Spacelike Singularities and Hidden Symmetries of Gravity

Henneaux, M.; Persson, D.; Spindel, Philippe

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Spacelike Singularities and Hidden Symmetries of Gravity

Henneaux, M.; Persson, D.; Spindel, Philippe

2008 • In Living Rev. Relativity, 11

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Abstract :

[en] 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 in nite) sequence of re ections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of in nite-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 in nite-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.

Disciplines :

Physics

Author, co-author :

Henneaux, M.

Persson, D.

Spindel, Philippe ; Université de Mons > Faculté des Sciences > Mécanique et gravitation

Language :

English

Title :

Spacelike Singularities and Hidden Symmetries of Gravity

Publication date :

01 January 2008

Journal title :

Living Rev. Relativity

ISSN :

1433-8351

Volume :

Peer reviewed :

Peer reviewed

Commentary :

http://www.livingreviews.2008-1. e-Print Archive: hep-th/0710.1818 Bibliographic Code:2008LRR....11....1H

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Bibliography

Alekseev, A., and Monnier, S., "Quantization of Wilson loops in Wess-Zumino-Witten models", J. High Energy Phys., 2007(08), 039, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0702174. 12
Andersson, L., "On the relation between mathematical and numerical relativity", Class. Quantum Grav., 23, S307-S318, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/ 0607065. 2.9
Andersson, L., and Rendall, A.D., "Quiescent cosmological singularities", Commun. Math. Phys., 218, 479-511, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/ 0001047. 2.7, 2.9
Apostol, T.M., Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, vol. 41, (Springer, New York, U.S.A., 1997), 2nd edition. 3.1.2
Araki, S., "On root systems and an infinitesimal classification of irreducible symmetric spaces", J. Math. Osaka City Univ., 13(1), 1-34, (1962). 2, 6.6, 6.6.4, 6.6.4, 6.6.4, 6.8
Argurio, R., Englert, F., and Houart, L., "Intersection rules for p-branes", Phys. Lett. B, 398, 61-68, (1997). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9701042. 10.3.4
Aurilia, A., Nicolai, H., and Townsend, P.K., "Hidden Constants: The Theta Parameter of QCD and the Cosmological Constant of N = 8 Supergravity", Nucl. Phys. B, 176, 509, (1980). 9.4.1
Baez, J.C., "The Octonions", (2001). URL (cited on 19 October 2007): http://arXiv.org/abs/math.ra/0105155. 10.3.3
Bagnoud, M., and Carlevaro, L., "Hidden Borcherds symmetries in Zn orbifolds of M-theory and magnetized D-branes in type 0′ orientifolds", J. High Energy Phys., 2006(11), 003, (2006). Related online version (cited on 01 November 2007): http://arXiv.org/abs/hep-th/0607136. 9.3.6
Bahls, P., The Isomorphism Problem in Coxeter Groups, (Imperial College Press, London, U.K., 2005). 3.2.2
Bao, L., Bielecki, J., Cederwall, M., Nilsson, B.E.W., and Persson, D., "U-Duality and the Compactified Gauss-Bonnet Term", (2007). URL (cited on 01 November 2007): http://arXiv.org/abs/0710.4907. 9.4.3
Bao, L., Cederwall, M., and Nilsson, B.E.W., "Aspects of higher curvature terms and Uduality", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0706.1183. 9.4.3
Bautier, K., Deser, S., Henneaux, M., and Seminara, D., "No cosmological D = 11 supergravity", Phys. Lett. B, 406, 49-53, (1997). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9704131. 9.4.1
Bekaert, X., Boulanger, N., and Henneaux, M., "Consistent deformations of dual formulations of linearized gravity: A no-go result", Phys. Rev. D, 67, 044010, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0210278. 9.3.6
Belinskii, V.A., and Khalatnikov, I.M., "Effect of scalar and vector fields on the nature of the cosmological singularity", Sov. Phys. JETP, 36, 591-597, (1973). 2.7, 9.4.3
