Skip to main content
SpringerLink
Log in

Membrane sigma-models and quantization of non-geometric flux backgrounds

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M . Starting from a suitable Courant sigma-model on an open membrane with target space M , regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma- model on the boundary of the membrane whose target space is the cotangent bundle T * M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich’s deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aldi and R. Heluani, Dilogarithms, OPE and twisted T-duality, arXiv:1105.4280 [INSPIRE].

  2. D. Andriot, M. Larfors, D. Lüst and P. Patalong, A ten-dimensional action for non-geometric fluxes, JHEP 09 (2011) 134 [arXiv:1106.4015] [INSPIRE].

    Article  ADS  Google Scholar 

  3. D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, A geometric action for non-geometric fluxes, Phys. Rev. Lett. 108 (2012) 261602 [arXiv:1202.3060] [INSPIRE].

    Article  ADS  Google Scholar 

  4. P. Aschieri, I. Bakovic, B. Jurčo and P. Schupp, Noncommutative gerbes and deformation quantization, J. Geom. Phys. 60 (2010) 1754 [hep-th/0206101] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. J.C. Baez and A.D. Lauda, Higher-dimensional algebra V: 2-groups, Theor. Appl. Categor. 12 (2004)423 [math.QA/0307200].

    MathSciNet  MATH  Google Scholar 

  6. J.C. Baez and A.S. Crans, Higher-dimensional algebra VI: Lie 2-Algebras, Theor. Appl. Categor. 12 (2004) 492 [math/0307263] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  7. R. Blumenhagen, Nonassociativity in string theory, arXiv:1112.4611 [INSPIRE].

  8. R. Blumenhagen and E. Plauschinn, Nonassociative gravity in string theory?, J. Phys. A A (2011)015401 [arXiv:1010.1263] [INSPIRE].

  9. R. Blumenhagen, A. Deser, E. Plauschinn and F. Rennecke, Bianchi identities for non-geometric fluxes — From quasi-Poisson structures to Courant algebroids, arXiv:1205.1522 [INSPIRE].

  10. R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].

    Article  ADS  Google Scholar 

  11. R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, Non-geometric fluxes, asymmetric strings and nonassociative geometry, J. Phys. A A 44 (2011) 385401 [arXiv:1106.0316] [INSPIRE].

    Article  ADS  Google Scholar 

  12. P. Bouwknegt and B. Jurčo, AKSZ construction of topological open p-brane action and Nambu brackets, arXiv:1110.0134 [INSPIRE].

  13. P. Bouwknegt and A.S. Pande, Topological T-duality and T-folds, Adv. Theor. Math. Phys. 13 (2009)1519 [arXiv:0810.4374] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  14. P. Bouwknegt, K.C. Hannabuss and V. Mathai, Nonassociative tori and applications to T-duality, Commun. Math. Phys. 264 (2006) 41 [hep-th/0412092] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. P. Bouwknegt, K.C. Hannabuss and V. Mathai, C ∗ -algebras in tensor categories, Clay Math. Proc. 12 (2011) 127 [math/0702802] [INSPIRE].

    MathSciNet  Google Scholar 

  16. J. Brodzki, V. Mathai, J.M. Rosenberg and R.J. Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys. 13 (2009) 497 [arXiv:0708.2648] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  17. A.S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212 (2000) 591 [math/9902090] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. A.S. Cattaneo and G. Felder, On the AKSZ formulation of the Poisson σ-model, Lett. Math. Phys. 56 (2001) 163 [math/0102108] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  19. A.S. Cattaneo and G. Felder, On the globalization of Kontsevich’s star product and the perturbative Poisson σ-model, Prog. Theor. Phys. Suppl. 144 (2001) 38 [hep-th/0111028] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. A. Chatzistavrakidis and L. Jonke, Matrix theory compactifications on twisted tori, Phys. Rev. D 85 (2012) 106013 [arXiv:1202.4310] [INSPIRE].

    ADS  Google Scholar 

  21. C. Condeescu, I. Florakis and D. Lüst, Asymmetric orbifolds, non-geometric fluxes and non-commutativity in closed string theory, JHEP 04 (2012) 121 [arXiv:1202.6366] [INSPIRE].

    Article  ADS  Google Scholar 

  22. L. Cornalba and R. Schiappa, Nonassociative star product deformations for D-brane world volumes in curved backgrounds, Commun. Math. Phys. 225 (2002) 33 [hep-th/0101219] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006)009 [hep-th/0512005] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. L. Davidovič and B. Sazdovič, Nongeometric background arising in the solution of Neumann boundary conditions, arXiv:1205.0921 [INSPIRE].

