Duality-Symmetric Approach to General Relativity and Supergravity
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

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We review the application of a duality-symmetric approach to gravity and supergravity with emphasizing benefits and disadvantages of the formulation. Contents of these notes includes: 1) Introduction with putting the accent on the role of dual gravity within $M$-theory; 2) Dualization of gravity with a cosmological constant in $\mathrm D=3$; 3) On-shell description of dual gravity in $\mathrm D>3$; 4) Construction of the duality-symmetric action for General Relativity with/without matter fields; 5) On-shell description of dual gravity in linearized approximation; 6) Brief summary of the paper.
Keywords: duality; gravity; supergravity.
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Alexei J. Nurmagambetov. Duality-Symmetric Approach to General Relativity and Supergravity. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a19/

[1] Aganagic M., Park J., Popescu C., Schwarz J. H., “World-volume action of the $M$ theory five-brane”, Nucl. Phys. B, 496 (1997), 191–214 ; arXiv:hep-th/9701166 | DOI | MR | Zbl

[2] Ajith K. M., Harikumar E., Sivakumar M., “Dual linearized gravity in arbitrary dimensions”, Class. Quant. Grav., 22 (2005), 5385–5396 ; arXiv:hep-th/0411202 | DOI | MR

[3] Arcos H. I., Pereira J. G., “Torsion gravity: a reappraisal”, Internat. J. Modern Phys. D, 13 (2004), 2193–2240 ; arXiv:gr-qc/0501017 | DOI | MR | Zbl

[4] Arnowitt R., Deser S., Misner C. W., “The dynamics of General Relativity”, Gravitation: an Introduction to Current Research, ed. L. Witten, Wiley, New York, 1962, 227–264 ; arXiv:gr-qc/0405109 | MR

[5] Bandos I., Berkovits N., Sorokin D., “Duality-symmetric eleven-dimensional supergravity and its coupling to M-branes”, Nucl. Phys. B, 522 (1998), 214–233 ; arXiv:hep-th/9711055 | DOI | MR | Zbl

[6] Bandos I., Lechner K., Nurmagambetov A., Pasti P., Sorokin D., Tonin M., “Covariant action for super-five-brane of $M$ theory”, Phys. Rev. Lett., 78 (1997), 4332–4335 ; arXiv:hep-th/9701149 | DOI | MR

[7] Bandos I. A., Nurmagambetov A. J., Sorokin D. P., “Various faces of type IIA supergravity”, Nucl. Phys. B, 676 (2004), 189–228 ; arXiv:hep-th/0307153 | DOI | MR | Zbl

[8] Banks T., Landskepticism: or why effective potentials don't count string models, arXiv:hep-th/0412129

[9] Bautier K., Deser S., Henneaux M., Seminara D., “No cosmological $\mathrm D=11$ supergravity”, Phys. Lett. B, 406 (1997), 49–53 ; arXiv:hep-th/9704131 | DOI | MR

[10] Bekaert X., Boulanger N., “Massless spin-two field S-duality”, Class. Quant. Grav., 20 (2003), S417–S424 ; arXiv:hep-th/0212131 | DOI | MR

[11] Bekaert X., Boulanger N., Henneaux M., “Consistent deformations of dual formulations of linearized gravity: a no-go result”, Phys. Rev. D, 67 (2003), 044010, 8 pp., ages ; arXiv:hep-th/0210278 | DOI | MR

[12] Bengtsson I., Kleppe A., “On chiral $p$-forms”, Int. J. Mod. Phys. A, 12 (1997), 3397–3412 ; arXiv:hep-th/9609102 | DOI | MR

[13] Bergshoeff E., Kallosh R., Ortin T., Roest D., Van Proeyen A., “New formulation of $\mathrm D=10$ supersymmetry and D8-O8 domain walls”, Class. Quant. Grav., 18 (2001), 3359–3382 ; arXiv:hep-th/0103233 | DOI | MR | Zbl

[14] Berkovits N., “Local actions with electric and magnetic sources”, Phys. Lett. B, 395 (1997), 28–35 ; arXiv:hep-th/9610134 | DOI | MR

[15] Blau M., “Killing vectors, constraints and conserved charges in Kaluza–Klein theories”, Class. Quant. Grav., 4 (1987), 1207–1221 | DOI | MR | Zbl

[16] Borisov A., Ogievetsky V., “Theory of dynamical affine and conformal symmetries as gravity theory”, Teoret. Mat. Fiz., 21:3 (1974), 329–342 (in Russian) | MR