Belinskii, V.A., Khalatnikov, I.M., and Lifshitz, E.M., "Oscillatory approach to a singular point in the relativistic cosmology", Adv. Phys., 19, 525-573, (1970). 1, 2.9, 9.4.3
Ben Messaoud, H., "Almost split real forms for hyperbolic Kac-Moody Lie algebras", J. Phys. A, 39, 13659-13690, (2006). 7.3
Berger, B.K., Garfinkle, D., Isenberg, J.A., Moncrief, V., and Weaver, M., "The singularity in generic gravitational collapse is spacelike, local, and oscillatory", Mod. Phys. Lett. A, 13, 1565-1574, (1998). Related online version (cited on 7 December 2007): http://arXiv.org/abs/gr-qc/ 9805063. 2.9
Bergshoeff, E.A., De Baetselier, I., and Nutma, T.A., "E 11 and the embedding tensor", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0705.1304. 8.2.3, 9.4.1
Bergshoeff, E.A., de Roo, M., de Wit, B., and van Nieuwenhuizen, P., "Ten-dimensional Maxwell-Einstein supergravity, its currents, and the issue of its auxiliary fields", Nucl. Phys. B, 195, 97-136, (1982). 2.1
Boulanger, N., Cnockaert, S., and Henneaux, M., "A note on spin-s duality", J. High Energy Phys., 2003(06), 060, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0306023. 9.3.6, 9.4.1, 9.4.4
Breitenlohner, P., Maison, D., and Gibbons, G.W., "4-Dimensional Black Holes from Kaluza-Klein Theories", Commun. Math. Phys., 120, 295-333, (1988). URL (cited on 24 April 2008): http://projecteuclid. org/euclid.cmp/1104177752. 5.4
Brown, J., Ganguli, S., Ganor, O.J., and Helfgott, C., "E10 orbifolds", J. High Energy Phys., 2005(06), 057, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0409037. 4.10.2, 9.3.6
Brown, J., Ganor, O.J., and Helfgott, C., "M-theory and E10: Billiards, branes, and imaginary roots", J. High Energy Phys., 2004(08), 063, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0401053. 9.3.6
Bunster, C., Cnockaert, S., Henneaux, M., and Portugues, R., "Monopoles for gravitation and for higher spin fields", Phys. Rev. D, 73, 105014, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0601222. 9.4.1, 9.4.4
Bunster, C., and Henneaux, M., "A monopole near a black hole", Proc. Natl. Acad. Sci. USA, 104, 12243-12249, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0703155. 9.4.4
Caprace, P.E., "Conjugacy of one-ended subgroups of Coxeter groups and parallel walls", (2005). URL (cited on 19 October 2007): http://arXiv.org/abs/math.GR/0508057. 42
Cartan, E., "Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple", Ann. Sci. Ecole Norm. Sup., 44, 345-467, (1927). 6.6.5
Chapline, G.F., and Manton, N.S., "Unification of Yang-Mills Theory and Supergravity in Ten Dimensions", Phys. Lett. B, 120, 105-109, (1983). 2.1
Chernoff, D.F., and Barrow, J.D., "Chaos in the Mixmaster Universe", Phys. Rev. Lett., 50, 134-137, (1983). 2.9
Chitre, D.M., Investigations of Vanishing of a Horizon for Bianchy Type X (the Mixmaster), Ph.D. Thesis, (University of Maryland, College Park, U.S.A., 1972). 1, 2.4
Cornish, N.J., and Levin, J.J., "Mixmaster universe: A chaotic Farey tale", Phys. Rev. D, 55, 7489-7510, (1997). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/9612066. 2.9
Cremmer, E., and Julia, B., "The N = 8 Supergravity Theory. 1. The Lagrangian", Phys. Lett. B, 80, 48, (1978). 5.3.3, 9
Cremmer, E., and Julia, B., "The SO(8) Supergravity", Nucl. Phys. B, 159, 141, (1979). 9
Cremmer, E., Julia, B., Lu, H., and Pope, C.N., "Dualisation of dualities. I", Nucl. Phys. B, 523, 73-144, (1998). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9710119. 5.3.1, 5.3.3, 5.3.3, 7.2.1, 9.3.6, 9.4.4
Cremmer, E., Julia, B., Lu, H., and Pope, C.N., "Dualisation of dualities. II: Twisted self-duality of doubled fields and superdualities", Nucl. Phys. B, 535, 242-292, (1998). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9806106. 9.3.6, 9.4.4
Cremmer, E., Julia, B., Lü, H., and Pope, C.N., "Higher- dimensional Origin of D = 3 Coset Symmetries", (1999). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9909099. 5.4
Cremmer, E., Julia, B., and Scherk, J., "Supergravity theory in 11 dimensions", Phys. Lett. B, 76, 409-412, (1978). 2.1
