Advertisement

Letters in Mathematical Physics

, Volume 102, Issue 1, pp 107–123 | Cite as

Nonassociative Strict Deformation Quantization of C*-Algebras and Nonassociative Torus Bundles

  • Keith C. Hannabuss
  • Varghese Mathai
Article
  • 124 Downloads

Abstract

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.

Mathematics Subject Classification (2010)

Primary 46L70 Secondary 47L70 46M15 18D10 46L08 46L55 

Keywords

nonassociative strict deformation quantization nonassociative torus nonassociative torus bundles T-duality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baez J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39, 145–205 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bouwknegt, P., Carey, A., Mathai, V., Murray, M., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–45 (2002). arXiv:hep-th/0106194Google Scholar
  3. 3.
    Bouwknegt, P., Evslin, J., Mathai, V.: T-duality: Topology change from H-flux. Commun. Math. Phys. 249, 383–415 (2004). arXiv:hep-th/0306062Google Scholar
  4. 4.
    Bouwknegt, P., Evslin, J., Mathai, V.: On the topology and H-flux of T-dual manifolds. Phys. Rev. Lett. 92, 181601 (2004). arXiv:hep-th/0312052Google Scholar
  5. 5.
    Bouwknegt, P., Hannabuss, K.C., Mathai, V.: Nonassociative tori and applications to T-duality. Commun. Math. Phys. 264, 41–69 (2006). arXiv:hep-th/0412092Google Scholar
  6. 6.
    Bouwknegt, P., Hannabuss, K.C., Mathai, V.: C*-algebras in tensor categories. Clay Math. Proc. 12, 127–165 (2011). arXiv:math.QA/0702802Google Scholar
  7. 7.
    Bouwknegt, P., Mathai, V.: D-branes, B-fields and twisted K-theory. J. High Energy Phys. 03, 007 (2000). arXiv:hep-th/0002023Google Scholar
  8. 8.
    Cornalba L., Schiappa R.: Nonassociative star product deformations for D-brane world-volumes in curved backgrounds. Commun. Math. Phys. 225(1), 33–66 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Echterhoff S., Nest R., Oyono-Oyono H.: Principal non-commutative torus bundles. Proc. Lond. Math. Soc. (3) 99, 1–31 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hannabuss, K.C., Mathai, V.: Noncommutative principal torus bundles via parametrised strict deformation quantization. AMS Proc. Symp. Pure Math. 81, 133–148 (2010). arXiv:0911.1886Google Scholar
  11. 11.
    Hannabuss, K.C., Mathai, V.: Parametrised strict deformation quantization of C*-bundles and Hilbert C*-modules. J. Aust. Math. Soc. 90(1), 25–38 (2011). arXiv:1007.4696Google Scholar
  12. 12.
    Herbst, M., Kling, A., Kreuzer, M.: Cyclicity of non-associative products on D-branes. J. High Energy Phys. (3), 003, 20p (2004)Google Scholar
  13. 13.
    Kasprzak P.: Rieffel deformation via crossed products. J. Funct. Anal. 257, 1288–1332 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Landstad M.B.: Duality theory for covariant systems. Trans. Am. Math. Soc. 248, 223–267 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Landstad M.B.: Quantization arising from abelian subgroups. Internat. J. Math. 5, 897–936 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Majid S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)zbMATHCrossRefGoogle Scholar
  17. 17.
    Mathai, V., Rosenberg, J.: T-duality for torus bundles via noncommutative topology. Commun. Math. Phys. 253, 705–721 (2005). arXiv:hep-th/0401168Google Scholar
  18. 18.
    Mathai, V., Rosenberg, J.: On mysteriously missing T-duals, H-flux and the T-duality group. In: Mo-Lin, G., Weiping, Z. (eds.) Differential Geometry and Physics, Nankai Tracts in Mathematics, Vol. 10, pp. 350–358. World Scientific, Singapore (2006). arXiv:hep-th/0409073Google Scholar
  19. 19.
    Mathai, V., Rosenberg, J.: 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, 123–158 (2006). arXiv:/hep-th/0508084Google Scholar
  20. 20.
    Nesterov A.I., Sabinin L.V.: Nonassociative geometry: Towards discrete structure of spacetime. Phys. Rev. D62, 081501 (2000)MathSciNetADSGoogle Scholar
  21. 21.
    Paal E.: Note on operadic non-associative deformations. J. Nonlinear Math. Phys. 13(suppl. 1), 87–92 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Raeburn, I., Williams, D.: Morita equivalence and continuous-trace C*-algebras. In: Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence, RI (1998)Google Scholar
  23. 23.
    Rieffel M.A.: Deformation quantization for actions of Rd. Mem. Am. Math. Soc. 106(506), 93 (1993)MathSciNetGoogle Scholar
  24. 24.
    Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nuclear Phys. B 479(1–2), 243–259 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Mathematical InstituteOxfordUK
  2. 2.Balliol CollegeOxfordUK
  3. 3.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia

Personalised recommendations