[17] Boulanger N., Cnockaert S., Henneaux M., “A note on spin-s duality”, JHEP, 06 (2003), 060, 18 pp., ages ; arXiv:hep-th/0306023 | DOI | MR

[18] Brown J., Ganguli S., Ganor O. J., Helfgott C., “$E_{10}$ orbifolds”, JHEP, 0506 (2005), 057, 60 pp., ages ; arXiv:hep-th/0409037 | DOI | MR

[19] Brown J., Ganor O. J., Helfgott C., “$M$-theory and $E_{10}$: billiards, branes, and imaginary roots”, JHEP, 0408 (2004), 063, 68 pp., ages ; arXiv:hep-th/0401053 | DOI | MR

[20] Chan H.-M., Faridani J., Tsou S. T., “Generalized dual symmetry for non-Abelian Yang–Mills fields”, Phys. Rev. D, 53 (1996), 7293–7305 ; arXiv:hep-th/9512173 | DOI | MR

[21] Chaudhuri S., Hidden symmetry unmasked: matrix theory and E(11), arXiv:hep-th/0404235

[22] Christensen S. M., Duff M. J., “Quantizing gravity with a cosmological constant”, Nucl. Phys. B, 170 (1980), 480–506 | DOI | MR

[23] Cremmer E., Julia B., “The SO(8) supergravity”, Nucl. Phys. B, 159 (1979), 141–212 | DOI | MR

[24] Cremmer E., Julia B., Lü H., Pope C. N., “Dualisation of dualities, I”, Nucl. Phys. B, 523 (1998), 73–144 ; arXiv:hep-th/9710119 | DOI | MR | Zbl

[25] Cremmer E., Julia B., Lü H., Pope C. N., “Dualisation of dualities. II: Twisted self-duality of doubled fields and superdualities”, Nucl. Phys. B, 535 (1998), 242–292 ; arXiv:hep-th/9806106 | DOI | MR | Zbl

[26] Cremmer E., Julia B., Scherk J., “Supergravity theory in 11 dimensions”, Phys. Lett. B, 76 (1978), 409–412 | DOI

[27] Damour T., Cosmological singularities, billiards and Lorentzian Kac–Moody algebras, arXiv:gr-qc/0412105 | MR

[28] Damour T., Cosmological singularities, Einstein billiards and Lorentzian Kac–Moody algebras, arXiv:gr-qc/0501064 | MR

[29] Damour T., Henneaux M., Nicolai H., “$E_{10}$ and a “small tension expansion” of $M$ theory”, Phys. Rev. Lett., 89 (2002), 221601, 4 pp., ages ; arXiv:hep-th/0207267 | DOI | MR

[30] Damour T., Nicolai H., “Higher order M theory corrections and the Kac–Moody algebra $E_{10}$”, Class. Quant. Grav., 22 (2005), 2849–2880 ; arXiv:hep-th/0504153 | DOI | MR

[31] de Andrade V. C., Barbosa A. L., Pereira J. G., Gravitation and duality symmetry, arXiv:gr-qc/0501037

[32] Deser S., Drechsler W., “Generalized gauge field copies”, Phys. Lett. B, 86 (1979), 189–192 | DOI | MR

[33] Deser S., Gibbons G. W., “Born–Infeld–Einstein actions?”, Class. Quant. Grav., 15 (1998), L35–L39 ; arXiv:hep-th/9803049 | DOI | MR | Zbl

[34] Deser S., Jackiw R., t'Hooft G., “Three-dimensional Einsten gravity: dynamics of flat space”, Ann. Phys., 152 (1984), 220–235 | DOI | MR

[35] Deser S., Jackiw R., “Three-dimensional cosmological gravity: dynamics of constant curvature”, Ann. Phys., 153 (1984), 405–416 | DOI | MR

[36] Deser S., Jackiw R., “Topologically massive gauge theories”, Ann. Phys., 140 (1982), 372–411 | DOI | MR

[37] Deser S., Nepomechie R. I., “Electric-magnetic duality of conformal gravitation”, Phys. Lett. A, 97 (1983), 329–332 | DOI | MR

[38] Deser S., Seminara D., “Free spin 2 duality invariance cannot be extended to general relativity”, Phys. Rev. D, 71 (2005), 081502, 7 pp., ages ; arXiv:hep-th/0503030 | DOI | MR