Curtright, T., "Generalized gauge fields", Phys. Lett. B, 165, 304-208, (1985). ADS: http://adsabs.harvard.edu/abs/1985PhLB. .165..304C. 9.3.6
Damour, T., and de Buyl, S., "Describing general cosmological singularities in Iwasawa variables", (2007). URL (cited on 01 November 2007): http://arXiv.org/abs/0710.5692. 2, 5, 2.9
Damour, T., de Buyl, S., Henneaux, M., and Schomblond, C., "Einstein billiards and overextensions of finite-dimensional simple Lie algebras", J. High Energy Phys., 2002(08), 030, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0206125. 1.1, 5.3.1, 5.3.3, 9.4.1
Damour, T., Hanany, A., Henneaux, M., Kleinschmidt, A., and Nicolai, H., "Curvature corrections and Kac-Moody compatibility conditions", Gen. Relativ. Gravit., 38, 1507-1528, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0604143. 7.2.2, 9.4.3
Damour, T., and Henneaux, M., "Chaos in superstring cosmology", Phys. Rev. Lett., 85, 920-923, (2000). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0003139. 2.3, 2.4
Damour, T., and Henneaux, M., "Oscillatory behaviour in homogeneous string cosmology models", Phys. Lett. B, 488, 108-116, (2000). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0006171. 2.3, 2, 2.7
Damour, T., and Henneaux, M., "E10,BE 10 and arithmetical chaos in superstring cosmology", Phys. Rev. Lett., 86, 4749-4752, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0012172. 1.1, 2.3, 4.10.2, 5, 5.2.1, 5.2.3, 7.2.2, 8.4, 10.2.1, 10.2.2
Damour, T., Henneaux, M., Julia, B., and Nicolai, H., "Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models", Phys. Lett. B, 509, 323-330, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0103094. 5, 5.2.1, 5.2.3, 5.2.3, 5.2.3, 8, 8.3, 8.3
Damour, T., Henneaux, M., and Nicolai, H., "E10 and a 'small tension expansion' of M theory", Phys. Rev. Lett., 89, 221601, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0207267. 1.1, 2.5, 7.2.2, 8.3.4, 8.4, 8.4.1, 8.4.2, 9, 9.2.2, 9.2.2, 9.3, 9.3.4, 9.3.5, 9.3.6, 9.3.6, 9.3.7, 9.4.1, 9.4.3, 11
Damour, T., Henneaux, M., and Nicolai, H., "Cosmological Billiards", Class. Quantum Grav., 20, R145-R200, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0212256. 1.2, 2, 2.2, 2.3, 3, 2.4, 5.2.1, 5.2.3, 5.2.3, 8.3, 9.2.2, 10.2.1, 10.2.2
Damour, T., Henneaux, M., Rendall, A.D., and Weaver, M., "Kasner-Like Behaviour for Subcritical Einstein-Matter Systems", Ann. Henri Poincare, 3, 1049-1111, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0202069. 1, 2.7, 2.9
Damour, T., Kleinschmidt, A., and Nicolai, H., "Hidden symmetries and the fermionic sector of eleven-dimensional supergravity", Phys. Lett. B, 634, 319-324, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0512163. 9.4.2
Damour, T., Kleinschmidt, A., and Nicolai, H., "K(E 10), supergravity and fermions", J. High Energy Phys., 2006(08), 046, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0606105. 9.4.2
Damour, T., Kleinschmidt, A., and Nicolai, H., "Constraints and the E10 Coset Model", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0709.2691. 2.8, 9.4.4
Damour, T., and Nicolai, H., "Eleven dimensional supergravity and the E10/K(E10) sigma-model at low A9 levels", (2004). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0410245. 8.4.1, 9.3.6
Damour, T., and Nicolai, H., "Higher order M theory corrections and the Kac-Moody algebra E10", Class. Quantum Grav., 22, 2849-2880, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0504153. 9.4.3, 9.4.3
de Buyl, S., Henneaux, M., Julia, B., and Paulot, L., "Cosmological billiards and oxidation", Fortschr. Phys., 52, 548-554, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0312251. 5.4
de Buyl, S., Henneaux, M., and Paulot, L., "Hidden symmetries and Dirac fermions", Class. Quantum Grav., 22, 3595-3622, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0506009. 9.4.2
de Buyl, S., Henneaux, M., and Paulot, L., "Extended E 8 invariance of 11-dimensional supergravity", J. High Energy Phys., 2006(02), 056, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0512292. 9.4.2
