Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes. (English) Zbl 1390.81434
Summary: We propose a non-associative phase space algebra for M-theory backgrounds with locally non-geometric fluxes based on the non-associative algebra of octonions. Our proposal is based on the observation that the non-associative algebra of the non-geometric \(R\)-flux background in string theory can be obtained by a proper contraction of the simple Malcev algebra generated by imaginary octonions. Furthermore, by studying a toy model of a four-dimensional locally non-geometric M-theory background which is dual to a twisted torus, we show that the non-geometric background is “missing” a momentum mode. The resulting seven-dimensional phase space can thus be naturally identified with the imaginary octonions. This allows us to interpret the full uncontracted algebra of imaginary octonions as the uplift of the string theory \(R\)-flux algebra to M-theory, with the contraction parameter playing the role of the string coupling constant \(g_{s}\).
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 83E30 | String and superstring theories in gravitational theory |
| 53Z05 | Applications of differential geometry to physics |
References:
| [1] | Graña, M., Flux compactifications in string theory: A comprehensive review, Phys. Rept., 423, 91, (2006) · doi:10.1016/j.physrep.2005.10.008 |
| [2] | Blumenhagen, R.; Körs, B.; Lüst, D.; Stieberger, S., Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept., 445, 1, (2007) · doi:10.1016/j.physrep.2007.04.003 |
| [3] | Lüst, D., T-duality and closed string non-commutative (doubled) geometry, JHEP, 12, 084, (2010) · Zbl 1294.81255 · doi:10.1007/JHEP12(2010)084 |
| [4] | Blumenhagen, R.; Plauschinn, E., Non-associative gravity in string theory?, J. Phys., A 44, 015401, (2011) · Zbl 1208.83101 |
| [5] | Blumenhagen, R.; Deser, A.; Lüst, D.; Plauschinn, E.; Rennecke, F., Non-geometric fluxes, asymmetric strings and non-associative geometry, J. Phys., A 44, 385401, (2011) · Zbl 1229.81220 |
| [6] | Bouwknegt, P.; Evslin, J.; Mathai, V., T duality: topology change from H flux, Commun. Math. Phys., 249, 383, (2004) · Zbl 1062.81119 · doi:10.1007/s00220-004-1115-6 |
| [7] | Bouwknegt, P.; Hannabuss, K.; Mathai, V., T duality for principal torus bundles, JHEP, 03, 018, (2004) · Zbl 1129.53013 · doi:10.1088/1126-6708/2004/03/018 |
| [8] | Bouwknegt, P.; Hannabuss, K.; Mathai, V., Non-associative tori and applications to T-duality, Commun. Math. Phys., 264, 41, (2006) · Zbl 1115.46063 · doi:10.1007/s00220-005-1501-8 |
| [9] | Bouwknegt, P.; Hannabuss, K.; Mathai, V., T-duality for principal torus bundles and dimensionally reduced Gysin sequences, Adv. Theor. Math. Phys., 9, 749, (2005) · Zbl 1129.53013 · doi:10.4310/ATMP.2005.v9.n5.a4 |
| [10] | Mathai, V.; Rosenberg, JM, T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group, Adv. Theor. Math. Phys., 10, 123, (2006) · Zbl 1111.81131 · doi:10.4310/ATMP.2006.v10.n1.a5 |
