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Non-commutativity parameters depend not only on the effective coordinate but on its T-dual as well. (English) Zbl 1298.81263
Summary: We extend our investigations of the open string propagation in the weakly curved background to the case when Kalb-Ramond field, beside the infinitesimal term linear in coordinate \(B_{{\mu}{\nu}{\rho}}x^{{\rho}}\), contains the constant term \(b_{{\mu}{\nu}} \neq 0\). In two previously investigated cases, for the flat background (\(b_{{\mu}{\nu}} \neq 0\) and \(B_{{\mu}{\nu}{\rho}}=0\)) and the weakly curved one (\(b_{{\mu}{\nu}} \neq 0\) and \(B_{{\mu}{\nu}{\rho}}=0\)) the effective metric is constant and the effective Kalb-Ramond field is zero. In the present article (\(b_{{\mu}{\nu}} \neq 0\) and \(B_{{\mu}{\nu}{\rho}}=0\)) the effective metric is coordinate dependent and there exists non-trivial effective Kalb-Ramond field. It depends on the \(\sigma\)-integral of the effective momentum \(P_{{\mu}}(\sigma)=\int_{0}^{{\sigma}}d\eta p_{{\mu}}(\eta)\), which is in fact T-dual of the effective coordinate, \( {P_\mu }=\kappa {g_{\mu \nu }}{\tilde{q}^\nu } \). Beside the standard coordinate dependent term \(\theta^{{\mu}{\nu}}(q)\), in the non-commutativity parameter, which is nontrivial only on the string end-points, there are additional \(P_{{\mu}}\) (or \( {\tilde{q}^\mu } \)) dependent terms which are nontrivial both at the string endpoints and at the string interior. The additional terms are infinitesimally small. The part of one of these terms has been obtained in ref. [The authors, Phys. Rev. D 83, No. 6, Article No. 066014, 15 p. (2011), arxiv:1004.4483] and the others are our improvements.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T75 Noncommutative geometry methods in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
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[1] Connes, A.; Douglas, MR; Schwarz, AS, Noncommutative geometry and matrix theory: compactification on tori, JHEP, 02, 003, (1998)
[2] Douglas, MR; Hull, CM, D-branes and the noncommutative torus, JHEP, 02, 008, (1998)
[3] Schomerus, V., D-branes and deformation quantization, JHEP, 06, 030, (1999)
[4] Schomerus, V., Lectures on branes in curved backgrounds, Class. Quant. Grav., 19, 5781, (2002)
[5] Ardalan, F.; Arfaei, H.; Sheikh-Jabbari, MM, Noncommutative geometry from strings and branes, JHEP, 02, 016, (1999)
[6] Chu, C-S; Ho, P-M, Noncommutative open string and D-brane, Nucl. Phys., B 550, 151, (1999)
[7] Seiberg, N.; Witten, E., String theory and noncommutative geometry, JHEP, 09, 032, (1999)
[8] Ardalan, F.; Arfaei, H.; Sheikh-Jabbari, MM, Dirac quantization of open strings and noncommutativity in branes, Nucl. Phys., B 576, 578, (2000)
[9] Chu, C-S; Ho, P-M, Constrained quantization of open string in background B field and noncommutative D-brane, Nucl. Phys., B 568, 447, (2000)
[10] Lee, T., Canonical quantization of open string and noncommutative geometry, Phys. Rev., D 62, 024022, (2000)
[11] Sazdović, B., Dilaton field induces commutative dp-brane coordinate, Eur. Phys. J., C 44, 599, (2005)
[12] Nikolić, B.; Sazdović, B., Gauge symmetries decrease the number of dp-brane dimensions, Phys. Rev., D 74, 045024, (2006)
[13] Nikolić, B.; Sazdović, B., Gauge symmetries decrease the number of dp-brane dimensions. II. inclusion of the Liouville term, Phys. Rev., D 75, 085011, (2007)
[14] Nikolić, B.; Sazdović, B., Noncommutativity in the space-time extended by Liouville field, Adv. Theor. Math. Phys., 14, 1, (2010)
[15] Boer, J.; Grassi, PA; Nieuwenhuizen, P., Non-commutative superspace from string theory, Phys. Lett., B 574, 98, (2003)
[16] Nikolić, B.; Sazdović, B., Type I background fields in terms of type IIB ones, Phys. Lett., B 666, 400, (2008)
[17] Nikolić, B.; Sazdović, B., D 5-brane type-I superstring background fields in terms of type IIB ones by canonical method and T -duality approach, Nucl. Phys., B 836, 100, (2010)
[18] Nikolić, B.; Sazdović, B., Noncommutativity relations in type IIB theory and their supersymmetry, JHEP, 08, 037, (2010)
[19] Cornalba, L.; Schiappa, R., Nonassociative star product deformations for D-brane world-volumes in curved backgrounds, Commun. Math. Phys., 225, 33, (2002)
[20] Herbst, M.; Kling, A.; Kreuzer, M., Star products from open strings in curved backgrounds, JHEP, 09, 014, (2001)
[21] Alekseev, AY; Recknagel, A.; Schomerus, V., Non-commutative world-volume geometries: branes on SU(2) and fuzzy spheres, JHEP, 09, 023, (1999)
[22] Ho, P-M; Yeh, Y-T, Noncommutative D-brane in non-constant NS-NS B field background, Phys. Rev. Lett., 85, 5523, (2000)
[23] Davidović, Lj; Sazdović, B., Noncommutativity in weakly curved background by canonical methods, Phys. Rev., D 83, 066014, (2011)
[24] Sazdović, B., Chiral symmetries of the WZNW model by Hamiltonian methods, Phys. Lett., B 352, 64, (1995)
[25] Fradkin, ES; Tseytlin, AA, Effective field theory from quantized strings, Phys. Lett., B 158, 316, (1985)
[26] Fradkin, ES; Tseytlin, AA, Quantum string theory effective action, Nucl. Phys., B 261, 1, (1985)
[27] Callan, CG; Martinec, EJ; Perry, MJ; Friedan, D., Strings in background fields, Nucl. Phys., B 262, 593, (1985)
[28] Banks, T.; Nemeschansky, D.; Sen, A., Dilaton coupling and BRST quantization of bosonic strings, Nucl. Phys., B 277, 67, (1986)
[29] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, Cambridge U.K. (1987).
[30] J. Polchinski, String theory, Cambridge University Press, Cambridge U.K. (1998).
[31] Zwiebach, A First Course in String Theory, Cambridge University Press, Cambridge U.K. (2004).
[32] Lj. Davidović and B. Sazdović, T-dual-coordinate dependence makes the effective Kalb-Ramond field nontrivial, arXiv:1105.2809 [SPIRES].
[33] Giveon, A.; Porrati, M.; Rabinovici, E., Target space duality in string theory, Phys. Rept., 244, 77, (1994)
[34] Sheikh-Jabbari, MM, A note on T -duality, open strings in B-field background and canonical transformations, Phys. Lett., B 474, 292, (2000)
[35] Kontsevich, M., Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., 66, 157, (2003)
[36] Sheikh-Jabbari, MM; Shirzad, A., Boundary conditions as Dirac constraints, Eur. Phys. J., C 19, 383, (2001)
[37] Fradkin, ES; Tseytlin, AA, Nonlinear electrodynamics from quantized strings, Phys. Lett., B 163, 123, (1985)
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