de Buyl, S., Pinardi, G., and Schomblond, C., "Einstein billiards and spatially homogeneous cosmological models", Class. Quantum Grav., 20, 5141-5160, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0306280. 2
de Buyl, S., and Schomblond, C., "Hyperbolic Kac Moody algebras and Einstein billiards", J. Math. Phys., 45, 4464-4492, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0403285. 4.8
Demaret, J., De Rop, Y., and Henneaux, M., "Chaos in Nondiagonal Spatially Homogeneous Cosmological Models in Space-time Dimensions ≥ 10", Phys. Lett. B, 211, 37-41, (1988). 2
Demaret, J., Hanquin, J.L., Henneaux, M., and Spindel, P., "Cosmological models in elevendimensional supergravity", Nucl. Phys. B, 252, 538-560, (1985). 2.3.1, 9.3.6, 9.3.6, 9.3.6, 10, 10.1, 10.1.1, 10.2.1, 10.3.2, 10.3.3, 10.3.3, 10.5
Demaret, J., Hanquin, J.L., Henneaux, M., Spindel, P., and Taormina, A., "The fate of the mixmaster behavior in vacuum inhomogeneous Kaluza-Klein cosmological models", Phys. Lett. B, 175, 129-132, (1986). 2.7
Demaret, J., Henneaux, M., and Spindel, P., "No Oscillatory Behavior in vacuum Kaluza-Klein cosmologies", Phys. Lett. B, 164, 27-30, (1985). 2.7, 9.4.3
Deser, S., Gomberoff, A., Henneaux, M., and Teitelboim, C., "Duality, self-duality, sources and charge quantization in abelian N-form theories", Phys. Lett. B, 400, 80-86, (1997). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 9702184. 9.4.4
Deser, S., and Teitelboim, C., "Duality Transformations of Abelian and Nonabelian Gauge Fields", Phys. Rev. D, 13, 1592-1597, (1976). 9.4.4
DeWitt, B.S., "Quantum Theory of Gravity. I. The Canonical Theory", Phys. Rev., 160, 1113-1148, (1967). 9.3.2
Dynkin, E.B., "Semisimple subalgebras of semisimple Lie algebras", Trans. Amer. Math. Soc., 6, 111, (1957). 4.10.2
Elskens, Y., and Henneaux, M., "Ergodic theory of the mixmaster model in higher space-time dimensions", Nucl. Phys. B, 290, 111-136, (1987). 2.9
Englert, F., Henneaux, M., and Houart, L., "From very-extended to overextended gravity and M-theories", J. High Energy Phys., 2005(02), 070, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0412184. 9.3.7
Englert, F., and Houart, L., "From brane dynamics to a Kac-Moody invariant formulation of M-theories", (2004). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0402076. 9.3.7
Englert, F., and Houart, L., "G+++ invariant formulation of gravity and M-theories: Exact BPS solutions", J. High Energy Phys., 2004(01), 002, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0311255. 9.3.7
Englert, F., and Houart, L., "G+++ invariant formulation of gravity and M-theories: Exact intersecting brane solutions", J. High Energy Phys., 2004(05), 059, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0405082. 9.3.7, 10.3.4
Englert, F., Houart, L., Kleinschmidt, A., Nicolai, H., and Tabti, N., "An E9 multiplet of BPS states", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0703285. 9.3.7, 9.3.7, 9.4.1
Englert, F., Houart, L., Taormina, A., and West, P.C., "The symmetry of M-theories", J. High Energy Phys., 2003(09), 020, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0304206. 9.3.7, 9.3.7, 11
Feingold, A.J., and Frenkel, I.B., "A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2", Math. Ann., 263, 87, (1983). 3.1.1, 3.1.2, 8.3, 31
Feingold, A.J., and Nicolai, H., "Subalgebras of Hyperbolic Kac-Moody Algebras", (2003). URL (cited on 19 October 2007): http://arXiv.org/abs/math.qa/0303179. 4.10.3
Fischbacher, T., "The structure of E10 at higher A9 levels: A first algorithmic approach", J. High Energy Phys., 2005(08), 012, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0504230. 8.4.1
Forte, L.A., and Sciarrino, A., "Standard and non-standard extensions of Lie algebras", J. Math. Phys., 47, 013513, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0506048. 4.9.4
Fré, P., Gargiulo, F., and Rulik, K., "Cosmic billiards with painted walls in non-maximal supergravities: A worked out example", Nucl. Phys. B, 737, 1-48, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0507256. 10.6