| [11] | Condeescu, C.; Florakis, I.; Lüst, D., Asymmetric orbifolds, non-geometric fluxes and non-commutativity in closed string theory, JHEP, 04, 121, (2012) · Zbl 1348.81362 · doi:10.1007/JHEP04(2012)121 |
| [12] | D. Andriot, M. Larfors, D. Lüst and P. Patalong, (Non-)commutative closed string on T-dual toroidal backgrounds, JHEP06 (2013) 021 [arXiv:1211.6437] [INSPIRE]. · Zbl 1342.81630 |
| [13] | Blair, CDA, Non-commutativity and non-associativity of the doubled string in non-geometric backgrounds, JHEP, 06, 091, (2015) · Zbl 1388.81489 · doi:10.1007/JHEP06(2015)091 |
| [14] | Bakas, I.; Lüst, D., T-duality, quotients and currents for non-geometric closed strings, Fortsch. Phys., 63, 543, (2015) · Zbl 1338.81311 · doi:10.1002/prop.201500031 |
| [15] | Blumenhagen, R.; Fuchs, M.; Haßler, F.; Lüst, D.; Sun, R., Non-associative deformations of geometry in double field theory, JHEP, 04, 141, (2014) · Zbl 1333.83126 · doi:10.1007/JHEP04(2014)141 |
| [16] | Mylonas, D.; Schupp, P.; Szabo, RJ, Membrane σ-models and quantization of non-geometric flux backgrounds, JHEP, 09, 012, (2012) · Zbl 1397.81409 · doi:10.1007/JHEP09(2012)012 |
| [17] | D. Mylonas, P. Schupp and R.J. Szabo, Non-associative geometry and twist deformations in non-geometric string theory, PoS(ICMP 2013)007 [arXiv:1402.7306] [INSPIRE]. |
| [18] | Lipkin, HJ; Weisberger, WI; Peshkin, M., Magnetic charge quantization and angular momentum, Annals Phys., 53, 203, (1969) · doi:10.1016/0003-4916(69)90279-6 |
| [19] | Grossman, B., A three cocycle in quantum mechanics, Phys. Lett., B 152, 93, (1985) · doi:10.1016/0370-2693(85)91146-3 |
| [20] | Jackiw, R., Three-cocycle in mathematics and physics, Phys. Rev. Lett., 54, 159, (1985) · doi:10.1103/PhysRevLett.54.159 |
| [21] | Jackiw, R., Magnetic sources and three cocycles (comment), Phys. Lett., B 154, 303, (1985) · doi:10.1016/0370-2693(85)90368-5 |
| [22] | Wu, Y-S; Zee, A., Cocycles and magnetic monopoles, Phys. Lett., B 152, 98, (1985) · doi:10.1016/0370-2693(85)91147-5 |
| [23] | Bakas, I.; Lüst, D., Three-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua, JHEP, 01, 171, (2014) · doi:10.1007/JHEP01(2014)171 |
| [24] | M. Günaydin and B. Zumino, Magnetic Charge and Non-Associative Algebras, in Old and New Problems in Fundamental Physics: Symposium in Honour of G.C. Wick, R.L. Cool, M. Jacob, E. Picasso and L.A.Radicati eds., Scuola Normale Superiore Publication (Quaderni), Pisa (1986), pp. 43-54, KISS preprint No. 198504333. |
| [25] | Günaydin, M.; Minic, D., Non-associativity, malcev algebras and string theory, Fortsch. Phys., 61, 873, (2013) · Zbl 1338.81323 |
| [26] | Günaydin, M.; Warner, NP, The G2 invariant compactifications in eleven-dimensional supergravity, Nucl. Phys., B 248, 685, (1984) · doi:10.1016/0550-3213(84)90618-7 |
| [27] | Wit, B.; Nicolai, H., the parallelizing S_{7}torsion in gauged N = 8 supergravity, Nucl. Phys., B 231, 506, (1984) · doi:10.1016/0550-3213(84)90517-0 |
| [28] | Cremmer, E.; Julia, B., the SO(8) supergravity, Nucl. Phys., B 159, 141, (1979) · doi:10.1016/0550-3213(79)90331-6 |
