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Connections Between Kac-Moody Algebras and M-Theory

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Connections Between Kac-Moody Algebras and M-Theory Connections between Kac-Moody algebras and M-theory Paul P. Cook∗ Department of Mathematics, King's College London, Strand, London WC2R 2LS, U.K. arXiv:0711.3498v1 [hep-th] 22 Nov 2007 Ph.D. Thesis Supervisor: Professor Peter West Submitted in September, 2006 ∗email: [email protected] Abstract We investigate the motivations and consequences of the conjecture that the Kac-Moody algebra E11 is the symmetry algebra of M-theory, and we develop methods to aid the further investigation of this idea. The definitions required to work with abstract root systems of Lie algebras are given in review leading up to the definition of a Kac-Moody algebra. The motivations for the E11 conjecture are reviewed and the nonlinear realisation of gravity relevant to the conjecture is explicitly described. The algebras of E11, relevant to M-theory, and K27, relevant to the bosonic string theory, along with their l1 representations are constructed. Tables of low level roots are produced for both the adjoint and l1 representations of these algebras. A solution generating element of the Kac-Moody algebra is given, and it is shown by construc- tion that it encodes all the known half-BPS brane solutions of the maximally oxidised supergravity theories. It is then used to look for higher level branes associated to the roots of the Kac-Moody algebra associated to the oxidised theory. The nature of how spacetime signature is encoded within the E11 formulation of M-theory is analysed. The effect of the multiple signatures that arise from the Weyl reflections of E11 on the solution generating group element is found and precise conditions of when an electric brane solution exists and in which signatures are given. As a corollary to these investigations the spacelike branes of M-theory are found associated to the solution generating element. The U-duality multiplets of exotic brane charges are shown to have a natural E11 origin. General formulae for finding the content of arbitrary brane charge multiplets are given and the exact content of the particle and string multiplets in dimensions 4; 5; 6; 7; 8 is related to the l1 representation of E11. 1 Acknowledgements It is a great pleasure to thank Peter West for his impressive insight, expert advice and humour throughout the course of my studies. His patience with my timekeeping was also laudable. Throughout my studies I have been supported by PPARC grant PPA/S/S/2003/03644, and I want to thank the "powers that be" for continuing to fund such research as mine, which I believe remains an important pursuit in the modern world. Studying an esoteric subject can be a lonely business, the past three years have been consid- erably more enjoyable than one might have imagined due to my incredible friends. In particular the occupants of Room 102 (and formerly 32F): Ant, Manuel, Carmen (who taught me that every- thing I begin in Latex I must also end), Sven, Vid, Alexis, Laura, Dan, Philipp, Peter, Our Mighty Emperor, Nick, Arnold, Jack and the hairy little man. I have also spent time with people from outside of my office, in particular I would like to thank the following for their company, support and good humour: Joe, Nima, Geoff, Sarah, Rosie, Alfie, George, Dennis, Lisa, My Long, Tam, Kevvy, the Andys, Katie, James, Trevor, Gina, Ben, Phil, Nadia, Dave, Nikos and all the hippos of the world. I would like to particularly thank Sam for always being able to cross every line anyone ever put down for him, and for making me laugh. Sometimes so hard I had to clean my computer afterwards. Finally this thesis would just be a twinkling in my eye if it weren't for the help, financial support, occasional laundry, hot meals and love of my parents. I owe them more than a thesis. But for now, with my thanks, that's all I have to offer them. 2 FOR MUM AND DAD 3 CONTENTS Contents 1 Introduction 8 2 Some Group Theory 13 2.1 What's in a group? . 13 2.2 Lie Groups . 14 2.3 Compact, Connected and Simply-Connected Groups . 15 2.4 The Exponential Map and the Lie Algebra . 17 2.5 The Adjoint and the Lie Bracket . 20 2.6 Campbell-Baker-Hausdorff and other identities . 21 2.7 Representations of SU(2) . 23 2.8 Weights and Roots . 26 2.8.1 The roots of so(5) . 28 2.9 Semisimple Lie Algebras . 33 2.10 From Semisimple Lie Groups to Kac-Moody Algebras . 36 3 Symmetries of M-theory 37 3.1 Nonlinear Realisations on a Coset Space . 37 3.2 The Diffeomorphism Group and Bosonic Supergravity . 40 3.3 An Awfully Big Conjecture . 