Fré, P., Gargiulo, F., Rulik, K., and Trigiante, M., "The general pattern of Kac-Moody extensions in supergravity and the issue of cosmic billiards", Nucl. Phys. B, 741, 42-82, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0507249. 10.6
Fré, P., Gargiulo, F., Sorin, A., Rulik, K., and Trigiante, M., "Cosmological backgrounds of superstring theory and solvable algebras: oxidation and branes", Nucl. Phys. B, 685, 3-64, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0309237. 10.6
Fré, P., Rulik, K., and Trigiante, M., "Exact solutions for Bianchi type cosmological metrics, Weyl orbits of E8(8) subalgebras and p-branes", Nucl. Phys. B, 694, 239-274, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0312189. 10.6
Fré, P., and Sorin, A.S., "The arrow of time and the Weyl group: all supergravity billiards are integrable", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0710.1059. 10.6
Fuchs, J., Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1992). 8.3.3
Fuchs, J., and Schweigert, C., Symmetries, Lie Algebras and Representations: A graduate course for physicists, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1997). 4.6.4, 4.9
Gaberdiel, M.R.and Olive, D.I., and West, P.C., "A class of Lorentzian Kac-Moody algebras", Nucl. Phys. B, 645, 403-437, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0205068. 4.9.4, 4.10.2
Garfinkle, D., "Numerical simulations of generic singuarities", Phys. Rev. Lett., 93, 161101, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0312117. 2.9
Green, M.B., and Vanhove, P., "Duality and higher derivative terms in M theory", J. High Energy Phys., 2006(01), 093, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0510027. 9.4.3
Gross, D.J., Harvey, J.A., Martinec, E.J., and Rohm, R., "The Heterotic String", Phys. Rev. Lett., 54, 502-505, (1985). 7.2.2
Gutperle, M., and Strominger, A., "Spacelike branes", J. High Energy Phys., 2002(04), 018, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0202210. 10.4, 40
Hawking, S.W., and Ellis, G.F.R., The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge, U.K., 1973). 1
Heinzle, J.M., Uggla, C., and Rohr, N., "The cosmological billiard attractor", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/ gr-qc/0702141. 2.9
Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, vol. 34, (American Mathematical Society, Providence, U.S.A., 2001). 4.9.2, 4.9.2, 4.9.2, 6, 6.4.2, 6.4.3, 6.4.3, 6.4.4, 6.4.5, 25, 6.6.4, 6.6.4, 6.6.4, 6.8, 7.1, 7.1, 9.1.3, B
Helminck, A.G., "Classification of real semisimple Lie algebras", in Koornwinder, T.H., ed., The structure of real semisimple Lie groups, MC Syllabus, vol. 49, pp. 113-136, (Math. Centrum, Amsterdam, Netherlands, 1982). 6
Henneaux, M., and Julia, B., "Hyperbolic billiards of pure D = 4 supergravities", J. High Energy Phys., 2003(05), 047, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0304233. 1.1, 4.9.3, 7, 7.1, 7.2, 7.3
Henneaux, M., Leston, M., Persson, D., and Spindel, P., "Geometric configurations, regular subalgebras of E10 and M-theory cosmology", J. High Energy Phys., 2006(10), 021, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0606123. 4.10, 1, 10, 10.3.3, 10.3.4, 10.5
Henneaux, M., Leston, M., Persson, D., and Spindel, P., "A Special Class of Rank 10 and 11 Coxeter Groups", J. Math. Phys., 48, 053512, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0610278. 10.6
Henneaux, M., Persson, D., and Wesley, D.H., "Chaos, Cohomology and Coxeter Groups", unknown status. Work in progress. 5.3.1
Henneaux, M., and Teitelboim, C., "The cosmological constant as a canonical variable", Phys. Lett. B, 143, 415-420, (1984). 9.4.1
Henneaux, M., and Teitelboim, C., "Dynamics of chiral (selfdual) p-forms", Phys. Lett. B, 206, 650, (1988). 9.4.4
Henneaux, M., and Teitelboim, C., "Duality in linearized gravity", Phys. Rev. D, 71, 024018, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0408101. 9.4.4
