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

SIGMA 16 (2020), 109, 10 pages      arXiv:2010.15367      https://doi.org/10.3842/SIGMA.2020.109
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Real Part of Twisted-by-Grading Spectral Triples

Manuele Filaci a and Pierre Martinetti b
a) Università di Genova - Dipartimento di Fisica and INFN sezione di Genova, Italy
b) Università di Genova - Dipartimento di Matematica and INFN sezione di Genova, Italy

Received September 03, 2020, in final form October 23, 2020; Published online October 29, 2020

Abstract
After a brief review on the applications of twisted spectral triples to physics, we adapt to the twisted case the notion of real part of a spectral triple. In particular, when one twists a usual spectral triple by its grading, we show that - depending on the $KO$ dimension - the real part is either twisted as well, or is the intersection of the initial algebra with its opposite. We illustrate this result with the spectral triple of the standard model.

Key words: noncommutative geometry; twisted spectral triple; standard model.

pdf (316 kb)   tex (17 kb)  

References

  1. Boyle L., Farnsworth S., The standard model, the Pati-Salam model, and 'Jordan geometry', New J. Phys. 22 (2020), 073023, 11 pages, arXiv:1910.11888.
  2. Brzeziński T., Ciccoli N., Dąbrowski L., Sitarz A., Twisted reality condition for Dirac operators, Math. Phys. Anal. Geom. 19 (2016), 16, 11 pages, arXiv:1601.07404.
  3. Brzeziński T., Dąbrowski L., Sitarz A., On twisted reality conditions, Lett. Math. Phys. 109 (2019), 643-659, arXiv:1804.07005.
  4. Chamseddine A.H., Connes A., Conceptual explanation for the algebra in the noncommutative approach to the standard model, Phys. Rev. Lett. 99 (2007), 191601, 4 pages, arXiv:0706.3690.
  5. Chamseddine A.H., Connes A., Why the standard model, J. Geom. Phys. 58 (2008), 38-47, arXiv:0706.3688.
  6. Chamseddine A.H., Connes A., Resilience of the spectral standard model, J. High Energy Phys. 2012 (2012), no. 9, 104, 11 pages, arXiv:1208.1030.
  7. Chamseddine A.H., Connes A., Marcolli M., Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991-1089, arXiv:hep-th/0610241.
  8. Chamseddine A.H., Connes A., van Suijlekom W.D., Beyond the spectral standard model: emergence of Pati-Salam unification, J. High Energy Phys. 2013 (2013), no. 11, 132, 36 pages, arXiv:1304.8050.
  9. Chamseddine A.H., Connes A., van Suijlekom W.D., Inner fluctuations in noncommutative geometry without the first order condition, J. Geom. Phys. 73 (2013), 222-234, arXiv:1304.7583.
  10. Chamseddine A.H., van Suijlekom W.D., A survey of spectral models of gravity coupled to matter, in Advances in Noncommutative Geometry, Editors A. Chamseddine, C. Consani, N. Higson, M. Khalkhali, H. Moscovici, G. Yu, Springer, Cham, 2019, 1-51, arXiv:1904.12392.
  11. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  12. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, Amer. Math. Soc., Providence, RI, 2008.
  13. Connes A., Moscovici H., Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 57-71, arXiv:math.OA/0609703.
  14. Devastato A., Farnsworth S., Lizzi F., Martinetti P., Lorentz signature and twisted spectral triples, J. High Energy Phys. 2018 (2018), no. 3, 089, 21 pages, arXiv:1710.04965.
  15. Devastato A., Lizzi F., Martinetti P., Grand symmetry, spectral action and the Higgs mass, J. High Energy Phys. 2014 (2014), no. 1, 042, 29 pages, arXiv:1304.0415.
  16. Devastato A., Martinetti P., Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry, Math. Phys. Anal. Geom. 20 (2017), 2, 43 pages, arXiv:1411.1320.
  17. Filaci M., Martinetti P., Minimal twist of almost commutative geometries, in preparation.
  18. Filaci M., Martinetti P., Pesco S., Twisted standard model in noncommutative geometry I: the field content, arXiv:2008.01629.
  19. Landi G., Martinetti P., On twisting real spectral triples by algebra automorphisms, Lett. Math. Phys. 106 (2016), 1499-1530, arXiv:1601.00219.
  20. Landi G., Martinetti P., Gauge transformations for twisted spectral triples, Lett. Math. Phys. 108 (2018), 2589-2626, arXiv:1704.06212.

Previous article  Next article  Contents of Volume 16 (2020)