×

Spacelike singularities and hidden symmetries of gravity. (English) Zbl 1194.83002

Summary: 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 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 selfcontained 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.

MSC:

83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
83E50 Supergravity
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML Link

References:

[1] Alekseev, A.; Monnier, S., Quantization of Wilson loops in Wess-Zumino-Witten models, J. High Energy Phys., 2007, 8, 039 (2007) · Zbl 1326.81172
[2] Andersson, L., On the relation between mathematical and numerical relativity, Class. Quantum Grav., 23, S307-S318 (2006) · Zbl 1191.83017
[3] Andersson, L.; Rendall, Ad, Quiescent cosmological singularities, Commun. Math. Phys., 218, 479-511 (2001) · Zbl 0979.83036
[4] Apostol, Tm, Modular Functions and Dirichlet Series in Number Theory (1997), New York, U.S.A.: Springer, New York, U.S.A.
[5] Araki, S., On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ., 13, 1, 1-34 (1962) · Zbl 0123.03002
[6] Argurio, R.; Englert, F.; Houart, L., Intersection rules for p-branes, Phys. Lett. B, 398, 61-68 (1997)
[7] Aurilia, A.; Nicolai, H.; Townsend, Pk, Hidden Constants: The Theta Parameter of QCD and the Cosmological Constant of N = 8 Supergravity, Nucl. Phys. B, 176, 509 (1980)
[8] Baez, John C., The octonions, Bulletin of the American Mathematical Society, 39, 2, 145-206 (2001) · Zbl 1026.17001
[9] Bagnoud, M.; Carlevaro, L., Hidden Borcherds symmetries in Z_n orbifolds of M-theory and magnetized D-branes in type 0’ orientifolds, J. High Energy Phys., 2006, 11, 003 (2006)
[10] Bahls, P., The Isomorphism Problem in Coxeter Groups (2005), London, U.K.: Imperial College Press, London, U.K. · Zbl 1113.20034
[11] 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
[12] 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
[13] Bautier, K.; Deser, S.; Henneaux, M.; Seminara, D., No cosmological D = 11 super-gravity, Phys. Lett. B, 406, 49-53 (1997)
[14] Bekaert, X.; Boulanger, N.; Henneaux, M., Consistent deformations of dual formulations of linearized gravity: A no-go result, Phys. Rev. D, 67, 044010 (2003)
[15] Belinskii, Va; Khalatnikov, Im, Effect of scalar and vector fields on the nature of the cosmological singularity, Sov. Phys. JETP, 36, 591-597 (1973)
[16] Belinskii, Va; Khalatnikov, Im; Lifshitz, Em, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys., 19, 525-573 (1970)
[17] Ben Messaoud, H., Almost split real forms for hyperbolic Kac-Moody Lie algebras, J. Phys. A, 39, 13659-13690 (2006) · Zbl 1124.17008
[18] Berger, Bk; Garfinkle, D.; Isenberg, Ja; Moncrief, V.; Weaver, M., The singularity in generic gravitational collapse is spacelike, local, and oscillatory, Mod. Phys. Lett. A, 13, 1565-1574 (1998)
[19] Bergshoeff, E.A., De Baetselier, I., and Nutma, T.A., “E11 and the embedding tensor”, (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0705.1304. 8.2.3, 9.4.1
[20] Bergshoeff, Ea; De Roo, M.; De Wit, B.; Van Nieuwenhuizen, P., Ten-dimensional Maxwell-Einstein supergravity, its currents, and the issue of its auxiliary fields, Nucl. Phys. B, 195, 97-136 (1982) · Zbl 0900.53034
[21] Boulanger, N.; Cnockaert, S.; Henneaux, M., A note on spin-s duality, J. High Energy Phys., 2003, 6, 060 (2003)
[22] Breitenlohner, P.; Maison, D.; Gibbons, Gw, 4-Dimensional Black Holes from Kaluza-Klein Theories, Commun. Math. Phys., 120, 295-333 (1988) · Zbl 0661.53064
[23] Brown, J.; Ganguli, S.; Ganor, Oj; Helfgott, C., E_10 orbifolds, J. High Energy Phys., 2005, 6, 057 (2005)
[24] Brown, J.; Ganor, Oj; Helfgott, C., M-theory and E10: Billiards, branes, and imaginary roots, J. High Energy Phys., 2004, 8, 063 (2004)
[25] Bunster, C.; Cnockaert, S.; Henneaux, M.; Portugues, R., Monopoles for gravitation and for higher spin fields, Phys. Rev. D, 73, 105014 (2006)
[26] Bunster, C.; Henneaux, M., A monopole near a black hole, Proc. Natl. Acad. Sci. USA, 104, 12243-12249 (2007) · Zbl 1190.83051
[27] 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
[28] Cartan, E., Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple, Ann. Sci. Ecole Norm. Sup., 44, 345-467 (1927) · JFM 53.0393.01
[29] Chapline, Gf; Manton, Ns, Unification of Yang-Mills Theory and Supergravity in Ten Dimensions, Phys. Lett. B, 120, 105-109 (1983)
[30] Chernoff, Df; Barrow, Jd, Chaos in the Mixmaster Universe, Phys. Rev. Lett., 50, 134-137 (1983)
[31] Chitre, Dm, Investigations of Vanishing of a Horizon for Bianchy Type X (the Mixmaster) (1972), College Park, U.S.A.: University of Maryland, College Park, U.S.A.
