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Generalized geometry and M theory. (English) Zbl 1298.81244
Summary: We reformulate the Hamiltonian form of bosonic eleven dimensional supergravity in terms of an object that unifies the three-form and the metric. For the case of four spatial dimensions, the duality group is manifest and the metric and C-field are on an equal footing even though no dimensional reduction is required for our results to hold. One may also describe our results using the generalized geometry that emerges from membrane duality. The relationship between the twisted Courant algebra and the gauge symmetries of eleven dimensional supergravity are described in detail.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30 String and superstring theories in gravitational theory
83E50 Supergravity
83E15 Kaluza-Klein and other higher-dimensional theories
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References:
[1] Julia, B.; Hawking, SW (ed.); Rocek, M. (ed.), Group disintegrations, (1981), Cambridge U.K.
[2] Julia, B.; Serdaroglu, M. (ed.); Inonu, E. (ed.), Gravity, supergravities and integrable systems, (1983), U.S.A.
[3] Thierry-Mieg, J.; Morel, B.; Hawking, SW (ed.); Rocek, M. (ed.), Superalgebras in exceptional gravity, (1981), Cambridge U.K.
[4] Cremmer, E.; Salam, A. (ed.); Sezgin, E. (ed.), Supergravities in 5 dimensions, (1989), Singapore
[5] Gibbons, GW; Hawking, SW, Classification of gravitational instanton symmetries, Commun. Math. Phys., 66, 291, (1979)
[6] Wit, B.; Nicolai, H., D = 11 supergravity with local SU(8) invariance, Nucl. Phys., B 274, 363, (1986)
[7] Nicolai, H., D = 11 supergravity with local SO(16) invariance, Phys. Lett., B 187, 316, (1987)
[8] Koepsell, K.; Nicolai, H.; Samtleben, H., An exceptional geometry for D = 11 supergravity?, Class. Quant. Grav., 17, 3689, (2000)
[9] Wit, B.; Nicolai, H., Hidden symmetries, central charges and all that, Class. Quant. Grav., 18, 3095, (2001)
[10] West, P., Generalised space-time and duality, Phys. Lett., B 693, 373, (2010)
[11] West, PC, \(E\)_{11} and M-theory, Class. Quant. Grav., 18, 4443, (2001)
[12] West, PC, \(E\)_{11}, SL(32) and central charges, Phys. Lett., B 575, 333, (2003)
[13] West, PC, \(E\)_{11} origin of brane charges and U-duality multiplets, JHEP, 08, 052, (2004)
[14] Kleinschmidt, A.; West, PC, Representations of G+++ and the role of space-time, JHEP, 02, 033, (2004)
[15] West, PC, Brane dynamics, central charges and \(E\)_{11}, JHEP, 03, 077, (2005)
[16] Obers, NA; Pioline, B., U-duality and M-theory, Phys. Rept., 318, 113, (1999)
[17] Riccioni, F.; West, PC, \(E\)_{11}-extended spacetime and gauged supergravities, JHEP, 02, 039, (2008)
[18] Nicolai, H.; Kleinschmidt, A., \(E\)_{10}: eine fundamentale symmetrie der physik?, Phys. Unserer Zeit, 3N, 134, (2010)
[19] Damour, T.; Henneaux, M.; Nicolai, H., \(E\)_{10} and a ’small tension expansion’ of M-theory, Phys. Rev. Lett., 89, 221601, (2002)
[20] Damour, T.; Henneaux, M.; Nicolai, H., Cosmological billiards, Class. Quant. Grav., 20, 145, (2003)
[21] D. Persson, Arithmetic and hyperbolic structures in string theory, arXiv:1001.3154 [SPIRES].
[22] Hillmann, C., Generalized \(E\)_{7(7)} coset dynamics and \(D\) = 11 supergravity, JHEP, 03, 135, (2009)
[23] Hitchin, N., Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser., 54, 281, (2003)
[24] N. Hitchin, Brackets, forms and invariant functionals, math/0508618.
[25] M. Gualtieri, Generalized complex geometry, math/0401221.
[26] Hull, CM, Generalised geometry for M-theory, JHEP, 07, 079, (2007)
[27] Pacheco, PP; Waldram, D., M-theory, exceptional generalised geometry and superpotentials, JHEP, 09, 123, (2008)
[28] Dirac, PAM, The theory of gravitation in Hamiltonian form, Proc. Roy. Soc. Lond., A 246, 333, (1958)
[29] Dirac, PAM, Fixation of coordinates in the Hamiltonian theory of gravitation, Phys. Rev., 114, 924, (1959)
[30] Arnowitt, RL; Deser, S.; Misner, CW, Dynamical structure and definition of energy in general relativity, Phys. Rev., 116, 1322, (1959)
[31] Deser, S.; Arnowitt, R.; Misner, CW, Consistency of canonical reduction of general relativity, J. Math Phys., 1, 434, (1960)
[32] Arnowitt, RL; Deser, S.; Misner, CW, Canonical variables for general relativity, Phys. Rev., 117, 1595, (1960)
[33] Arnowitt, RL; Deser, S.; Misner, CW; Witten, L. (ed.), The dynamics of general relativity, (1962), U.S.A.
