×

zbMATH — the first resource for mathematics

(Non-)commutative closed string on T-dual toroidal backgrounds. (English) Zbl 1342.81630
Summary: We investigate the connection between (non-)geometry and (non-)commutativity of the closed string. To this end, we solve the classical string on three T-dual toroidal backgrounds: a torus with \(H\)-flux, a twisted torus and a non-geometric background with \(Q\)-flux. In all three situations we work under the assumption of a dilute flux and consider quantities to linear order in the flux density. Furthermore, we perform the first steps of a canonical quantization for the twisted torus, to derive commutators of the string expansion modes. We use them as well as T-duality to determine, in the non-geometric background, a commutator of two string coordinates, which turns out to be non-vanishing. We relate this non-commutativity to the closed string boundary conditions, and the non-geometric \(Q\)-flux.

MSC:
81T75 Noncommutative geometry methods in quantum field theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Hellerman, S.; McGreevy, J.; Williams, B., Geometric constructions of nongeometric string theories, JHEP, 01, 024, (2004) · Zbl 1243.81156
[2] Dabholkar, A.; Hull, C., Duality twists, orbifolds and fluxes, JHEP, 09, 054, (2003)
[3] Shelton, J.; Taylor, W.; Wecht, B., Nongeometric flux compactifications, JHEP, 10, 085, (2005)
[4] Andriot, D.; Larfors, M.; Lüst, D.; Patalong, P., A ten-dimensional action for non-geometric fluxes, JHEP, 09, 134, (2011) · Zbl 1301.81178
[5] Dabholkar, A.; Hull, C., Generalised T-duality and non-geometric backgrounds, JHEP, 05, 009, (2006)
[6] Hull, C.; Zwiebach, B., Double field theory, JHEP, 09, 099, (2009)
[7] Hull, C.; Zwiebach, B., The gauge algebra of double field theory and Courant brackets, JHEP, 09, 090, (2009)
[8] Hohm, O.; Hull, C.; Zwiebach, B., Background independent action for double field theory, JHEP, 07, 016, (2010) · Zbl 1290.81069
[9] Hohm, O.; Hull, C.; Zwiebach, B., Generalized metric formulation of double field theory, JHEP, 08, 008, (2010) · Zbl 1291.81255
[10] Andriot, D.; Hohm, O.; Larfors, M.; Lüst, D.; Patalong, P., A geometric action for non-geometric fluxes, Phys. Rev. Lett., 108, 261602, (2012)
[11] 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
[12] Chu, C-S; Ho, P-M, Noncommutative open string and D-brane, Nucl. Phys., B 550, 151, (1999) · Zbl 0947.81136
[13] Schomerus, V., D-branes and deformation quantization, JHEP, 06, 030, (1999) · Zbl 0961.81066
[14] Seiberg, N.; Witten, E., String theory and noncommutative geometry, JHEP, 09, 032, (1999) · Zbl 0957.81085
[15] F. Ardalan, H. Arfaei and M. Sheikh-Jabbari, Mixed branes and M(atrix) theory on noncommutative torus, hep-th/9803067 [INSPIRE]. · Zbl 0965.81124
[16] Ardalan, F.; Arfaei, H.; Sheikh-Jabbari, M., Noncommutative geometry from strings and branes, JHEP, 02, 016, (1999) · Zbl 0965.81124
[17] Ardalan, F.; Arfaei, H.; Sheikh-Jabbari, M., Dirac quantization of open strings and noncommutativity in branes, Nucl. Phys., B 576, 578, (2000) · Zbl 1056.81539
[18] Blumenhagen, R.; Plauschinn, E., Nonassociative gravity in string theory?, J. Phys., A 44, 015401, (2011) · Zbl 1208.83101
[19] Lüst, D., T-duality and closed string non-commutative (doubled) geometry, JHEP, 12, 084, (2010) · Zbl 1294.81255
[20] Blumenhagen, R.; Deser, A.; Lüst, D.; Plauschinn, E.; Rennecke, F., Non-geometric fluxes, asymmetric strings and nonassociative geometry, J. Phys., A 44, 385401, (2011) · Zbl 1229.81220
[21] 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
[22] Bouwknegt, P.; Mathai, V., D-branes, B fields and twisted k-theory, JHEP, 03, 007, (2000) · Zbl 0959.81037
[23] Mathai, V.; Rosenberg, JM, T duality for torus bundles with H fluxes via noncommutative topology, Commun. Math. Phys., 253, 705, (2004) · Zbl 1078.58006
[24] V. Mathai and J.M. Rosenberg, On mysteriously missing T-duals, H-flux and the T-duality group, hep-th/0409073 [INSPIRE]. · Zbl 1121.81107
[25] Bouwknegt, P.; Hannabuss, K.; Mathai, V., Nonassociative tori and applications to T-duality, Commun. Math. Phys., 264, 41, (2006) · Zbl 1115.46063