  25. J. DeBellis, C. Sämann and R.J. Szabo, Quantized Nambu-Poisson manifolds and n-Liealgebras, J. Math. Phys. 51 (2010) 122303 [arXiv:1001.3275] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. J. DeBellis, C. Sämann and R.J. Szabo, Quantized Nambu-Poisson manifolds in a 3-Liealgebra reduced model, JHEP 04 (2011) 075 [arXiv:1012.2236] [INSPIRE].

    Article  ADS  Google Scholar 

  27. M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. I. Ellwood and A. Hashimoto, Effective descriptions of branes on non-geometric tori, JHEP 12 (2006)025 [hep-th/0607135] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. D. Fiorenza, C.L. Rogers and U. Schreiber, A higher Chern-Weil derivation of AKSZ σ-models, arXiv:1108.4378 [INSPIRE].

  30. E. Getzler, Lie theory for nilpotent L ∞ algebras, Ann. Math. 170 (2009) 271 [math/0404003].

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006)91 [hep-th/0509003] [INSPIRE].

    Article  ADS  Google Scholar 

  32. P. Grange and S. Schäfer-Nameki, T-duality with H-flux: non-commutativity, T-folds and G × G structure,Nucl. Phys. B 770(2007)123 [hep-th/0609084][INSPIRE].

    Article  ADS  Google Scholar 

  33. N. Halmagyi, Non-geometric string backgrounds and worldsheet algebras, JHEP 07 (2008) 137 [arXiv:0805.4571] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. N. Halmagyi, Non-geometric backgrounds and the first order string σ-model, arXiv:0906.2891 [INSPIRE].

  35. K.C. Hannabuss and V. Mathai, Nonassociative strict deformation quantization of C ∗ -algebras and nonassociative torus bundles, arXiv:1012.2274 [INSPIRE].

  36. A. Henriques, Integrating L ∞ algebras, Compos. Math. 144 (2008) 1017 [math/0603563].

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Herbst, A. Kling and M. Kreuzer, Star products from open strings in curved backgrounds, JHEP 09 (2001) 014 [hep-th/0106159] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. M. Herbst, A. Kling and M. Kreuzer, Cyclicity of nonassociative products on D-branes, JHEP 03 (2004) 003 [hep-th/0312043] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. P.-M. Ho, Making nonassociative algebra associative, JHEP 11 (2001) 026 [hep-th/0103024] [INSPIRE].

    Article  ADS  Google Scholar 

  40. P.-M. Ho and Y.-T. Yeh, Noncommutative D-brane in nonconstant NS-NS B field background, Phys. Rev. Lett. 85 (2000) 5523 [hep-th/0005159] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  41. C. Hofman and J.-S. Park, Topological open membranes, hep-th/0209148 [INSPIRE].

  42. C. Hofman and J.-S. Park, BV quantization of topological open membranes, Commun. Math. . 249 (2004) 249 [hep-th/0209214] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. O. Hohm, T-duality versus gauge symmetry, Prog. Theor. Phys. Suppl. 188 (2011) 116 [arXiv:1101.3484] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  44. C. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. C. Hull and R. Reid-Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 09 (2009) 014 [arXiv:0902.4032] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435 [hep-th/9312059] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  48. N. Ikeda, Chern-Simons gauge theory coupled with BF theory, Int. J. Mod. Phys. A 18 (2003)2689 [hep-th/0203043] [INSPIRE].

    ADS  Google Scholar 

  49. R. Jackiw, 3-cocycle in mathematics and physics, Phys. Rev. Lett. 54 (1985) 159 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. D. Joyce, On manifolds with corners, arXiv:0910.3518.

  51. B. Jurčo and P. Schupp, Noncommutative Yang-Mills from equivalence of star products, Eur. Phys. J. C 14 (2000) 367 [hep-th/0001032] [INSPIRE].

    ADS  Google Scholar 

  52. B. Jurčo and P. Schupp, Nambu-σ-model and effective membrane actions, Phys. Lett. B 713 (2012)313 [arXiv:1203.2910] [INSPIRE].

    ADS  Google Scholar 

  53. B. Jurčo, P. Schupp and J. Wess, Noncommutative gauge theory for Poisson manifolds, Nucl. B 584 (2000) 784 [hep-th/0005005] [INSPIRE].

    ADS  Google Scholar 

  54. B. Jurčo, P. Schupp and J. Wess, Non-abelian noncommutative gauge theory via noncommutative extra dimensions, Nucl. Phys. B 604 (2001) 148 [hep-th/0102129] [INSPIRE].