| [29] | Günaydin, M.; Sierra, G.; Townsend, PK, Exceptional supergravity theories and the MAGIC square, Phys. Lett., B 133, 72, (1983) · doi:10.1016/0370-2693(83)90108-9 |
| [30] | Günaydin, M.; Koepsell, K.; Nicolai, H., Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys., 221, 57, (2001) · Zbl 0992.22016 · doi:10.1007/PL00005574 |
| [31] | Günaydin, M.; Pavlyk, O., Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups, JHEP, 08, 101, (2005) · doi:10.1088/1126-6708/2005/08/101 |
| [32] | Günaydin, M., Lectures on spectrum generating symmetries and U-duality in supergravity, extremal black holes, quantum attractors and harmonic superspace, Springer Proc. Phys., 134, 31, (2010) · doi:10.1007/978-3-642-10736-8_2 |
| [33] | Blair, CDA; Malek, E., geometry and fluxes of SL(5) exceptional field theory, JHEP, 03, 144, (2015) · Zbl 1388.83749 · doi:10.1007/JHEP03(2015)144 |
| [34] | Kachru, S.; Schulz, MB; Tripathy, PK; Trivedi, SP, New supersymmetric string compactifications, JHEP, 03, 061, (2003) · doi:10.1088/1126-6708/2003/03/061 |
| [35] | Hull, CM, A geometry for non-geometric string backgrounds, JHEP, 10, 065, (2005) · doi:10.1088/1126-6708/2005/10/065 |
| [36] | Shelton, J.; Taylor, W.; Wecht, B., Nongeometric flux compactifications, JHEP, 10, 085, (2005) · doi:10.1088/1126-6708/2005/10/085 |
| [37] | Dabholkar, A.; Hull, C., Generalised T-duality and non-geometric backgrounds, JHEP, 05, 009, (2006) · doi:10.1088/1126-6708/2006/05/009 |
| [38] | Graña, M.; Minasian, R.; Petrini, M.; Waldram, D., T-duality, generalized geometry and non-geometric backgrounds, JHEP, 04, 075, (2009) · doi:10.1088/1126-6708/2009/04/075 |
| [39] | Andriot, D.; Larfors, M.; Lüst, D.; Patalong, P., A ten-dimensional action for non-geometric fluxes, JHEP, 09, 134, (2011) · Zbl 1301.81178 · doi:10.1007/JHEP09(2011)134 |
| [40] | Andriot, D.; Hohm, O.; Larfors, M.; Lüst, D.; Patalong, P., A geometric action for non-geometric fluxes, Phys. Rev. Lett., 108, 261602, (2012) · Zbl 1255.83123 · doi:10.1103/PhysRevLett.108.261602 |
| [41] | Andriot, D.; Hohm, O.; Larfors, M.; Lüst, D.; Patalong, P., Non-geometric fluxes in supergravity and double field theory, Fortsch. Phys., 60, 1150, (2012) · Zbl 1255.83123 · doi:10.1002/prop.201200085 |
| [42] | Blumenhagen, R.; Deser, A.; Plauschinn, E.; Rennecke, F.; Schmid, C., The intriguing structure of non-geometric frames in string theory, Fortsch. Phys., 61, 893, (2013) · Zbl 1338.81315 |
| [43] | D. Andriot and A. Betz, β-supergravity: a ten-dimensional theory with non-geometric fluxes and its geometric framework, JHEP12 (2013) 083 [arXiv:1306.4381] [INSPIRE]. |
| [44] | Chatzistavrakidis, A.; Deser, A.; Jonke, L., T-duality without isometry via extended gauge symmetries of 2D σ-models, JHEP, 01, 154, (2016) · Zbl 1388.81300 · doi:10.1007/JHEP01(2016)154 |
| [45] | Chatzistavrakidis, A.; Deser, A.; Jonke, L.; Strobl, T., Beyond the standard gauging: gauge symmetries of Dirac σ-models, JHEP, 08, 172, (2016) · Zbl 1390.81313 · doi:10.1007/JHEP08(2016)172 |
| [46] | A. Chatzistavrakidis, Non-isometric T-duality from gauged σ-models, arXiv:1604.03739 [INSPIRE]. |
