Advertisement

Letters in Mathematical Physics

, Volume 108, Issue 10, pp 2293–2301 | Cite as

Nearly associative deformation quantization

  • Dmitri Vassilevich
  • Fernando Martins Costa Oliveira
Article
  • 100 Downloads

Abstract

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.

Keywords

Deformation quantization Nonassociative algebras Nongeometric string backgrounds 

Mathematics Subject Classification

81S99 17A20 

Notes

Acknowledgements

We are grateful to Ivan Shestakov, Vlad Kupriyanov and Richard Szabo for fruitful discussions. One of the authors (D.V.) was supported in part by the Grant 2016/03319-6 of the São Paulo Research Foundation (FAPESP), by the Grant 303807/2016-4 of CNPq, by the RFBR Project 18-02-00149-a, and by the Tomsk State University Competitiveness Improvement Program. The other author (F.M.C.O.) was supported by CAPES.

References

  1. 1.
    Anderson, T.: A note on derivations of commutative algebras. Proc. Am. Math. Soc. 17, 1199–1202 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakas, I., Lüst, D.: 3-Cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua. JHEP 1401, 171 (2014).  https://doi.org/10.1007/JHEP01(2014)171. arXiv:1309.3172 [hep-th]ADSCrossRefGoogle Scholar
  3. 3.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. 1. Deformations of symplectic structures. Ann. Phys. 111, 61 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benkart, G.M., Osborn, J.M.: Flexible Lie-admissible algebras. J. Algebra 71, 11–31 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blumenhagen, R., Fuchs, M.: Towards a theory of nonassociative gravity. JHEP 1607, 019 (2016).  https://doi.org/10.1007/JHEP07(2016)019. arXiv:1604.03253 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blumenhagen, R., Plauschinn, E.: Nonassociative gravity in string theory? J. Phys. A 44, 015401 (2011).  https://doi.org/10.1088/1751-8113/44/1/015401. arXiv:1010.1263 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bojowald, M., Brahma, S., Buyukcam, U., Strobl, T.: States in non-associative quantum mechanics: uncertainty relations and semiclassical evolution. JHEP 1503, 093 (2015).  https://doi.org/10.1007/JHEP03(2015)093. arXiv:1411.3710 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bojowald, M., Brahma, S., Buyukcam, U., Strobl, T.: Monopole star products are non-alternative. JHEP 1704, 028 (2017).  https://doi.org/10.1007/JHEP04(2017)028. arXiv:1610.08359 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cornalba, L., Schiappa, R.: Nonassociative star product deformations for D-brane world volumes in curved backgrounds. Commun. Math. Phys. 225, 33 (2002).  https://doi.org/10.1007/s002201000569. arXiv:hep-th/0101219 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dito, G., Sternheimer, D.: Deformation quantization: genesis, developments and metamorphoses. In: IRMA Lectures in Mathematics and Theoretical Physics, vol. 1, pp. 9–54. de Gruyter, Berlin (2002). arXiv:math/0201168
  11. 11.
    Gunaydin, M., Lust, D., Malek, E.: Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes. JHEP 1611, 027 (2016).  https://doi.org/10.1007/JHEP11(2016)027. arXiv:1607.06474 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Günaydin, M., Minic, D.: Nonassociativity, Malcev algebras and string theory. Fortschr. Phys. 61, 873 (2013).  https://doi.org/10.1002/prop.201300010. arXiv:1304.0410 [hep-th]MathSciNetzbMATHGoogle Scholar
  13. 13.
    Herbst, M., Kling, A., Kreuzer, M.: Cyclicity of nonassociative products on D-branes. JHEP 0403, 003 (2004).  https://doi.org/10.1088/1126-6708/2004/03/003. arXiv:hep-th/0312043 ADSCrossRefGoogle Scholar
  14. 14.
    Jackiw, R.: 3-Cocycle in mathematics and physics. Phys. Rev. Lett. 54, 159 (1985).  https://doi.org/10.1103/PhysRevLett.54.159 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kontsevich, M.: Deformation quantization of Poisson manifolds. 1. Lett. Math. Phys. 66, 157 (2003).  https://doi.org/10.1023/B:MATH.0000027508.00421.bf. arXiv:q-alg/9709040 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kupriyanov, V.G.: Weak associativity and deformation quantization. Nucl. Phys. B 910, 240 (2016).  https://doi.org/10.1016/j.nuclphysb.2016.07.004. arXiv:1606.01409 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kupriyanov, V.G., Szabo, R.J.: \(\text{ G }_{2}\)-structures and quantization of non-geometric M-theory backgrounds. JHEP 1702, 099 (2017).  https://doi.org/10.1007/JHEP02(2017)099. arXiv:1701.02574 [hep-th]ADSCrossRefGoogle Scholar
  18. 18.
    Kupriyanov, V.G., Vassilevich, D.V.: Nonassociative Weyl star products. JHEP 1509, 103 (2015).  https://doi.org/10.1007/JHEP09(2015)103. arXiv:1506.02329 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lüst, D.: T-duality and closed string non-commutative (doubled) geometry. JHEP 1012, 084 (2010).  https://doi.org/10.1007/JHEP12(2010)084. arXiv:1010.1361 [hep-th]CrossRefzbMATHGoogle Scholar
  20. 20.
    Mylonas, D., Schupp, P., Szabo, R.J.: Membrane sigma-models and quantization of non-geometric flux backgrounds. JHEP 1209, 012 (2012).  https://doi.org/10.1007/JHEP09(2012)012. arXiv:1207.0926 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Mylonas, D., Schupp, P., Szabo, R.J.: Non-geometric fluxes, quasi-Hopf twist deformations and nonassociative quantum mechanics. J. Math. Phys. 55, 122301 (2014).  https://doi.org/10.1063/1.4902378. arXiv:1312.1621 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shestakov, I.P.: Speciality problem for Malcev algebras and Poisson Malcev algebras. In: Costa, R., et al. (eds.) IV Conference on Non-associative Algebra and Its Applications, São Paulo. Marcel Dekker, New York (2000)Google Scholar
  23. 23.
    Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Birkhäuser, Basel (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Springer, Berlin (2007)zbMATHGoogle Scholar
  25. 25.
    Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18, 523–557 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhevlakov, K.A., Slin’ko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic Press, New York (1982)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMCCUniversidade Federal do ABCSanto AndréBrazil
  2. 2.Department of PhysicsTomsk State UniversityTomskRussia

Personalised recommendations