Lorentz signature and twisted spectral triples
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Springer
Received: October 20, Revised: January 30, Accepted: March 7, Published: March 15,
2017 2018 2018 2018
A. Devastato,a S. Farnsworth,b F. Lizzia,c,d and P. Martinettie a
INFN sezione di Napoli, C.U. Monte S. Angelo, via Cintia, 80126 Napoli, Italy b Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨ uhlenberg 1, 14476 Potsdam-Golm, Germany c Dipartimento di Fisica “E. Pancini”, Universit` a di Napoli Federico II, C.U. Monte S. Angelo, via Cintia, 80126 Napoli, Italy d Institut de C´ıencies del Cosmos (ICCUB), Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Catalonia, Spain e Dipartimento di Matematica, Universit` a di Genova, via Dodecaneso 35, 16146 Genova, Italy
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twists and Wick rotation. Keywords: Non-Commutative Geometry, Differential and Algebraic Geometry, SpaceTime Symmetries ArXiv ePrint: 1710.04965
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)089
JHEP03(2018)089
Lorentz signature and twisted spectral triples
Contents 1
2 Twisted spectral geometry for the standard model
3
3 Twist and Lorentz structure 3.1 Twisted inner product 3.2 Lorentzian signature and Krein space
7 7 8
4 Actions 4.1 Fermionic action 4.2 Bosonic action
10 11 15
5 Conclusions and outlook
16
A Adjoint action
17
1
Introduction
Noncommutative differential geometry (NCG) provides a unified framework from which to describe both Einstein-Hilbert gravity (in Euclidean signature) and classical gauge theories [1]. In particular, it gives an elegant description of the full Standard Model of particle physics in all of its detail, including the Higgs mechanism and neutrino mixing, as gravity on a certain “almost commutative manifold” [2, 3]. A recent and comprehensive review can be found in [4]. The main benefit of the NCG approach to physics is that it offers a more constrained description of gauge theories than the usual effective field theory approach. Indeed, the added geometric constraints impose a range of successful and phenomenologically accurate restrictions on the allowed particle content of the Standard Model of particle physics [5–8]. Despite this success, an early estimate for the Higgs mass was also furnished at mH ≃ 170 Gev. This prediction was disfavored by the Tevatron data, and has since been ruled out by the LHC [9, 10]. While falling short of an accurate comparison with experiment, this prediction depended on a number of assumptions including the big desert hypothesis, as well as the presence of a scale at which the coupling constants of the three gauge interactions unify. In light of the many successes of
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