Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 143, 28 pages      arXiv:2007.01241      https://doi.org/10.3842/SIGMA.2020.143
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Riemannian Geometry of a Discretized Circle and Torus

Arkadiusz Bochniak, Andrzej Sitarz and Paweł Zalecki
Institute of Theoretical Physics, Jagiellonian University, prof. Stanisława Łojasiewicza 11, 30-348 Kraków, Poland

Received July 03, 2020, in final form December 15, 2020; Published online December 23, 2020

Abstract
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and metric compatibility condition in general and show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert $C^\ast$-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits.

Key words: noncommutative Riemannian geometry; linear connections; curvature.

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References

  1. Argota-Quiroz J.N., Majid S., Quantum gravity on polygons and $\mathbb{R}\times \mathbb{Z}_n$ FLRW model, arXiv:2005.13999.
  2. Balachandran A.P., Bimonte G., Ercolessi E., Landi G., Lizzi F., Sparano G., Teotonio-Sobrinho P., Noncommutative lattices as finite approximations, J. Geom. Phys. 18 (1996), 163-194, arXiv:hep-th/9510217.
  3. Barrett J.W., Matrix geometries and fuzzy spaces as finite spectral triples, J. Math. Phys. 56 (2015), 082301, 25 pages, arXiv:1502.05383.
  4. Barrett J.W., Gaunt J., Finite spectral triples for the fuzzy torus, arXiv:1908.06796.
  5. Beggs E., Majid S., $*$-compatible connections in noncommutative Riemannian geometry, J. Geom. Phys. 61 (2011), 95-124, arXiv:0904.0539.
  6. Beggs E., Majid S., Gravity induced from quantum spacetime, Classical Quantum Gravity 31 (2014), 035020, 39 pages, arXiv:1305.2403.
  7. Beggs E., Majid S., Spectral triples from bimodule connections and Chern connections, J. Noncommut. Geom. 11 (2017), 669-701, arXiv:1508.04808.
  8. Beggs E., Majid S., Quantum Riemannian geometry, Grundlehren der mathematischen Wissenschaften, Vol. 355, Springer, Cham, 2020.
  9. Bhowmick J., Goswami D., Joardar S., A new look at Levi-Civita connection in noncommutative geometry, arXiv:1606.08142.
  10. Bhowmick J., Goswami D., Landi G., Levi-Civita connections and vector fields for noncommutative differential calculi, Internat. J. Math. 31 (2020), 2050065, 23 pages, arXiv:2001.01545.
  11. Bhowmick J., Goswami D., Landi G., On the Koszul formula in noncommutative geometry, Rev. Math. Phys. 32 (2020), 2050032, 33 pages, arXiv:1910.09306.
  12. Bhowmick J., Goswami D., Mukhopadhyay S., Levi-Civita connections for a class of spectral triples, Lett. Math. Phys. 110 (2020), 835-884, arXiv:1809.06721.
  13. Bresser K., Müller-Hoissen F., Dimakis A., Sitarz A., Non-commutative geometry of finite groups, J. Phys. A: Math. Gen. 29 (1996), 2705-2735, arXiv:q-alg/9509004.
  14. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  15. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, Amer. Math. Soc., Providence, RI, 2008.
  16. Dąbrowski L., Sitarz A., An asymmetric noncommutative torus, SIGMA 11 (2015), 075, 11 pages, arXiv:1406.4645.
  17. Dimakis A., Müller-Hoissen F., Differential calculus and gauge theory on finite sets, J. Phys. A: Math. Gen. 27 (1994), 3159-3178, arXiv:hep-th/9401149.
  18. Dubois-Violette M., Madore J., Masson T., Mourad J., On curvature in noncommutative geometry, J. Math. Phys. 37 (1996), 4089-4102, arXiv:q-alg/9512004.
  19. Dubois-Violette M., Michor P.W., Dérivations et calcul différentiel non commutatif. II, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 927-931, arXiv:hep-th/9406166.
  20. Fathizadeh F., Khalkhali M., Scalar curvature for the noncommutative two torus, J. Noncommut. Geom. 7 (2013), 1145-1183, arXiv:1110.3511.
  21. Fathizadeh F., Khalkhali M., Curvature in noncommutative geometry, in Advances in Noncommutative Geometry: on the Occasion of Alain Connes' 70th Birthday, Editors A. Chamseddine, C. Consani, N. Higson, M. Khalkhali, H. Moscovici, G. Yu, Springer, Cham, 2019, 321-420, arXiv:1901.07438.
  22. Floricel R., Ghorbanpour A., Khalkhali M., The Ricci curvature in noncommutative geometry, J. Noncommut. Geom. 13 (2019), 269-296, arXiv:1612.06688.
  23. Glaser L., Scaling behaviour in random non-commutative geometries, J. Phys. A: Math. Theor. 50 (2017), 275201, 18 pages, arXiv:1612.00713.
  24. Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics. New Series m: Monographs, Vol. 51, Springer-Verlag, Berlin, 1997.
  25. Madore J., Masson T., Mourad J., Linear connections on matrix geometries, Classical Quantum Gravity 12 (1995), 1429-1440, arXiv:hep-th/9506183.
  26. Majid S., Noncommutative Riemannian geometry on graphs, J. Geom. Phys. 69 (2013), 74-93, arXiv:1011.5898.
  27. Majid S., Quantum gravity on a square graph, Classical Quantum Gravity 36 (2019), 245009, 23 pages, arXiv:1810.10831.
  28. Majid S., Quantum Riemannian geometry and particle creation on the integer line, Classical Quantum Gravity 36 (2019), 135011, 22 pages, arXiv:1811.06264.
  29. Mourad J., Linear connections in non-commutative geometry, Classical Quantum Gravity 12 (1995), 965-974, arXiv:hep-th/9410201.
  30. Pérez-Sánchez C.I., Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models, arXiv:1912.13288.
  31. Sitarz A., Metric on quantum spaces, Lett. Math. Phys. 31 (1994), 35-39.
  32. Torres E.L., Majid S., Quantum gravity and Riemannian geometry on the fuzzy sphere, arXiv:2004.14363.
  33. Woronowicz S.L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125-170.

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