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Levi-Civita connections for a class of spectral triples

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Abstract

We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence–uniqueness result for a class of modules of one-forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes–Landi deformations of spectral triples on the Connes–Dubois- Violette–Rieffel deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in Frohlich et al. (Commun Math Phys 203:119–184, 1999) failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frohlich et al. (1999).

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References

  1. Arnlind, J., Wilson, M.: Riemannian curvature of the noncommutative 3-sphere. J. Noncommut. Geom. 11(2), 507–536 (2017)

    Article  MathSciNet  Google Scholar 

  2. Arnlind, J., Wilson, M.: On the Gauss–Chern–Bonnet theorem for the noncommutative 4-sphere. J. Geom. Phys. 111, 126–141 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  3. Arnlind, J., Landi, G.: Projections, modules and connections for the noncommutative cylinder. arXiv:1901.07276

  4. Bassett, M.E., Majid, S.: Finite noncommutative geometries related to \(F_p[x]\). arXiv:1603.00426

  5. Beggs, E.J., Majid, S.: \(\ast \)-Compatible connections in noncommutative Riemannian geometry. J. Geom. Phys. 61, 95–124 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  6. Beggs, E.J., Majid, S.: Spectral triples from bimodule connections and Chern connections. J. Noncommut. Geom. 11(2), 669–701 (2017)

    Article  MathSciNet  Google Scholar 

  7. Beggs, E.J., Majid, S.: Quantum Riemannian Geometry, Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2019)

    Google Scholar 

  8. Bhowmick, J., Goswami, D., Joardar, S.: A new look at Levi-Civita connection in noncommutative geometry. arXiv:1606.08142

  9. Bhowmick, J., Goswami, D., Landi, G.: On the Koszul formula in noncommutative geometry (in preparation)

  10. Bhowmick, J.: Quantum isometry groups, Ph.D. Thesis. arXiv:0907.0618

  11. Chakraborty, P.S., Sinha, K.B.: Geometry on the quantum Heisenberg manifold. J. Funct. Anal. 203, 425–452 (2003)

    Article  MathSciNet  Google Scholar 

  12. Cipriani, F., Sauvagoet, J.-L.: Derivations as square roots of Dirichlet forms. J. Funct. Anal. 201, 78–120 (2003)

    Article  MathSciNet  Google Scholar 

  13. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  14. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230(3), 539–579 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  15. Connes, A., Rieffel, M.A.: Yang–Mills for non-commutative two-tori. Contemp. Math. 62, 237–266 (1987)

    Article  Google Scholar 

  16. Connes, A., Landi, G.: Noncommutative manifolds the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  17. Connes, A., Tretkoff, P.: The Gauss–Bonnet theorem for the noncommutative two torus. In: Consani, C., Connes, A. (eds.) Noncommutative Geometry, Arithmetic, and Related Topics, pp. 141–158. Johns Hopkins University Press, Baltimore (2011)

  18. Connes, A., Moscovici, H.: Modular curvature for noncommutative two-tori. J. Am. Math. Soc. 27, 639–684 (2014)

    Article  MathSciNet  Google Scholar 

  19. Connes, A., Fathizadeh, F.: The term \(a_4\) in the heat kernel expansion of noncommutative tori. arXiv:1611.09815

  20. Dabrowski, L., Sitarz, A.: Curved noncommutative torus and Gauss–Bonnet. J. Math. Phys. 54, 013518 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  21. Dabrowski, L., Sitarz, A.: An asymmetric noncommutative torus. SIGMA 11, 075 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Dubois-Violette, M., Michor, P.W.: Derivation et calcul differentiel non commutatif II. C. R. Acad. Sci. Paris Ser. I Math. 319, 927–931 (1994)

    MATH  Google Scholar 

  23. Fathizadeh, F., Khalkhali, M.: The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure. J. Noncommut. Geom. 6, 457–480 (2012)

    Article  MathSciNet  Google Scholar 

  24. Fathizadeh, F., Khalkhali, M.: Scalar curvature for the noncommutative two torus. J. Noncommut. Geom. 7, 1145–1183 (2013)