[32] Cornish, Nj; Levin, Jj, Mixmaster universe: A chaotic Farey tale, Phys. Rev. D, 55, 7489-7510 (1997)
[33] Cremmer, E.; Julia, B., The N = 8 Supergravity Theory. 1. The Lagrangian, Phys. Lett. B, 80, 48 (1978)
[34] Cremmer, E.; Julia, B., The SO(8) Supergravity, Nucl. Phys. B, 159, 141 (1979)
[35] Cremmer, E.; Julia, B.; Lu, H.; Pope, Cn, Dualisation of dualities. I, Nucl. Phys. B, 523, 73-144 (1998) · Zbl 1031.81599
[36] Cremmer, E.; Julia, B.; Lu, H.; Pope, Cn, Dualisation of dualities. II: Twisted self-duality of doubled fields and superdualities, Nucl. Phys. B, 535, 242-292 (1998) · Zbl 1080.81598
[37] 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
[38] Cremmer, E.; Julia, B.; Scherk, J., Supergravity theory in 11 dimensions, Phys. Lett. B, 76, 409-412 (1978)
[39] Curtright, T., Generalized gauge fields, Phys. Lett. B, 165, 304-208 (1985)
[40] 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
[41] Damour, T.; De Buyl, S.; Henneaux, M.; Schomblond, C., Einstein billiards and overextensions of finite-dimensional simple Lie algebras, J. High Energy Phys., 2002, 8, 030 (2002) · Zbl 1226.83083
[42] Damour, T.; Hanany, A.; Henneaux, M.; Kleinschmidt, A.; Nicolai, H., Curvature corrections and Kac-Moody compatibility conditions, Gen. Relativ. Gravit., 38, 1507-1528 (2006) · Zbl 1106.83017
[43] Damour, T.; Henneaux, M., Chaos in superstring cosmology, Phys. Rev. Lett., 85, 920-923 (2000) · Zbl 0978.83055
[44] Damour, T.; Henneaux, M., Oscillatory behaviour in homogeneous string cosmology models, Phys. Lett. B, 488, 108-116 (2000) · Zbl 1006.83068
[45] Damour, T.; Henneaux, M., E_10, BE_10 and arithmetical chaos in superstring cosmology, Phys. Rev. Lett., 86, 4749-4752 (2001)
[46] Damour, T.; Henneaux, M.; Julia, B.; Nicolai, H., Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B, 509, 323-330 (2001) · Zbl 0977.83075
[47] Damour, T.; Henneaux, M.; Nicolai, H., E_10 and a ‘small tension expansion’ of M theory, Phys. Rev. Lett., 89, 221601 (2002) · Zbl 1267.83103
[48] Damour, T.; Henneaux, M.; Nicolai, H., Cosmological Billiards, Class. Quantum Grav., 20, R145-R200 (2003) · Zbl 1138.83306
[49] Damour, T.; Henneaux, M.; Rendall, Ad; Weaver, M., Kasner-Like Behaviour for Subcritical Einstein-Matter Systems, Ann. Henri Poincare, 3, 1049-1111 (2002) · Zbl 1011.83038
[50] Damour, T.; Kleinschmidt, A.; Nicolai, H., Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B, 634, 319-324 (2006) · Zbl 1247.83225
[51] Damour, T.; Kleinschmidt, A.; Nicolai, H., K(E_10), supergravity and fermions, J. High Energy Phys., 2006, 8, 046 (2006)
[52] Damour, T., Kleinschmidt, A., and Nicolai, H., “Constraints and the E_10 Coset Model”, (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0709.2691. 2.8, 9.4.4 · Zbl 1130.83014
[53] Damour, T., and Nicolai, H., “Eleven dimensional supergravity and the E_10/K(E_10) sigma-model at low A_9 levels”, (2004). URL (cited on 19 October 2007):http://arXiv.org/abs/hep-th/0410245. 8.4.1, 9.3.6