Henry-Labordere, P., Julia, B., and Paulot, L., "Borcherds symmetries in M-theory", J. High Energy Phys., 2002(04), 049, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/ abs/hep-th/0203070. 11
Henry-Labordere, P., Julia, B., and Paulot, L., "Real Borcherds superalgebras and M-theory", J. High Energy Phys., 2003(04), 060, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0212346. 11
Henry-Labordere, P., Julia, B., and Paulot, L., "Symmetries in M-theory: Monsters, Inc", Cargese 2002, conference paper, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0303178. 11
Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, (Chelsea Pub. Co., New York, U.S.A., 1952). 10.1.1, 37
Hillmann, C., and Kleinschmidt, A., "Pure type I supergravity and DE10", Gen. Relativ. Gravit., 38, 1861-1885, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0608092. 4.10.2
Humphreys, J.E., Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1990). 3.1.1, 3.1.2, 3.2, 3.2.5, 3.2.6, 3.3, 3.4, 8, 3.5, 3.6
Isenberg, J.A., and Moncrief, V., "Asymptotic behavior of polarized and half-polarized U(1) symmetric vacuum spacetimes", Class. Quantum Grav., 19, 5361-5386, (2002). Related online version (cited on 7 December 2007): http://arXiv.org/abs/gr-qc/0203042. 2.9
Ivashchuk, V.D., Kirillov, A.A., and Melnikov, V.N., "Stochastic behavior of multidimensional cosmological models near a singularity", Russ. Phys. J., 37, 1102-1106, (1994). 3
Ivashchuk, V.D., and Melnikov, V.N., "Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity", Class. Quantum Grav., 12, 809-826, (1995). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/ 9407028. 3
Ivashchuk, V.D., and Melnikov, V.N., "Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity", J. Math. Phys., 41, 6341-6363, (2000). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9904077. 3, 2.4
Ivashchuk, V.D., Melnikov, V.N., and Kirillov, A.A., "Stochastic properties of multidimensional cosmological models near a singular point", J. Exp. Theor. Phys. Lett., 60, 235-239, (1994). 3
Julia, B., "Group disintegrations", in Hawking, S.W., and Roček, M., eds., Superspace and Supergravity, Nuffield Gravity Workshop, Cambridge, England, June 22-July 12, 1980, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1981). 1.1, 7.2.1, 7.2.1, 8, 9, 11
Julia, B., Levie, J., and Ray, S., "Gravitational duality near de Sitter space", J. High Energy Phys., 2005(11), 025, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0507262. 9.4.4
Kac, V.G., "Laplace operators of infinite-dimensional Lie algebras and theta functions", Proc. Natl. Acad. Sci. USA, 81, 645-647, (1984). 12
Kac, V.G., Infinite dimensional Lie algebras, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1990), 3rd edition. 3.1.1, 3.1.2, 3.5, 3.5, 9, 4.1, 4.3, 4.5, 12, 4.6.1, 4.6.2, 4.6.4, 4.7, 4.8.2, 4.8.3, 4.9.1, 4.9.3, 4.9.3, 4.9.3, 1, 2, 8.2.2, 8.2.3, A, 2
Kantor, S., "Die Configurationen (3, 3)10", Sitzungsber. Akad. Wiss. Wien, 84(II), 1291-1314, (1881). 10.1.1
Keurentjes, A., "The group theory of oxidation", Nucl. Phys. B, 658, 303-347, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0210178. 5.4
Keurentjes, A., "The group theory of oxidation. II: Cosets of non-split groups", Nucl. Phys. B, 658, 348-372, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0212024. 5.4
Keurentjes, A., "Poincare duality and G+++ algebras", (2005). URL (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0510212. 9.1.6
Khalatnikov, I.M., Lifshitz, E.M., Khanin, K.M., Shchur, L.N., and Sinai, Y.G., "On the Stochasticity in Relativistic Cosmology", J. Stat. Phys., 38, 97-114, (1985). 2.9
Kirillov, A.A., "The Nature of the Spatial Distribution of Metric Inhomogeneities in the General Solution of the Einstein Equations near a Cosmological Singularity", J. Exp. Theor. Phys., 76, 355-358, (1993). 3
Kirillov, A.A., and Melnikov, V.N., "Dynamics of inhomogeneities of metric in the vicinity of a singularity in multidimensional cosmology", Phys. Rev. D, 52, 723-729, (1995). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/9408004. 3