    Article  MathSciNet  Google Scholar 

  25. Fathizadeh, F., Khalkhali, M.: Scalar curvature for noncommutative four-tori. J. Noncommut. Geom. 9(2), 473–503 (2015)

    Article  MathSciNet  Google Scholar 

  26. Floricel, R., Ghorbanpour, A., Khalkhali, M.: The Ricci curvature in noncommutative geometry. J. Noncommut. Geom. arXiv:1612.06688

  27. Frohlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and differential geometry. Commun. Math. Phys. 193, 527–594 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  28. Frohlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and non-commutative geometry. Commun. Math. Phys. 203, 119–184 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  29. Goswami, D.: Quantum group of isometries in classical and noncommutative geometry. Commun. Math. Phys. 285, 141–160 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  30. Heckenberger, I., Schmuedgen, K.: Levi-Civita connections on the quantum groups \({\rm SL}_q(N)\) \(O_q(N)\) and \({\rm Sp}_q(N)\). Commun. Math. Phys. 185, 177–196 (1997)

    Article  ADS  Google Scholar 

  31. Kajiwara, T., Pinzari, C., Watatani, Y.: Jones index theory for Hilbert C\({}^*\)-bimodules and its equivalence with conjugation theory. J. Funct. Anal. 215, 1–49 (2004)

    Article  MathSciNet  Google Scholar 

  32. Kang, S.: The Yang–Mills functional and Laplace’s equation on quantum Heisenberg manifolds. J. Funct. Anal. 258(1), 307–327 (2010)

    Article  MathSciNet  Google Scholar 

  33. Kang, S., Luef, F., Packer, J.A.: Yang–Mills connections on quantum Heisenberg manifolds. arXiv:1810.08486

  34. Landi, G.: An Introduction to Noncommutative Spaces and their Geometries. Lecture notes in Physics monographs, vol. 51. Springer, Berlin (1997)

    Google Scholar 

  35. Lesch, M., Moscovici, H.: Modular curvature and Morita equivalence. Geom. Funct. Anal. 26(3), 818–873 (2016)

    Article  MathSciNet  Google Scholar 

  36. Li, H.: \( \theta \)-Deformations as compact quantum metric spaces. Commun. Math. Phys. 256, 213–238 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  37. Liu, Y.: Modular curvature for toric noncommutative manifolds. J. Noncommut. Geom. 12(2), 511–575 (2018)

    Article  MathSciNet  Google Scholar 

  38. Majid, S.: Noncommutative Riemannian and spin geometry of the standard q-sphere. Commun. Math. Phys. 256, 255–285 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  39. Peterka, M.A., Sheu, A.J.L.: On noncommutative Levi-Civita connections. Int. J. Geom. Methods. Mod. Phys. 14(5), 1750071 (2017)

    Article  MathSciNet  Google Scholar 

  40. Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531–562 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  41. Rieffel, M.A.: Deformation Quantization for Actions of \({\mathbb{R}}^{d}\), vol. 106(506). Memoirs of the American Mathematical Society, New York (1993)

    MATH  Google Scholar 

  42. Rosenberg, J.: Levi-Civita’s theorem for noncommutative tori. SIGMA 9, 071 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Skeide, M.: Hilbert modules in quantum electrodynamics and quantum probability. Commun. Math. Phys. 192(3), 569–604 (1998)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

D.G. is partially supported by the J.C. Bose National Fellowship.

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  1. Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India

    Jyotishman Bhowmick, Debashish Goswami & Sugato Mukhopadhyay

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  1. Jyotishman Bhowmick
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  2. Debashish Goswami
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  3. Sugato Mukhopadhyay
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Correspondence to Jyotishman Bhowmick.

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Bhowmick, J., Goswami, D. & Mukhopadhyay, S. Levi-Civita connections for a class of spectral triples. Lett Math Phys 110, 835–884 (2020). https://doi.org/10.1007/s11005-019-01241-4

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