[54] Damour, T.; Nicolai, H., Higher order M theory corrections and the Kac-Moody algebra E_10, Class. Quantum Grav., 22, 2849-2880 (2005) · Zbl 1129.81339
[55] De Buyl, S.; Henneaux, M.; Julia, B.; Paulot, L., Cosmological billiards and oxidation, Fortschr. Phys., 52, 548-554 (2004) · Zbl 1052.81036
[56] De Buyl, S.; Henneaux, M.; Paulot, L., Hidden symmetries and Dirac fermions, Class. Quantum Grav., 22, 3595-3622 (2005) · Zbl 1153.83371
[57] De Buyl, S.; Henneaux, M.; Paulot, L., Extended E_8 invariance of 11-dimensional supergravity, J. High Energy Phys., 2006, 2, 056 (2006)
[58] De Buyl, S.; Pinardi, G.; Schomblond, C., Einstein billiards and spatially homogeneous cosmological models, Class. Quantum Grav., 20, 5141-5160 (2003) · Zbl 1057.83027
[59] De Buyl, S.; Schomblond, C., Hyperbolic Kac Moody algebras and Einstein billiards, J. Math. Phys., 45, 4464-4492 (2004) · Zbl 1064.83053
[60] Demaret, J.; De Rop, Y.; Henneaux, M., Chaos in Nondiagonal Spatially Homogeneous Cosmological Models in Space-time Dimensions ≤ 10, Phys. Lett. B, 211, 37-41 (1988)
[61] Demaret, J.; Hanquin, Jl; Henneaux, M.; Spindel, P., Cosmological models in eleven-dimensional supergravity, Nucl. Phys. B, 252, 538-560 (1985)
[62] Demaret, J.; Hanquin, Jl; Henneaux, M.; Spindel, P.; Taormina, A., The fate of the mixmaster behavior in vacuum inhomogeneous Kaluza-Klein cosmological models, Phys. Lett. B, 175, 129-132 (1986)
[63] Demaret, J.; Henneaux, M.; Spindel, P., No Oscillatory Behavior in vacuum Kaluza-Klein cosmologies, Phys. Lett. B, 164, 27-30 (1985)
[64] Deser, S.; Gomberoff, A.; Henneaux, M.; Teitelboim, C., Duality, self-duality, sources and charge quantization in abelian N-form theories, Phys. Lett. B, 400, 80-86 (1997)
[65] Deser, S.; Teitelboim, C., Duality Transformations of Abelian and Nonabelian Gauge Fields, Phys. Rev. D, 13, 1592-1597 (1976)
[66] Dewitt, Bs, Quantum Theory of Gravity. I. The Canonical Theory, Phys. Rev., 160, 1113-1148 (1967) · Zbl 0158.46504
[67] Dynkin, Eb, Semisimple subalgebras of semisimple Lie algebras, Trans. Amer. Math. Soc., 6, 111 (1957) · Zbl 0077.03404
[68] Elskens, Y.; Henneaux, M., Ergodic theory of the mixmaster model in higher space-time dimensions, Nucl. Phys. B, 290, 111-136 (1987)
[69] Englert, F.; Henneaux, M.; Houart, L., From very-extended to overextended gravity and M-theories, J. High Energy Phys., 2005, 2, 070 (2005)
[70] 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 · Zbl 1243.81152
[71] Englert, F.; Houart, L., G+++ invariant formulation of gravity and M-theories: Exact BPS solutions, J. High Energy Phys., 2004, 1, 002 (2004) · Zbl 1243.81152
[72] Englert, F.; Houart, L., G+++ invariant formulation of gravity and M-theories: Exact intersecting brane solutions, J. High Energy Phys., 2004, 5, 059 (2004)
[73] Englert, F., Houart, L., Kleinschmidt, A., Nicolai, H., and Tabti, N., “An E_9 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