[39] Deser S., Teitelboim C., “Duality transformations of Abelian and non-Abelian gauge fields”, Phys. Rev. D, 13 (1976), 1592–1597 | DOI | MR

[40] Deser S., Wilczek F., “Non-uniqueness of gauge-field potentials”, Phys. Lett. B, 65 (1976), 391–393 | DOI | MR

[41] de Wit B., Nicolai H., “Hidden symmetries, central charges and all that”, Class. Quant. Grav., 18 (2001), 3095–3112 ; arXiv:hep-th/0011239 | DOI | MR | Zbl

[42] de Wit B., Nicolai H., “Hidden symmetries in $\mathrm D=11$ supergravity”, Phys. Lett. B, 155 (1985), 47–53 | DOI | MR

[43] de Wit B., Nicolai H., “$\mathrm D=11$ supergravity with local SU(8) invariance”, Nucl. Phys. B, 274 (1986), 363–400 | DOI | MR

[44] Douglas M. R., “Basic results in vacuum statictics”, Comptes Rendus Physique, 5 (2004), 965–977 ; arXiv:hep-th/0409207 | DOI | MR

[45] Fradkin E. S., Gitman D. M., Shvartsman S. M., Quantum electrodynamics with unstable vacuum, Springer, Berlin, 1991

[46] Fradkin E. S., Tseytlin A.A., “Quantum equivalence of dual field theories”, Ann. Phys., 162 (1985), 31–48 | DOI | MR

[47] Francia D., Hull C. M., Higher-spin gauge fields and duality, arXiv:hep-th/0501236

[48] Fubini S., Veneziano G., “Algebraic treatment of subsidiary conditions in dual resonance model”, Ann. Phys., 63 (1971), 12–27 | DOI

[49] Gaberdiel M. R., Olive D. I., West P. C., “A class of Lorentzian Kac–Moody algebras”, Nucl. Phys. B, 645 (2002), 403–437 ; arXiv:hep-th/0205068 | DOI | MR | Zbl

[50] Garcia-Compean H., Obregon O., Plebansky J. F., Ramirez C., “Towards a gravitational analogue to $S$-duality in non-Abelian gauge theories”, Phys. Rev. D, 57 (1998), 7501–7506 ; arXiv:hep-th/9711115 | DOI | MR

[51] Garcia-Compean H., Obregon O., Ramirez C., “Gravitational duality in MacDowell–Mansouri gauge theory”, Phys. Rev. D, 58 (1998), 104012, 3 pp., ages ; arXiv:hep-th/9802063 | DOI | MR

[52] Green M. B., Schwarz J. H., Witten E., Superstring theory, Cambridge University Press, Cambridge, 1987

[53] Gross D. J., “High-energy symmetries of string theory”, Phys. Rev. Lett., 60 (1988), 1229–1232 | DOI | MR

[54] Henneaux M., Teitelboim C., “Duality in linearized gravity”, Phys. Rev. D, 71 (2005), 024018, 8 pp., ages ; arXiv:gr-qc/0408101 | DOI | MR

[55] Henry-Labordere P., Julia B., Paulot L., “Borcherds symmetries in $M$-theory”, JHEP, 04 (2002), 049, 31 pp., ages ; arXiv:hep-th/0212346 | DOI | MR

[56] Henry-Labordere P., Julia B., Paulot L., “Real Borcherds superalgebras and $M$-theory”, JHEP, 04 (2003), 060, 21 pp., ages ; arXiv:hep-th/0203070 | DOI | MR

[57] Hull C. M., “Gravitational duality, branes and charges”, Nucl. Phys. B, 509 (1998), 216–251 ; arXiv:hep-th/9705162 | DOI | MR | Zbl

[58] Hull C. M., “Strongly coupled gravity and duality”, Nucl. Phys. B, 583 (2000), 237–259 ; arXiv:hep-th/0004195 | DOI | MR | Zbl

[59] Hull C. M., “Duality in gravity and higher spin gauge fields”, JHEP, 09 (2001), 027, 25 pp., ages ; arXiv:hep-th/0107149 | DOI | MR

[60] Hull C. M., Townsend P. K., “Unity of superstring dualities”, Nucl. Phys. B, 438 (1995), 109–137 ; arXiv:hep-th/9410167 | DOI | MR | Zbl

[61] Iqbal A., Neitzke A., Vafa C., “A mysterious duality”, Adv. Theor. Math. Phys., 5 (2002), 769–808 ; arXiv:hep-th/0111068 | MR