| [47] | Boer, J.; Shigemori, M., Exotic branes and non-geometric backgrounds, Phys. Rev. Lett., 104, 251603, (2010) · doi:10.1103/PhysRevLett.104.251603 |
| [48] | Boer, J.; Shigemori, M., Exotic branes in string theory, Phys. Rept., 532, 65, (2013) · Zbl 1356.81193 · doi:10.1016/j.physrep.2013.07.003 |
| [49] | Hassler, F.; Lüst, D., Non-commutative/non-associative IIA (IIB) Q- and R-branes and their intersections, JHEP, 07, 048, (2013) · Zbl 1342.81634 · doi:10.1007/JHEP07(2013)048 |
| [50] | I. Bakhmatov, A. Kleinschmidt and E.T. Musaev, Non-geometric branes are DFT monopoles, arXiv:1607.05450 [INSPIRE]. · Zbl 1390.83368 |
| [51] | Condeescu, C.; Florakis, I.; Kounnas, C.; Lüst, D., Gauged supergravities and non-geometric Q/R-fluxes from asymmetric orbifold CFT‘s, JHEP, 10, 057, (2013) · Zbl 1342.83463 · doi:10.1007/JHEP10(2013)057 |
| [52] | Günaydin, M.; Gursey, F., Quark structure and octonions, J. Math. Phys., 14, 1651, (1973) · Zbl 0338.17004 · doi:10.1063/1.1666240 |
| [53] | D. Lüst, Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry, PoS(CORFU2011)086 [arXiv:1205.0100] [INSPIRE]. |
| [54] | Wecht, B., Lectures on nongeometric flux compactifications, Class. Quant. Grav., 24, s773, (2007) · Zbl 1128.81024 · doi:10.1088/0264-9381/24/21/S03 |
| [55] | Diaconescu, D-E; Moore, GW; Witten, E., \(E\)_{8}gauge theory and a derivation of k-theory from M-theory, Adv. Theor. Math. Phys., 6, 1031, (2003) · doi:10.4310/ATMP.2002.v6.n6.a2 |
| [56] | Günaydin, M., Exceptional realizations of Lorentz group: supersymmetries and leptons, Nuovo Cim., A 29, 467, (1975) · doi:10.1007/BF02734524 |
| [57] | M. Günaydin and O. Pavlyk, Quasiconformal Realizations of E_{6(6)}, E_{7(7)}, E_{8(8)}and SO(\(n\) + 3, m + 3), N ≥ 4 Supergravity and Spherical Vectors, Adv. Theor. Math. Phys.13 (2009)1895 [arXiv:0904.0784]. · Zbl 1208.83105 |
| [58] | Hull, C.; Zwiebach, B., Double field theory, JHEP, 09, 099, (2009) · doi:10.1088/1126-6708/2009/09/099 |
| [59] | Roček, M.; Verlinde, EP, Duality, quotients and currents, Nucl. Phys., B 373, 630, (1992) |
| [60] | Blumenhagen, R.; Deser, A.; Plauschinn, E.; Rennecke, F., Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids, JHEP, 02, 122, (2013) · Zbl 1342.81402 · doi:10.1007/JHEP02(2013)122 |
| [61] | G. Aldazabal, W. Baron, D. Marqués and C. Núñez, The effective action of Double Field Theory, JHEP11 (2011) 052 [Erratum ibid.11 (2011) 109] [arXiv:1109.0290] [INSPIRE]. · Zbl 1306.81178 |
| [62] | Aldazabal, G.; Graña, M.; Marqués, D.; Rosabal, JA, Extended geometry and gauged maximal supergravity, JHEP, 06, 046, (2013) · Zbl 1342.83439 · doi:10.1007/JHEP06(2013)046 |
| [63] | Berman, DS; Blair, CDA; Malek, E.; Perry, MJ, The OD, D geometry of string theory, Int. J. Mod. Phys., A 29, 1450080, (2014) · Zbl 1295.83052 · doi:10.1142/S0217751X14500808 |
| [64] | Berman, DS; Godazgar, H.; Godazgar, M.; Perry, MJ, The local symmetries of M-theory and their formulation in generalised geometry, JHEP, 01, 012, (2012) · Zbl 1306.81196 · doi:10.1007/JHEP01(2012)012 |
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