47 3.4 A Return to Three Dimensions . 52 4 Constructing the Algebra 59 4.1 The Algebra of M-theory . 59 4.2 The Bosonic String Theory Algebra . 66 4.3 The Translation Generator . 70 4.4 The Central Charges of M-theory . 73 4.5 The l1 Representation of the Bosonic String . 76 5 A Solution Generating Group Element 79 5.1 M-theory, IIA and IIB brane solutions . 79 5.2 The Commutators . 82 5.3 The Dilaton Generator . 85 5.4 Solutions in Generic Gravity Theories . 86 5.5 Simply Laced Groups . 88 5.5.1 Very Extended Dn−3 ............................... 88 5.6 Non-Simply Laced Groups . 93 5.6.1 Very Extended Bn−3 ............................... 94 4 CONTENTS 5.6.2 Very Extended F4 ................................ 99 5.7 Higher Level Branes . 106 6 Multiple Signatures of M-theory 109 6.1 General Signature Formulation of the Einstein Equations . 109 6.2 New Solutions from Signature Change . 112 6.3 Spacetime Signature and Weyl Reflections . 117 6.4 Invariant Roots and Signature Orbits . 120 6.4.1 Membrane Solution Signatures . 121 6.4.2 Fivebrane Solution Signatures . 124 6.4.3 pp-Wave Solution Signatures . 124 6.5 S-Branes from a Choice of Local Sub-Algebra . 126 6.6 A Naive Interpretation for Signature Change . 133 7 Kac-Moody Algebras and U-duality Charge Multiplets 136 7.1 General Decomposition . 136 7.2 Rank p topological charges . 139 7.3 Representations of the fundamental weights of E11−d . 140 7.4 Representations of Ad−1 ⊗ E11−d with fundamental E11−d weights . 141 7.4.1 The Particle Multiplet . 141 7.4.2 The String Multiplet . 143 7.5 More Exotic Charges from a general weight of E11−d . 144 A Tables of low level roots of E11, K27 and their l1 representations 147 B Solutions of M-theory 182 B.1 Electric Branes . 182 B.2 The M2-Brane . 182 B.3 The M5-Brane . 184 B.4 The pp-Wave ....................................... 185 B.5 Spacelike Brane Solutions . 185 B.6 The S2-Brane . 186 B.7 The S5-Brane . 187 B.8 The Spp-Wave ...................................... 189 C Electric Branes from Kac-Moody Algebras 190 C.1 Very Extended E6 .................................... 190 C.2 Very Extended E7 - The 10-dimensional theory . 194 C.3 Very Extended G2 .................................... 195 5 LIST OF TABLES D Dynkin Diagrams of the Very-Extended Semisimple Lie Groups 199 List of Tables 1 The hidden symmetries emergent upon dimensional reduction of D=11 supergravity 10 2 Examples of Compact Matrix Lie Groups . 16 3 Examples of connected matrix Lie groups . 16 4 Examples of simply-connected matrix Lie groups . 17 5 The roots and root vectors of so(5) . 30 6 The roots of so(5) .................................... 31 7 The low level roots of E11 ................................ 67 8 The first few roots of K27 ................................ 70 9 The signature orbit of the root associated to R91011 . 123 10 The signature orbit of the root associated to R123 . 123 11 The signature orbit of the root associated to R67891011 . 124 12 The signature orbit of the root associated to R123456 . 125 13 The signature orbit of the root associated to highest weight pp-wave . 125 14 The signature orbit of the root associated to lowest weight pp-wave . 126 91011 15 The signature orbit of S2-brane from E11 from R . 128 123 16 The signature orbit of S2-brane from E11 from R . 129 67891011 17 The signature orbit of S5-brane from E11 generator R . 130 123456 18 The signature orbit of S5-brane from E11 generator R . 131 19 The signature orbits of the root associated to highest weight Spp-wave . 132 20 The composition of two particle signatures . 135 21 The various compositions of two M2 branes signatures . 135 22 Representations of Ad−1 ⊗ E11−d with fundamental E11−d weights . 141 23 The particle charge multiplet in d = 4 . 142 24 The particle charge multiplet in d = 5 . 142 25 The particle charge multiplet in d = 6 . 142 26 The particle charge multiplet in d = 7 . 143 27 The string charge multiplet in d =4 .......................... 143 28 The string charge multiplet in d =5 .......................... 143 29 The string charge multiplet in d =6 .......................... 144 30 The string charge multiplet in d =7 .......................... 144 31 Representations of Ad−1 ⊗ E11−d with with λi + λi+1 E11−d weights . 145 32 The roots of E11 to level 12 . 147 33 The roots of K27 to level (3,3) . 157 6 LIST OF FIGURES 34 The roots of the l1 representation of E11 to level 12 . 163 35 The roots of the l1 representation of K27 to level (3,3) . 178 List of Figures 1 An arbitrary path, A(t), through a connected matrix Lie group. 19 2 The root diagram of so(5) with the fundamental weights superposed .
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