Kleinschmidt, A., Indefinite Kac-Moody Algebras in String Theory, Ph.D. Thesis, (Cambridge University, Cambridge, 2004). 8.2, 9.2.2, 9.4.1
Kleinschmidt, A., and Nicolai, H., "E10 and SO(9, 9) invariant supergravity", J. High Energy Phys., 2004(07), 041, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0407101. 4.10.2, 8.4, 8.4.3
Kleinschmidt, A., and Nicolai, H., "IIB supergravity and E10", Phys. Lett. B, 606, 391-402, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0411225. 8.4, 8.4.3
Kleinschmidt, A., and Nicolai, H., "E10 cosmology", J. High Energy Phys., 2006(01), 137, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0511290. 10.3.1, 10.3.2, 10.3.2, 10.3.2, 39
Kleinschmidt, A., Nicolai, H., and Palmkvist, J., "K(E 9) from K(E10)", J. High Energy Phys., 2007(06), 051, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0611314. 9.4.2
Knapp, A.W., Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140, (Birkhäuser, Boston, U.S.A., 2002), 2nd edition. 6, 6.4.2, 6.4.3, 6.4.3, 6.4.4, 6.4.5, 6.4.6, 6.4.8, 6.5.2, 6.5.4, 6.5.4, 6.5.6, B
Lambert, N., and West, P.C., "Enhanced coset symmetries and higher derivative corrections", Phys. Rev. D, 74, 065002, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0603255. 9.4.3
Lambert, N., and West, P.C., "Duality groups, automorphic forms and higher derivative corrections", Phys. Rev. D, 75, 066002, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/ hep-th/0611318. 9.4.3
Lifshitz, E.M., Lifshitz, I.M., and Khalatnikov, I.M., "Asymptotic analysis of oscillatory mode of approach to a singularity in homogeneous cosmological models", Sov. Phys. JETP, 32, 173, (1971). 2.9
Loos, O., Symmetric Spaces, 2 vols., (W.A. Benjamin, New York, U.S.A., 1969). 6
Marcus, N., and Schwarz, J.H., "Three-Dimensional Supergravity Theories", Nucl. Phys. B, 228, 145, (1983). 5.3.3, 17, 9
Margulis, G.A., "Applications of ergodic theory to the investigation of manifolds of negative curvature", Funct. Anal. Appl., 4, 335-336, (1969). 2.4
Michel, Y., and Pioline, B., "Higher Derivative Corrections, Dimensional Reduction and Ehlers Duality", J. High Energy Phys., 2007(09), 103, (2007). Related online version (cited on 19 October 2007): http://arXiv.org/abs/0706.1769. 9.4.3
Misner, C.W., "Mixmaster Universe", Phys. Rev. Lett., 22, 1071-1074, (1969). 1, 2.7
Misner, C.W., "The Mixmaster cosmological metrics", (1994). URL (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/9405068. 1, 2.4
Nicolai, H., "d = 11 Supergravity With Local SO(16) Invariance", Phys. Lett. B, 187, 316, (1987). 8
Nicolai, H., "The integrability of N = 16 supergravity", Phys. Lett. B, 194, 402, (1987). 9.3.7
Nicolai, H., and Fischbacher, T., "Low level representations for E10 and E11", (2003). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0301017. 8.4.2, 9.4.1
Obers, N.A., and Pioline, B., "U-duality and M-theory", Phys. Rep., 318, 113-225, (1999). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9809039. 5, 5.4
Ohta, N., "Accelerating cosmologies from S-branes", Phys. Rev. Lett., 91, 061303, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0303238. 10.3.2
Ohta, N., "Intersection rules for S-branes", Phys. Lett. B, 558, 213-220, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0301095. 10.3.4
Page, W., and Dorwart, H.L., "Numerical Patterns and geometric Configurations", Math. Mag., 57(2), 82-92, (1984). 10.1.1, 10.5
Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149, (Springer, New York, U.S.A., 1994). 2.4
Rendall, A.D., "The Nature of Spacetime Singularities", in Ashtekar, A., ed., 100 Years of Relativity. Space-Time Structure: Einstein and Beyond, (World Scientific, Singapore, 2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0503112. 2.9
Riccioni, F., Steele, D., and West, P., "Duality Symmetries and G+++ Theories", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0706.3659. 9.4.4
