Noncommutative Geometry and the Standard Model
Kraków, November 8-9, 2019
The conference on the recent develepments of the noncommutative approach to fundamental particle interactions and gravity, organized at the Institute of Physics, Jagiellonian University.
Organizers: A.Sitarz, A.Bochniak, P.Zalecki.
Time to think about time Some considerations on the applications of spectral noncommutative geometry to physical systems with Lorentz signature.
Lorentzian fermionic action by twisting euclidean spectral triples The twist (in the sense of Connes-Moscovici) of the spectral triple of the Standard Model yields to new kinds of fields: a scalar which permits to solve the Higgs mass problem and the metastability of the electroweak vacuum, and a pseudo-vector whose meaning was unclear so far. In addition, the twist induces on the Hilbert space of euclidean spinors a new inner product, which coincides with the inner product of Lorentzian spinors. We will show how a similar transition form the euclidean to the Lorentzian occurs at the level of the fermionic action, thanks to the pseudo vector field. In particular, we get the Weyl and the Dirac equations in Lorentzian signature by twisting, respectively, the euclidean spectral triples of a doubled manifold and of electrodynamics.
The standard model, the Pati-Salam model, and ,,Jordan geometry’’. I will describe how, by thinking about the description in the standard model in terms of noncommutative geometry, we are led to a related description of the standard model (and the Pati-Salam) in terms of 'Jordan geometry'. I will explain some of the virtues of this perspective, as well as some puzzles and challenges for the future.
Jordan geometry I will discuss some of the motivations coming from non-commutative geometry for considering geometries coordinatized by Jordan algebras. I will also discuss some of the recent progress in this direction, focusing in particular on the Jordan almost-associative geometries underlying gauge theories.
Exceptional quantum algebra for the Standard Model of particle physics The exceptional euclidean Jordan algebra consisting of 3x3 hermitian octonionic matrices, appears to be tailor made for the internal space of the three generations of quarks and leptons. The talk is based on the paper of Michel Dubois-Violette and on ongoing work with him and with Svetla Drenska.
Geometric notions from spectral action Given a Riemannian manifold, one can retrieve from the heat kernel asymptotics of the Laplacian a few geometric notions on this manifold like its dimension, volume, scalar curvature, etc. I will review some aspects on the same problematic in noncommutative geometry revisiting the influence of the metric, scalar curvature etc. The influence of torsion will also be emphasized.
Algebraic backgrounds and the B-L symmetry In this talk I will argue that Spectral Triples are an inadequate setting for gravity (or Kaluza-Klein) theories, and that one should use instead "algebraic backgrounds". This new structure contains the same objects as a Spectral Triple, with the exception of the Dirac operator and the addition of a bimodule of "1-forms". The application of this idea to the Standard Model yields unexpected connections with gauged B-L symmetry.
Some comments on Morita equivalence and spectral triples I will discuss some algebraic conditions on spectral triples that are inspired by the examples of Hodge-Dirac operator on an oriented Riemannian manifold and Dirac operator on a spin manifold. I will discuss in detail the case of finite-dimensional spectral triples, and study the behaviour of such conditions under products of spectral triples. (Based on joint works with L. Dabrowski and A. Sitarz.)
Noncommutative inner geometry of the Standard Model A non-commutative C*-algebra is commonly regarded as the algebra of continuous functions on a 'quantum space'. Its smooth and metric structures can be described in terms of a spectral triple which involves an analogue of the Dirac operator. The Standard Model of fundamental particles in physics can be described as the almost commutative geometry, the inner part of which can be interpreted as a quantum analogue of the de-Rham-Hodge spectral triple.
Progress in spectral triples with twisted real structure In addition to the twisted spectral triples of Connes-Moscovici, it is also possible to twist a real spectral triple by making various modifications to its real structure using an additional twist operator’. I will present ongoing progress made in the investigation of such spectral triples with twisted real structures, particularly with regards to the fluctuation of the Dirac operator and applications to the so-called second-order condition.