On a classification of irreducible almost commutative geometries, a second helping

Abstract

We extend a classification of irreducible, almost-commutative geometries whose spectral action is dynamically non-degenerate, to internal algebras that have six simple summands. We find essentially four particle models: An extension of the standard model by a new species of fermions with vectorlike coupling to the gauge group and gauge invariant masses, two versions of the electro-strong model and a variety of the electro-strong model with Higgs mechanism. PACS-92: 11.15 Gauge field theories MSC-91: 81T13 Yang-Mills and other gauge theories 1 Privatgelehrter, Kiel (Germany) 2 christophstephan@gmx.de

A Reconstruction Theorem for Almost-Commutative Spectral Triples

“…Such spectral triples are defined as the noncommutative-geometric Cartesian product of the canonical spectral triple of a compact spin manifold, the prototypical commutative spectral triple, with a finite spectral triple, namely, a spectral triple with finite-dimensional Hilbert space (see [5,31,35] for the general theory, and [26][27][28][29][30] for classification results).…”

Moduli Spaces of Dirac Operators for Finite Spectral Triples

“…In particular, once a suitable ordering is fixed on the spectrum of a finite-dimensional real C * -algebra A, the study of finite real spectral triples with algebra A reduces completely to the study of the appropriate multiplicity matrices and of certain moduli spaces constructed using those matrices. This reduction is what has allowed for the success of Krajewski's diagrammatic approach [18, §4] in the cases dealt with by Iochum, Jureit, Schücker, and Stephan [12][13][14][15][16][17]22]. We have also seen how to apply this theory both to the "finite geometries" of the current version of the NCG Standard Model [4,7,8] and to Chamseddine and Connes's framework [2,3] for deriving the same finite geometries.…”

“…These algebraic consequences of quasi-orientability, which were derived from the stronger condition of orientability in the original papers [20] and [18], are key to the formalism developed by Krajewski and Paschke-Sitarz, and hence to the later work by Iochum, Jureit, Schücker, and Stephan [12][13][14]22]. We can now characterise orientable bimodules amongst quasi-orientable bimodules:…”

“…Just as in the odd case, the above discussion and Corollary 3.5 imply that we can identify D 0 (A, H, γ) with In the quasi-orientable case, the picture simplifies considerably, as all components ∆ αβ αβ necessarily vanish. One is then left, essentially, with the situation described by Krajewski [12][13][14][15][16]22], offer a concise, diagrammatic approach to the study of quasi-orientable even bilateral spectral triples that strongly emphasizes the underlying combinatorics. Though they do admit ready generalisation to the non-quasiorientable case, we will not discuss them here.…”

“…Krajewski, in particular, defined what are now called Krajewski diagrams to facilitate the classification of such spectral triples. Iochum, Jureit, Schücker, and Stephan have since undertaken a programme of classifying Krajewski diagrams for finite spectral triples satisfying certain additional physically desirable assumptions [12][13][14]22] using combinatorial computations [17], with the aim of fixing the finite spectral triple of the Standard Model amongst all other such triples.…”