action . 10 2.2 Hamiltonian description . 10 2.2.1 Action in canonical form . 11 2.2.2 Iwasawa change of variables . 12 2.3 Decoupling of spatial points close to a spacelike singularity . 14 2.3.1 Spatially homogeneous models . 15 2.3.2 The ultralocal Hamiltonian . 16 2.4 Dynamics as a billiard in hyperbolic space . 16 2.5 Rules for deriving the wall forms from the Lagrangian { Summary . 19 2.6 More on the free motion: The Kasner solution . 20 2.7 Chaos and billiard volume . 20 2.8 A note on the constraints . 21 2.9 On the validity of the BKL conjecture { A status report . 21 3 Hyperbolic Coxeter Groups 23 3.1 Preliminary example: The BKL billiard (vacuum D = 4 gravity) . 23 3.1.1 Billiard reflections . 23 3.1.2 On the group P GL(2; Z)............................. 25 3.2 Coxeter groups { The general theory . 26 3.2.1 Examples . 26 3.2.2 Definition . 29 3.2.3 The length function . 30 3.2.4 Geometric realization . 31 3.2.5 Positive and negative roots . 32 3.2.6 Fundamental domain . 33 3.3 Finite Coxeter groups . 34 3.4 Affine Coxeter groups . 34 3.5 Lorentzian and hyperbolic Coxeter groups . 37 3.6 Crystallographic Coxeter groups . 43 4 Lorentzian Kac{Moody Algebras 47 4.1 Definitions . 47 4.2 Roots . 49 4.3 The Chevalley involution . 51 4.4 Three examples . 51 4.5 The affine case . 53 4.6 The invariant bilinear form . 53 4.6.1 Definition . 53 4.6.2 Real and imaginary roots . 56 4.6.3 Fundamental weights and the Weyl vector . 57 4.6.4 The generalized Casimir operator . 57 4.7 The Weyl group . 58 4.8 Hyperbolic Kac{Moody algebras . 61 4.8.1 The fundamental domain F ........................... 62 2 4.8.2 Roots and the root lattice . 62 4.8.3 Examples . 62 4.9 Overextensions of finite-dimensional Lie algebras . 64 4.9.1 Untwisted overextensions . 65 4.9.2 Root systems in Euclidean space . 66 4.9.3 Twisted overextensions . 67 4.9.4 Algebras of Gaberdiel{Olive{West type . 68 4.10 Regular subalgebras of Kac{Moody algebras . 69 4.10.1 Definitions . 69 4.10.2 Examples { Regular subalgebras of E10 .................... 70 4.10.3 Further properties . 72 5 Kac{Moody Billiards I { The Case of Split Real Forms 74 5.1 More on Coxeter billiards . 74 5.1.1 The Coxeter billiard of pure gravity in D spacetime dimensions . 74 5.1.2 The Coxeter billiard for the coupled gravity-3-Form system . 75 5.2 Dynamics in the Cartan subalgebra . 77 5.2.1 Billiard dynamics in the Cartan subalgebra . 78 5.2.2 The fundamental Weyl chamber and the billiard table . 80 5.2.3 Hyperbolicity implies chaos . 80 5.3 Understanding the emerging Kac{Moody algebra . 81 5.3.1 Invariance under toroidal dimensional reduction . 82 5.3.2 Iwasawa decomposition for split real forms . 83 5.3.3 Starting at the bottom { Overextensions of finite-dimensional Lie algebras . 84 5.4 Models associated with split real forms . 86 6 Finite-Dimensional Real Lie Algebras 88 6.1 Definitions . 88 6.2 A preliminary example: sl(2; C)............................. 89 6.2.1 Real forms of sl(2; C) .............................. 89 6.2.2 Cartan subalgebras . 90 6.2.3 The Killing form . 90 6.2.4 The compact real form su(2) . 91 6.2.5 su(2) and sl(2; R) compared and contrasted { The Cartan involution . 91 6.2.6 Concluding remarks . 92 6.3 The compact and split real forms of a semi-simple Lie algebra . 93 6.4 Classical decompositions . 95 6.4.1 Real forms and conjugations . 95 6.4.2 The compact real form aligned with a given real form . 95 6.4.3 Cartan involution and Cartan decomposition . 95 6.4.4 Restricted roots . 97 6.4.5 Iwasawa and KAK decompositions . 98 6.4.6 θ-stable Cartan subalgebras . 98 6.4.7 Real roots { Compact and non-compact imaginary roots . 99 6.4.8 Jumps between Cartan subalgebras { Cayley transformations . 100 6.5 Vogan diagrams . 101 6.5.1 Illustration { The sl(5; C) case . 101 6.5.2 The Borel and de Siebenthal theorem . 105 6.5.3 Cayley transformations in su(3; 2) . 105 6.5.4 Reconstruction . 107 3 6.5.5 Illustrations: sl(4; R) versus sl(2; H) . 107 6.5.6 A pictorial summary { All real simple Lie algebras (Vogan diagrams) . 108 6.6 Tits{Satake diagrams . 116 6.6.1 Example 1: su(3; 2) . 116 6.6.2 Example 2: su(4; 1) . 118 6.6.3 Tits{Satake diagrams: Definition . 120 6.6.4 Formal considerations . 120 6.6.5 Illustration: F4 .................................. 121 6.6.6 Some more formal considerations . 122 6.7 The real semi-simple algebras so(k; l) . 124 6.7.1 Dimensions l = 2 q + 1 < k = 2 p . 125 6.7.2 Dimensions l = 2 q + 1 > k = 2 p . 126 6.7.3 Dimensions l = 2 q; k = 2 p . 126 6.8 Summary { Tits{Satake diagrams for non-compact real forms . 127 7 Kac{Moody Billiards II { The Case of Non-Split Real Forms 134 7.1 The restricted Weyl group and the maximal split \subalgebra" . 134 7.2 \Split symmetry controls chaos" . 136 7.2.1 (BC)1 and N = 2, D = 4 pure supergravity . 136 7.2.2 Heterotic supergravity and so(8; 24) . 138 7.3 Models associated with non-split real forms . ..