[74] Englert, F.; Houart, L.; Taormina, A.; West, Pc, The symmetry of M-theories, J. High Energy Phys., 2003, 9, 020 (2003)
[75] Feingold, Aj; Frenkel, Ib, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2, Math. Ann., 263, 87 (1983) · Zbl 0489.17008
[76] 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 · Zbl 1050.17021
[77] Fischbacher, T., The structure of E_10 at higher A_9 levels: A first algorithmic approach, J. High Energy Phys., 2005, 8, 012 (2005)
[78] Forte, La; Sciarrino, A., Standard and non-standard extensions of Lie algebras, J. Math. Phys., 47, 013513 (2006) · Zbl 1111.17005
[79] Fré, P.; Gargiulo, F.; Rulik, K., Cosmic billiards with painted walls in non-maximal supergravities: A worked out example, Nucl. Phys. B, 737, 1-48 (2006) · Zbl 1109.83013
[80] Fré, P.; Gargiulo, F.; Rulik, K.; Trigiante, M., The general pattern of Kac-Moody extensions in supergravity and the issue of cosmic billiards, Nucl. Phys. B, 741, 42-82 (2006) · Zbl 1214.83044
[81] Fré, P.; Gargiulo, F.; Sorin, A.; Rulik, K.; Trigiante, M., Cosmological backgrounds of superstring theory and solvable algebras: oxidation and branes, Nucl. Phys. B, 685, 3-64 (2004) · Zbl 1107.83320
[82] Fré, P.; Rulik, K.; Trigiante, M., Exact solutions for Bianchi type cosmological metrics, Weyl orbits of E_8(8) subalgebras and p-branes, Nucl. Phys. B, 694, 239-274 (2004) · Zbl 1151.83363
[83] 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
[84] Fuchs, J., Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory (1992), Cambridge, U.K.; New York, U.S.A.: Cambridge University Press, Cambridge, U.K.; New York, U.S.A. · Zbl 0925.17031
[85] Fuchs, J.; Schweigert, C., Symmetries, Lie Algebras and Representations: A graduate course for physicists (1997), Cambridge, U.K.; New York, U.S.A.: Cambridge University Press, Cambridge, U.K.; New York, U.S.A. · Zbl 0923.17001
[86] Gaberdiel, Mr; Olive, Di; West, Pc, A class of Lorentzian Kac-Moody algebras, Nucl. Phys. B, 645, 403-437 (2002) · Zbl 0999.17033
[87] Garfinkle, D., Numerical simulations of generic singuarities, Phys. Rev. Lett., 93, 161101 (2004)
[88] Green, Mb; Vanhove, P., Duality and higher derivative terms in M theory, J. High Energy Phys., 2006, 1, 093 (2006)
[89] Gross, Dj; Harvey, Ja; Martinec, Ej; Rohm, R., The Heterotic String, Phys. Rev. Lett., 54, 502-505 (1985)
[90] Gutperle, M.; Strominger, A., Spacelike branes, J. High Energy Phys., 2002, 4, 018 (2002)
[91] Hawking, Sw; Ellis, Gfr, The Large Scale Structure of Space-Time (1973), Cambridge, U.K.: Cambridge University Press, Cambridge, U.K. · Zbl 0265.53054
[92] 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 · Zbl 1170.83003
[93] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (2001), Providence, U.S.A.: American Mathematical Society, Providence, U.S.A. · Zbl 0177.50601