[26] Brodzki, J.; Mathai, V.; Rosenberg, JM; Szabo, RJ, Noncommutative correspondences, duality and D-branes in bivariant k-theory, Adv. Theor. Math. Phys., 13, 497, (2009) · Zbl 1166.81032
[27] C. Sämann and R.J. Szabo, Groupoid quantization of loop spaces, PoS (CORFU2011) 046 [arXiv:1203.5921] [INSPIRE].
[28] Sämann, C.; Szabo, RJ, Groupoids, loop spaces and quantization of 2-plectic manifolds, Rev. Math. Phys., 25, 1330005, (2003) · Zbl 1276.81085
[29] Grange, P.; Schäfer-Nameki, S., T-duality with H-flux: non-commutativity, T-folds and G×G structure, Nucl. Phys., B 770, 123, (2007) · Zbl 1117.81351
[30] Mylonas, D.; Schupp, P.; Szabo, RJ, Membrane σ-models and quantization of non-geometric flux backgrounds, JHEP, 09, 012, (2012)
[31] Chatzistavrakidis, A.; Jonke, L., Matrix theory origins of non-geometric fluxes, JHEP, 02, 040, (2013) · Zbl 1342.81410
[32] Kachru, S.; Schulz, MB; Tripathy, PK; Trivedi, SP, New supersymmetric string compactifications, JHEP, 03, 061, (2003)
[33] Lowe, DA; Nastase, H.; Ramgoolam, S., Massive IIA string theory and matrix theory compactification, Nucl. Phys., B 667, 55, (2003) · Zbl 1059.81153
[34] Buscher, T., A symmetry of the string background field equations, Phys. Lett., B 194, 59, (1987)
[35] Buscher, T., Path integral derivation of quantum duality in nonlinear σ-models, Phys. Lett., B 201, 466, (1988)
[36] Hull, C., A geometry for non-geometric string backgrounds, JHEP, 10, 065, (2005)
[37] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Vol. 1: Introduction, Cambridge University Press, Cambridge U.K. (1987), pg. 469. · Zbl 0619.53002
[38] Marchesano, F.; Schulgin, W., Non-geometric fluxes as supergravity backgrounds, Phys. Rev., D 76, 041901, (2007)
[39] Giddings, SB; Kachru, S.; Polchinski, J., Hierarchies from fluxes in string compactifications, Phys. Rev., D 66, 106006, (2002)
[40] Davidovic, L.; Sazdovic, B., Nongeometric background arising in the solution of Neumann boundary conditions, Eur. Phys. J., C 72, 2199, (2012)
[41] L. Davidovic and B. Sazdovic, T-duality in the weakly curved background, arXiv:1205.1991 [INSPIRE].
[42] D.C. Thompson, T-duality invariant approaches to string theory, arXiv:1012.4393 [INSPIRE].
[43] Groot Nibbelink, S.; Patalong, P., A Lorentz invariant doubled worldsheet theory, Phys. Rev., D 87, 041902, (2013)
[44] Duff, M., Duality rotations in string theory, Nucl. Phys., B 335, 610, (1990) · Zbl 0967.81519
[45] Andriot, D.; Goi, E.; Minasian, R.; Petrini, M., Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory, JHEP, 05, 028, (2011) · Zbl 1296.81075
[46] Todorov, I., ’quantization is a mystery’, Bulg. J. Phys., 39, 107, (2012)
[47] C. Esposito, Lectures on deformation quantization of Poisson manifolds, arXiv:1207.3287.
[48] Axenides, M.; Floratos, E., Nambu-Lie 3-algebras on fuzzy 3-manifolds, JHEP, 02, 039, (2009) · Zbl 1245.81285
[49] Nikolić, B.; Sazdović, B., Fermionic T-duality and momenta noncommutativity, Phys. Rev., D 84, 065012, (2011)
[50] Shirzad, A.; Bakhshi, A.; Koohsarian, Y., Symplectic quantization of massive bosonic string in background B-field, Mod. Phys. Lett., A 27, 1250073, (2012) · Zbl 1257.81043
[51] Schulz, MB, T-folds, doubled geometry and the SU(2) WZW model, JHEP, 06, 158, (2012)
[52] Alvarez, E.; Álvarez-Gaumé, L.; Lozano, Y., A canonical approach to duality transformations, Phys. Lett., B 336, 183, (1994)
[53] Graña, M.; Minasian, R.; Petrini, M.; Waldram, D., T-duality, generalized geometry and non-geometric backgrounds, JHEP, 04, 075, (2009)
[54] Blumenhagen, R.; Deser, A.; Plauschinn, E.; Rennecke, F., A bi-invariant Einstein-Hilbert action for the non-geometric string, Phys. Lett., B 720, 215, (2013) · Zbl 1372.83056
[55] 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
[56] Dall’Agata, G.; Prezas, N.; Samtleben, H.; Trigiante, M., Gauged supergravities from twisted doubled tori and non-geometric string backgrounds, Nucl. Phys., B 799, 80, (2008) · Zbl 1292.83052
[57] Aldazabal, G.; Baron, W.; Marques, D.; Núñez, C., The effective action of double field theory, JHEP, 11, 052, (2011) · Zbl 1306.81178
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.