On a classification of irreducible almost commutative geometries

Abstract

We classify all irreducible, almost commutative geometries whose spectral action is dynamically non-degenerate. Heavy use is made of Krajewski's diagrammatic language. The motivation for our definition of dynamical non-degeneracy stems from particle physics where the fermion masses are non-degenerate.

On globally non-trivial almost-commutative manifolds

“…They were later used in [Con96,CCM07] to geometrically describe YangMills theories and the Standard Model of elementary particles. The name almost-commutative manifolds was coined in [ISS04], their classification started in [Kra98,PS98]. Let M be a smooth compact even-dimensional Riemannian spin manifold.…”

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.…”