[94] Helminck, Ag; Koornwinder, Th, Classification of real semisimple Lie algebras, The structure of real semisimple Lie groups, 113-136 (1982), Amsterdam, Netherlands: Math. Centrum, Amsterdam, Netherlands · Zbl 0496.22023
[95] Henneaux, M.; Julia, B., Hyperbolic billiards of pure D = 4 supergravities, J. High Energy Phys., 2003, 5, 047 (2003)
[96] Henneaux, M.; Leston, M.; Persson, D.; Spindel, P., Geometric configurations, regular subalgebras of E_10 and M-theory cosmology, J. High Energy Phys., 2006, 10, 021 (2006)
[97] Henneaux, M.; Leston, M.; Persson, D.; Spindel, P., A Special Class of Rank 10 and 11 Coxeter Groups, J. Math. Phys., 48, 053512 (2007) · Zbl 1144.81354
[98] Henneaux, M., Persson, D., and Wesley, D.H., “Chaos, Cohomology and Coxeter Groups”, unknown status. Work in progress. 5.3.1
[99] Henneaux, M.; Teitelboim, C., The cosmological constant as a canonical variable, Phys. Lett. B, 143, 415-420 (1984)
[100] Henneaux, M.; Teitelboim, C., Dynamics of chiral (selfdual) p-forms, Phys. Lett. B, 206, 650 (1988)
[101] Henneaux, M.; Teitelboim, C., Duality in linearized gravity, Phys. Rev. D, 71, 024018 (2005)
[102] Henry-Labordere, P.; Julia, B.; Paulot, L., Borcherds symmetries in M-theory, J. High Energy Phys., 2002, 4, 049 (2002)
[103] Henry-Labordere, P.; Julia, B.; Paulot, L., Real Borcherds superalgebras and M-theory, J. High Energy Phys., 2003, 4, 060 (2003)
[104] 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
[105] Hilbert, D.; Cohn-Vossen, S., Geometry and the Imagination (1952), New York, U.S.A.: Chelsea Pub. Co., New York, U.S.A. · Zbl 0047.38806
[106] Hillmann, C.; Kleinschmidt, A., Pure type I supergravity and DE_10, Gen. Relativ. Gravit., 38, 1861-1885 (2006) · Zbl 1157.83351
[107] Humphreys, Je, Reflection Groups and Coxeter Groups (1990), Cambridge, U.K.; New York, U.S.A.: Cambridge University Press, Cambridge, U.K.; New York, U.S.A. · Zbl 0725.20028
[108] Isenberg, Ja; Moncrief, V., Asymptotic behavior of polarized and half-polarized U(1) symmetric vacuum spacetimes, Class. Quantum Grav., 19, 5361-5386 (2002) · Zbl 1025.83008
[109] Ivashchuk, Vd; Kirillov, Aa; Melnikov, Vn, Stochastic behavior of multidimensional cosmological models near a singularity, Russ. Phys. J., 37, 1102-1106 (1994) · Zbl 0946.83509
[110] Ivashchuk, Vd; Melnikov, Vn, Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity, Class. Quantum Grav., 12, 809-826 (1995) · Zbl 0821.53076
[111] Ivashchuk, Vd; Melnikov, Vn, Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity, J. Math. Phys., 41, 6341-6363 (2000) · Zbl 0977.83078
[112] Ivashchuk, Vd; Melnikov, Vn; Kirillov, Aa, Stochastic properties of multidimensional cosmological models near a singular point, J. Exp. Theor. Phys. Lett., 60, 235-239 (1994)
[113] Julia, B.; Hawking, Sw; Roček, M., Group disintegrations, Superspace and Supergravity (1981), Cambridge, U.K.; New York, U.S.A.: Cambridge University Press, Cambridge, U.K.; New York, U.S.A.