    ADS  Google Scholar 

  55. B. Jurčo, P. Schupp and J. Wess, Noncommutative line bundle and Morita equivalence, Lett. Math. Phys. 61 (2002) 171 [hep-th/0106110] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  56. V. Kathotia, Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff Formula, I, Int. J. Math. 11 (2000) 523 [math/9811174].

    MathSciNet  MATH  Google Scholar 

  57. C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43 (2002) 341 [math/0104189] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. M. Kontsevich, Deformation quantization of Poisson manifolds. 1, Lett. Math. Phys. 66 (2003)157 [q-alg/9709040] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles, arXiv:0711.4106 [INSPIRE].

  60. G. Landi, F. Lizzi and R.J. Szabo, String geometry and the noncommutative torus, Commun. Math. Phys. 206 (1999) 603 [hep-th/9806099] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. F. Lizzi and R.J. Szabo, Noncommutative geometry and string duality, PoS(corfu98)073 (1998) [hep-th/9904064] [INSPIRE].

  62. Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom. 45 (1997)547 [math.DG/9508013].

    MathSciNet  Google Scholar 

  63. D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].

    Article  Google Scholar 

  64. D. Lüst, Twisted Poisson structures and non-commutative/non-associative closed string geometry, arXiv:1205.0100 [INSPIRE].

  65. D. Manchon, Poisson bracket, deformed bracket and gauge group actions in Kontsevich deformation quantization, math/0003004 [INSPIRE].

  66. V. Mathai and J.M. Rosenberg, T duality for torus bundles with H fluxes via noncommutative topology, Commun. Math. Phys. 253 (2004) 705 [hep-th/0401168] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  67. A.I. Nesterov, Three-cocycles, nonassociative gauge transformations and Dirac’s monopole, Phys. Lett. A 328 (2004) 110 [hep-th/0406073] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. J.-S. Park, Topological open p-branes, in Symplectic geometry and mirror symmetry, K. Fukaya et al. eds., World Scientific, Singapore (2001), p. 311, hep-th/0012141 [INSPIRE].

  69. M.A. Rieffel, Lie group convolution algebras as deformation quantizations of linear Poisson structures, Amer. J. Math. 112 (1990) 657.

    Article  MathSciNet  MATH  Google Scholar 

  70. D. Roytenberg, A note on quasi Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002) 123 [math/0112152] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  71. D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, math/0203110 [INSPIRE].

  72. D. Roytenberg, AKSZ-BV formalism and Courant algebroid-induced topological field theories, Lett. Math. Phys. 79 (2007) 143 [hep-th/0608150] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys. 46 (1998) 81 [math/9802118] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  74. C. Sämann and R.J. Szabo, Quantization of 2-plectic manifolds, arXiv:1106.1890 [INSPIRE].

  75. C. Sämann and R.J. Szabo, Groupoid quantization of loop spaces, arXiv:1203.5921 [INSPIRE].

  76. L.J. Santharoubane, Cohomology of Heisenberg Lie algebras, Proc. Amer. Math. Soc. 87 (1983)23.

    Article  MathSciNet  MATH  Google Scholar 

  77. P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9 (1994) 3129 [hep-th/9405110] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  78. N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  79. P. Ševera and A. Weinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  80. J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  81. B. Shoikhet, On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure, math/9903036.

  82. R.J. Szabo, Quantum field theory on noncommutative spaces, Phys. Rept. 378 (2003) 207 [hep-th/0109162] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  83. R.J. Szabo, Symmetry, gravity and noncommutativity, Class. Quant. Grav. 23 (2006) R199 [hep-th/0606233] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  84. L. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994)295 [hep-th/9301111] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  85. T. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005)133 [math/0304038].

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh, EH14 4AS, U.K.

    Dionysios Mylonas, Peter Schupp & Richard J. Szabo

  2. Maxwell Institute for Mathematical Sciences, Edinburgh, U.K.

    Dionysios Mylonas, Peter Schupp & Richard J. Szabo

  3. Jacobs University Bremen, 28759, Bremen, Germany

    Peter Schupp

Authors
  1. Dionysios Mylonas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Schupp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Richard J. Szabo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dionysios Mylonas.

Additional information

ArXiv ePrint: 1207.0926

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mylonas, D., Schupp, P. & Szabo, R.J. Membrane sigma-models and quantization of non-geometric flux backgrounds. J. High Energ. Phys. 2012, 12 (2012). https://doi.org/10.1007/JHEP09(2012)012

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP09(2012)012

Keywords

Search

Navigation