[34] DeWitt, BS, Quantum theory of gravity. 1. the canonical theory, Phys. Rev., 160, 1113, (1967)
[35] Hull, CM, Duality and the signature of space-time, JHEP, 11, 017, (1998)
[36] Duff, MJ, Duality rotations in string theory, Nucl. Phys., B 335, 610, (1990)
[37] Tseytlin, AA, Duality symmetric formulation of string world sheet dynamics, Phys. Lett., B 242, 163, (1990)
[38] Tseytlin, AA, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys., B 350, 395, (1991)
[39] Hull, CM, A geometry for non-geometric string backgrounds, JHEP, 10, 065, (2005)
[40] Hull, CM, Global aspects of T-duality, gauged σ-models and T-folds, JHEP, 10, 057, (2007)
[41] Hull, CM, Doubled geometry and T-folds, JHEP, 07, 080, (2007)
[42] Hull, C.; Zwiebach, B., Double field theory, JHEP, 09, 099, (2009)
[43] Hull, C.; Zwiebach, B., The gauge algebra of double field theory and Courant brackets, JHEP, 09, 090, (2009)
[44] Hohm, O.; Hull, C.; Zwiebach, B., Background independent action for double field theory, JHEP, 07, 016, (2010)
[45] Duff, MJ; Lu, JX, Duality rotations in membrane theory, Nucl. Phys., B 347, 394, (1990)
[46] Berman, DS; Copland, NB, The string partition function in hull’s doubled formalism, Phys. Lett., B 649, 325, (2007)
[47] Berman, DS; Copland, NB; Thompson, DC, Background field equations for the duality symmetric string, Nucl. Phys., B 791, 175, (2008)
[48] Berman, DS; Thompson, DC, Duality symmetric strings, dilatons and \(O\)(\(d\), \(d\)) effective actions, Phys. Lett., B 662, 279, (2008)
[49] Avramis, SD; Derendinger, J-P; Prezas, N., Conformal chiral boson models on twisted doubled tori and non-geometric string vacua, Nucl. Phys., B 827, 281, (2010)
[50] Bonelli, G.; Zabzine, M., From current algebras for p-branes to topological M-theory, JHEP, 09, 015, (2005)
[51] Bonelli, G.; Tanzini, A.; Zabzine, M., On topological M-theory, Adv. Theor. Math. Phys., 10, 239, (2006)
[52] Bonelli, G.; Tanzini, A.; Zabzine, M., Topological branes, p-algebras and generalized Nahm equations, Phys. Lett., B 672, 390, (2009)
[53] Aldazabal, G.; Andres, E.; Camara, PG; Graña, M., U-dual fluxes and generalized geometry, JHEP, 11, 083, (2010)
[54] Graña, M.; Minasian, R.; Petrini, M.; Waldram, D., T-duality, generalized geometry and non-geometric backgrounds, JHEP, 04, 075, (2009)
[55] Witten, E., String theory dynamics in various dimensions, Nucl. Phys., B 443, 85, (1995)
[56] Hull, CM; Townsend, PK, Unity of superstring dualities, Nucl. Phys., B 438, 109, (1995)
[57] R.A. Reid-Edwards, Bi-algebras, generalised geometry and T-duality, arXiv:1001.2479 [SPIRES].
[58] N. Halmagyi, Non-geometric backgrounds and the first order string σ-model, arXiv:0906.2891 [SPIRES].
[59] J. McOrist, D.R. Morrison and S. Sethi, Geometries, non-geometries and fluxes, arXiv:1004.5447 [SPIRES].
[60] Boer, J.; Shigemori, M., Exotic branes and non-geometric backgrounds, Phys. Rev. Lett., 104, 251603, (2010)
[61] Moncrief, V.; Teitelboim, C., Momentum constraints as integrability conditions for the Hamiltonian constraint in general relativity, Phys. Rev., D 6, 966, (1972)
[62] Gibbons, GW; Hawking, SW; Perry, MJ, Path integrals and the indefiniteness of the gravitational action, Nucl. Phys., B 138, 141, (1978)
[63] Buchdahl, HA, Reciprocal static solutions of the equations of the gravitational field, Austral. J. Phys., 9, 13, (1956)
[64] J. Ehlers, Konstruktionen und Charakterisierung von Losungen der Einsteinschen Gravitationsfeldgleichungen, Ph.D. thesis, University of Hamburg, Hamburg, Germany (1957).
[65] Geroch, RP, A method for generating solutions of einstein’s equations, J. Math. Phys., 12, 918, (1971)
[66] Geroch, RP, A method for generating new solutions of einstein’s equation. 2, J. Math. Phys., 13, 394, (1972)
[67] Thierry-Mieg, J., BRS structure of the antisymmetric tensor gauge theories, Nucl. Phys., B 335, 334, (1990)
[68] Baulieu, L.; Henneaux, M., P forms and diffeomorphisms: Hamiltonian formulation, Phys. Lett., B 194, 81, (1987)
[69] Courant, T., Dirac manifolds, Trans. Amer. Math. Soc., 319, 631, (1990)
[70] Bergshoeff, E.; Sezgin, E.; Townsend, PK, Properties of the eleven-dimensional super membrane theory, Ann. Phys., 185, 330, (1988)
[71] Hohm, O.; Hull, C.; Zwiebach, B., Generalized metric formulation of double field theory, JHEP, 08, 008, (2010)
[72] Hull, CM; Julia, B., Duality and moduli spaces for time-like reductions, Nucl. Phys., B 534, 250, (1998)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.