[114] Julia, B.; Levie, J.; Ray, S., Gravitational duality near de Sitter space, J. High Energy Phys., 2005, 11, 025 (2005)
[115] Kac, Vg, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Natl. Acad. Sci. USA, 81, 645-647 (1984) · Zbl 0532.17008
[116] Kac, Vg, Infinite dimensional Lie algebras (1990), Cambridge, U.K.; New York, U.S.A.: Cambridge University Press, Cambridge, U.K.; New York, U.S.A. · Zbl 0716.17022
[117] Kantor, S., Die Configurationen (3, 3)_10, Sitzungsber. Akad. Wiss. Wien, 84, II, 1291-1314 (1881) · JFM 13.0460.05
[118] Keurentjes, A., The group theory of oxidation, Nucl. Phys. B, 658, 303-347 (2003) · Zbl 1017.83025
[119] Keurentjes, A., The group theory of oxidation. II: Cosets of non-split groups, Nucl. Phys. B, 658, 348-372 (2003) · Zbl 1017.83026
[120] Keurentjes, A., “Poincare duality and G+++ algebras”, (2005). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0510212. 9.1.6 · Zbl 1220.83041
[121] Khalatnikov, Im; Lifshitz, Em; Khanin, Km; Shchur, Ln; Sinai, Yg, On the Stochasticity in Relativistic Cosmology, J. Stat. Phys., 38, 97-114 (1985)
[122] Kirillov, Aa, 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)
[123] Kirillov, Aa; Melnikov, Vn, Dynamics of inhomogeneities of metric in the vicinity of a singularity in multidimensional cosmology, Phys. Rev. D, 52, 723-729 (1995)
[124] Kleinschmidt, A., Indefinite Kac-Moody Algebras in String Theory (2004), Cambridge: Cambridge University, Cambridge
[125] Kleinschmidt, A.; Nicolai, H., E_10 and SO(9, 9) invariant supergravity, J. High Energy Phys., 2004, 7, 041 (2004)
[126] Kleinschmidt, A.; Nicolai, H., IIB supergravity and E_10, Phys. Lett. B, 606, 391-402 (2005) · Zbl 1247.83230
[127] Kleinschmidt, A.; Nicolai, H., E_10 cosmology, J. High Energy Phys., 2006, 1, 137 (2006) · Zbl 1241.83071
[128] Kleinschmidt, A.; Nicolai, H.; Palmkvist, J., K(E_9) from K(E_10), J. High Energy Phys., 2007, 6, 051 (2007)
[129] Knapp, Aw, Lie Groups Beyond an Introduction (2002), Boston, U.S.A.: Birkhäuser, Boston, U.S.A. · Zbl 1075.22501
[130] Lambert, N.; West, Pc, Enhanced coset symmetries and higher derivative corrections, Phys. Rev. D, 74, 065002 (2006)
[131] Lambert, N.; West, Pc, Duality groups, automorphic forms and higher derivative corrections, Phys. Rev. D, 75, 066002 (2007)
[132] Lifshitz, Em; Lifshitz, Im; Khalatnikov, Im, Asymptotic analysis of oscillatory mode of approach to a singularity in homogeneous cosmological models, Sov. Phys. JETP, 32, 173 (1971)
[133] Loos, O., Symmetric Spaces (1969), New York, U.S.A.: W.A. Benjamin, New York, U.S.A. · Zbl 0175.48601
[134] Marcus, N.; Schwarz, Jh, Three-Dimensional Supergravity Theories, Nucl. Phys. B, 228, 145 (1983)
[135] Margulis, Ga, Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl., 4, 335-336 (1969) · Zbl 0207.20305
[136] Michel, Y.; Pioline, B., Higher Derivative Corrections, Dimensional Reduction and Ehlers Duality, J. High Energy Phys., 2007, 9, 103 (2007)
[137] Misner, Cw, Mixmaster Universe, Phys. Rev. Lett., 22, 1071-1074 (1969) · Zbl 0177.28701
[138] Misner, C.W., “The Mixmaster cosmological metrics”, (1994). URL (cited on 19 October 2007): http://arXiv.org/abs/gr-qc/9405068. 1, 2.4
[139] Nicolai, H., d = 11 Supergravity With Local SO(16) Invariance, Phys. Lett. B, 187, 316 (1987)
[140] Nicolai, H., The integrability of N = 16 supergravity, Phys. Lett. B, 194, 402 (1987)
[141] Nicolai, H., and Fischbacher, T., “Low level representations for E_10 and E_11”, (2003). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0301017. 8.4.2, 9.4.1 · Zbl 1042.17022
[142] Obers, Na; Pioline, B., U-duality and M-theory, Phys. Rep., 318, 113-225 (1999)
[143] Ohta, N., Accelerating cosmologies from S-branes, Phys. Rev. Lett., 91, 061303 (2003)
[144] Ohta, N., Intersection rules for S-branes, Phys. Lett. B, 558, 213-220 (2003) · Zbl 1011.83031