Riccioni, F., and West, P.C., "Dual fields and E 11", Phys. Lett. B, 645, 286-292, (2007). Related online version (cited on 7 December 2007): http://arXiv.org/abs/hep-th/0612001. 8.4.2
Riccioni, F., and West, P.C., "The E11 origin of all maximal supergravities", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0705.0752. 9.4.1
Ringström, H., "The Bianchi IX attractor", Ann. Henri Poincare, 2, 405-500, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0006035. 2.9
Russo, J.G., and Tseytlin, A.A., "One-loop four-graviton amplitude in eleven-dimensional supergravity", Nucl. Phys. B, 508, 245-259, (1997). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9707134. 9.4.3
Ruuska, V., "On purely hyperbolic Kac-Moody algebras", in Mickelsson, J., and Pekonen, O., eds., Topological and Geometrical Methods in Field Theory, Proceedings of the conference, Turku, Finland, 26 May-1 June 1991, pp. 359-369, (World Scientific, Singapore; River Edge, U.S.A., 1992). 4.9.4
Saçlioǧlu, C., "Dynkin diagrams for hyperbolic Kac-Moody algebras", J. Phys. A, 22, 3753, (1989). 4.8
Satake, I., "On Representations and Compactifications of Symmetric Riemannian Spaces", Ann. Math. (2), 71(1), 77-110, (1960). 6.6
Schnakenburg, I., and West, P.C., "Kac-Moody symmetries of IIB supergravity", Phys. Lett. B, 517, 421-428, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0107181. 11
Schnakenburg, I., and West, P.C., "Massive IIA supergravity as a non-linear realisation", Phys. Lett. B, 540, 137-145, (2002). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/ 0204207. 9.4.1, 11
Schnakenburg, I., and West, P.C., "Kac-Moody symmetries of ten-dimensional non-maximal supergravity theories", J. High Energy Phys., 2004(05), 019, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0401196. 11
Schwarz, J.H., and Sen, A., "Duality symmetric actions", Nucl. Phys. B, 411, 35-63, (1994). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/9304154. 9.4.4
Spindel, P., and Zinque, M., "Asymptotic behavior of the Bianchi IX cosmological models in the R2 theory of gravity", Int. J. Mod. Phys. D, 2, 279-294, (1993). 2.7
Tits, J., "Classification of algebraic semisimple groups", in Borel, A., and Mostow, G.D., eds., Algebraic Groups and Discontinuous Subgroups, Proceedings of the Symposium in Pure Mathematics of the AMS held at the University of Colorado, Boulder, Colorado, July 5-August 6, 1965, Proc. Symp. Pure Math., vol. 9, pp. 33-62, (American Mathematical Society, Providence, U.S.A., 1966). 6.6
Uggla, C., "The Nature of Generic Cosmological Singularities", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0706.0463. 2.9
Uggla, C., van Elst, H., Wainwright, J., and Ellis, G.F.R., "The past attractor in inhomogeneous cosmology", Phys. Rev. D, 68, 103502, (2003). Related online version (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/0304002. 2.9
Vinberg, E.B., ed., Geometry II: Spaces of Constant Curvature, Encyclopaedia of Mathematical Sciences, vol. 29, (Springer, Berlin, Germany; New York, U.S.A., 1993). 3.5
Wesley, D.H., "Kac-Moody algebras and controlled chaos", Class. Quantum Grav., 24, F7-F13, (2006). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0610322. 5.3.1
Wesley, D.H., Steinhardt, P.J., and Turok, N., "Controlling chaos through compactification in cosmological models with a collapsing phase", Phys. Rev. D, 72, 063513, (2005). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0502108. 5.3.1
West, P.C., "E11 and M theory", Class. Quantum Grav., 18, 4443-4460, (2001). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0104081. 1.1, 8, 9, 9.3.6, 9.3.7, 11
West, P.C., "The IIA, IIB and eleven dimensional theories and their common E11 origin", Nucl. Phys. B, 693, 76-102, (2004). Related online version (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0402140. 11
West, P.C., "E11 and Higher Spin Theories", (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0701026. 9.4.1
Zimmer, R.J., Ergodic Theory and Semisimple Groups, Monographs in Mathematics, vol. 81, (Birkhäuser, Boston, U.S.A., 1984). 2.4