[62] Julia B., “Kac–Moody symmetry of gravitational and supergravity theories”, Lect. Appl. Math., 21 (1985), 355–375 | MR

[63] Julia B., “Supergeometry and Kac–Moody algebras”, Vertex Operators in Mathematics and Physics, Springer, New York, 1985, 393–410 | MR

[64] Julia B., Levie J., Ray S., “Gravitational duality near de Sitter space”, JHEP, 11 (2005), 025, 13 pp., ages ; arXiv:hep-th/0507262 | DOI | MR

[65] Kac V., Infinite dimensional Lie algebras, Birkhäuser, Boston, 1983 | MR | Zbl

[66] Kaku M., Introduction to superstrings and $M$-theory, 2nd ed., Springer, Berlin, 1999 | MR

[67] Koepsell K., Nicolai H., Samtleben H., “An exceptional geometry for $d=11$ supergravity?”, Class. Quant. Grav., 17 (2000), 3689–3702 ; arXiv:hep-th/0006034 | DOI | MR | Zbl

[68] Linde A., “Inflation and string cosmology”, J. Phys. Conf. Ser., 24 (2005), 151–160 ; arXiv:hep-th/0503195 | DOI | MR

[69] Lü H., Pope C. N., Sezgin E., Stelle K. S., , Nucl. Phys. B, 456, 1995 arXiv:hep-th/9508042 | DOI | MR | Zbl

[70] Marcus N., Schwarz J. H., “Field theories that have no manifestly Lorentz-invariant formulation”, Phys. Lett. B, 115 (1982), 111–114 | DOI | MR

[71] Martin I., Restuccia A., “Duality symmetric actions and canonical quantization”, Phys. Lett. B, 323 (1994), 311–315 | DOI | MR

[72] Maznytsia A., Preitschopf C. R., Sorokin D. P., “Duality of self-dual actions”, Nucl. Phys. B, 539 (1999), 438–452 ; arXiv:hep-th/9805110 | DOI | MR | Zbl

[73] McClain B., Yu F., Wu Y. S., “Covariant quantization of chiral bosons and $\mathrm{Osp}(1,1\vert2)$ symmetry”, Nucl. Phys. B, 343 (1990), 689–704 | DOI | MR

[74] Mizoguchi S., “$E_{10}$ Symmetry in one-dimensional supergravity”, Nucl. Phys. B, 528 (1998), 238–264 ; arXiv:hep-th/9703160 | DOI | MR | Zbl

[75] Mkrtchyan H., Mkrtchyan R., Remarks on $E_{11}$ approach, arXiv:hep-th/0507183 | MR

[76] Moore G., Finite in all directions, arXiv:hep-th/9305139

[77] Nicolai H., “$\mathrm D=11$ supergravity with local SO(16) invariance”, Phys. Lett. B, 187 (1987), 316–320 | DOI | MR

[78] Nicolai H., “A hyperbolic Lie algebra from supergravity”, Phys. Lett. B, 276 (1992), 333–340 | DOI | MR

[79] Nicolai H., Peeters K., Zamaklar M., “Loop quantum gravity: an outside review”, Class. Quant. Grav., 22 (2005), R193–R247 ; arXiv:hep-th/0501114 | DOI | MR | Zbl

[80] Nicolai H., Warner N. P., “The structure of $N=16$ supergravity in two dimensions”, Comm. Math. Phys., 125 (1989), 369–384 | DOI | MR | Zbl

[81] Nieto J. A., “$S$-duality for linearized gravity”, Phys. Lett. A, 262 (1999), 274–281 ; arXiv:hep-th/9910049 | DOI | MR | Zbl

[82] Nurmagambetov A. J., “On the sigma-model structure of type IIA supergravity action in doubled field approach”, JETP Lett., 79 (2004), 191–195 ; arXiv:hep-th/0403100 | DOI

[83] Nurmagambetov A. J., Duality-symmetric gravity and supergravity: testing the PST approach, arXiv:hep-th/0407116 | MR

[84] Obers N. A., Pioline B., “$U$-duality and $M$-theory”, Phys. Rep., 318 (1999), 113–225 ; arXiv:hep-th/9809039 | DOI | MR

[85] Ogievetsky V. I., “Infinite-dimensional algebra of general covariance group as the closure of finite dimensional algebras of conformal and linear groups”, Lett. Nuov. Cim., 8 (1973), 988–990 | DOI | MR