[145] Page, W.; Dorwart, Hl, Numerical Patterns and geometric Configurations, Math. Mag., 57, 2, 82-92 (1984) · Zbl 0537.51003
[146] Ratcliffe, Jg, Foundations of Hyperbolic Manifolds (1994), New York, U.S.A.: Springer, New York, U.S.A. · Zbl 0809.51001
[147] Rendall, Ad; Ashtekar, A., The Nature of Spacetime Singularities, 100 Years of Relativity. Space-Time Structure: Einstein and Beyond (2005), Singapore: World Scientific, Singapore · Zbl 1084.83003
[148] 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 · Zbl 1190.83098
[149] Riccioni, F.; West, Pc, Dual fields and E11, Phys. Lett. B, 645, 286-292 (2007) · Zbl 1256.83032
[150] Riccioni, F., and West, P.C., “The E_11 origin of all maximal supergravities”, (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0705.0752. 9.4.1 · Zbl 1256.83032
[151] Ringström, H., The Bianchi IX attractor, Ann. Henri Poincare, 2, 405-500 (2001) · Zbl 0985.83002
[152] Russo, Jg; Tseytlin, Aa, One-loop four-graviton amplitude in eleven-dimensional supergravity, Nucl. Phys. B, 508, 245-259 (1997) · Zbl 0925.83112
[153] Ruuska, V.; Mickelsson, J.; Pekonen, O., On purely hyperbolic Kac-Moody algebras, Topological and Geometrical Methods in Field Theory, 359-369 (1992), Singapore; River Edge, U.S.A.: World Scientific, Singapore; River Edge, U.S.A. · Zbl 1103.17306
[154] Saçlioğlu, C., Dynkin diagrams for hyperbolic Kac-Moody algebras, J. Phys. A, 22, 3753 (1989) · Zbl 0725.17029
[155] Satake, I., On Representations and Compactifications of Symmetric Riemannian Spaces, Ann. Math. (2), 71, 1, 77-110 (1960) · Zbl 0094.34603
[156] Schnakenburg, I.; West, Pc, Kac-Moody symmetries of IIB supergravity, Phys. Lett. B, 517, 421-428 (2001) · Zbl 0971.83085
[157] Schnakenburg, I.; West, Pc, Massive IIA supergravity as a non-linear realisation, Phys. Lett. B, 540, 137-145 (2002) · Zbl 0996.83062
[158] Schnakenburg, I.; West, Pc, Kac-Moody symmetries often-dimensional non-maximal supergravity theories, J. High Energy Phys., 2004, 5, 019 (2004)
[159] Schwarz, Jh; Sen, A., Duality symmetric actions, Nucl. Phys. B, 411, 35-63 (1994) · Zbl 0999.81503
[160] Spindel, P.; Zinque, M., Asymptotic behavior of the Bianchi IX cosmological models in the R^2 theory of gravity, Int. J. Mod. Phys. D, 2, 279-294 (1993) · Zbl 0935.83522
[161] Tits, J.; Borel, A.; Mostow, Gd, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups, 33-62 (1966), Providence, U.S.A.: American Mathematical Society, Providence, U.S.A. · Zbl 0238.20052
[162] Uggla, C., “The Nature of Generic Cosmological Singularities”, (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/0706.0463. 2.9 · Zbl 1162.83002
[163] Uggla, C.; Van Elst, H.; Wainwright, J.; Ellis, Gfr, The past attractor in inhomo-geneous cosmology, Phys. Rev. D, 68, 103502 (2003)
[164] Vinberg, Eb, Geometry II: Spaces of Constant Curvature (1993), Berlin, Germany; New York, U.S.A.: Springer, Berlin, Germany; New York, U.S.A.
[165] Wesley, Dh, Kac-Moody algebras and controlled chaos, Class. Quantum Grav., 24, F7-F13 (2006) · Zbl 1206.83152
[166] Wesley, Dh; Steinhardt, Pj; Turok, N., Controlling chaos through compactification in cosmological models with a collapsing phase, Phys. Rev. D, 72, 063513 (2005)
[167] West, Pc, E_11 and M theory, Class. Quantum Grav., 18, 4443-4460 (2001) · Zbl 0992.83079
[168] West, Pc, The IIA, IIB and eleven dimensional theories and their common E11 origin, Nucl. Phys. B, 693, 76-102 (2004) · Zbl 1151.81379
[169] West, P.C., “E_11 and Higher Spin Theories”, (2007). URL (cited on 19 October 2007): http://arXiv.org/abs/hep-th/0701026. 9.4.1 · Zbl 1248.81190
[170] Zimmer, Rj, Ergodic Theory and Semisimple Groups (1984), Boston, U.S.A.: Birkhäuser, Boston, U.S.A. · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.