[86] Pasti P., Sorokin D. P., Tonin M., “Note on manifest Lorentz and general coordinate invariance in duality symmetric models”, Phys. Lett. B, 352 (1995), 59–63 ; arXiv:hep-th/9503182 | DOI

[87] Pasti P., Sorokin D. P., Tonin M., “Duality symmetric actions with manifest space-time symmetries”, Phys. Rev. D, 52 (1995), 4277–4281 ; arXiv:hep-th/9506109 | DOI | MR

[88] Pasti P., Sorokin D. P., Tonin M., “On Lorentz invariant actions for chiral $p$-forms”, Phys. Rev. D, 55 (1997), 6292–6298 ; arXiv:hep-th/9611100 | DOI | MR

[89] Peldan P., “Actions for gravity, with generalizations: a review”, Class. Quant. Grav., 11 (1994), 1087–1132 ; arXiv:gr-qc/9305011 | DOI | MR

[90] Polchinski J., String theory, Cambridge University Press, Cambridge, 1998

[91] Pope C. N., Lectures on Kaluza–Klein, http://faculty.physics.tamu.edu/pope/

[92] Salam A., Sezgin E., Supergravities in diverse dimensions, Vol. 1, World Scientific, Singapore, 1989 | Zbl

[93] Schnakenburg I., West P., “Kac–Moody symmetries of IIB supergravity”, Phys. Lett. B, 517 (2001), 421–428 ; arXiv:hep-th/0107181 | DOI | MR | Zbl

[94] Schnakenburg I., West P., “Massive IIA supergravity as a non-linear realisation”, Phys. Lett. B, 540 (2002), 137–145 ; arXiv:hep-th/0204207 | DOI | MR | Zbl

[95] Schwarz J. H., Sen A., “Duality symmetric actions”, Nucl. Phys. B, 411 (1994), 35–63 ; arXiv:hep-th/9304154 | DOI | MR | Zbl

[96] Sorokin D., “Lagrangian description of duality-symmetric fields”, Advances in the Interplay between Quantum and Gravity Physics, eds. P. G. Bergmann and V. de Sabbata, Kluwer Academic Pub., 2002, 365–385 | Zbl

[97] Schrödinger E., Space-time structure, Cambridge Univ. Press, 1950 | MR | Zbl

[98] Susskind L., The anthropic landscape of string theory, arXiv:hep-th/0302219

[99] Thirring W., A course on mathematical physics. Vol. 2. Field theory, 2nd ed., Springer, Berlin, 1986 | MR

[100] Townsend P. K., “The eleven-dimensional supermembrane revisited”, Phys. Lett. B, 350 (1995), 184–187 ; arXiv:hep-th/9501068 | DOI | MR

[101] Virasoro M. A., “Subsidiary conditions and ghosts in dual-resonance model”, Phys. Rev. D, 1 (1970), 2933–2936 | DOI

[102] West P., “Hidden superconformal symmetry in $M$ theory”, JHEP, 08 (2000), 007, 21 pp., ages ; arXiv:hep-th/0005270 | DOI | MR | Zbl

[103] West P., “$\mathrm{E}_{11}$ and $M$ theory”, Class. Quant. Grav., 18 (2001), 4443–4460 ; . arXiv:hep-th/0104081 | DOI | MR | Zbl

[104] West P., “Very extended $\mathrm{E}_8$ and $\mathrm{A}_8$ at low levels, gravity and supergravity”, Class. Quant. Grav., 20 (2002), 2393–2406 ; arXiv:hep-th/0212291 | DOI | MR

[105] Witten E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311:1 (1988/89), 46–78 | DOI | MR

[106] Witten E., “String theory dynamics in various dimensions”, Nucl. Phys. B, 443 (1995), 85–126 ; arXiv:hep-th/9503124 | DOI | MR | Zbl

[107] Wotzasek C., “The Wess–Zumino term for chiral bosons”, Phys. Rev. Lett., 66 (1991), 129–132 | DOI | MR | Zbl

[108] Wu T. T., Yang C. N., “Some remarks about unquantized non-Abelian gauge fields”, Phys. Rev. D, 12 (1975), 3843–3844 | DOI | MR

[109] Zwanziger D., “Local Lagrangian quantum field theory of electric and magnetic charges”, Phys. Rev. D, 3 (1971), 880–890 | DOI | MR