Classification of Finite Spectral Triples

Abstract

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings.

Latex, 29 pages with 2 figures, reference added

Cited by in corpus (60)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Dark matter with invisible light from heavy double charged leptons of almost-commutative geometry?
• Almost-Commutative Geometries Beyond the Standard Model
• On a Classification of Irreducible Almost-Commutative Geometries IV
• Riemannian manifolds in noncommutative geometry
• On a Classification of Irreducible Almost Commutative Geometries
• Particle Physics from Almost Commutative Spacetimes
• Distances in Finite Spaces from Noncommutative Geometry
• Matrix geometries and fuzzy spaces as finite spectral triples
• New Scalar Fields in Noncommutative Geometry
• Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics
• Can (noncommutative) geometry accommodate leptoquarks?
• Supersymmetric QCD and noncommutative geometry
• On a Classification of Irreducible Almost Commutative Geometries, A Second Helping
• A survey of spectral models of gravity coupled to matter
• On a Classification of Irreducible Almost Commutative Geometries III
• Bootstrapping Dirac Ensembles
• Almost-Commutative Geometries Beyond the Standard Model II: New Colours
• Spectral action and big desert
• Almost-Commutative Geometries Beyond the Standard Model III: Vector Doublets
• The mass of simple and higher-order networks
• Finite Pseudo-Riemannian Spectral Triples and The Standard Model
• Moduli spaces of Dirac operators for finite spectral triples
• Gauge networks in noncommutative geometry
• A reconstruction theorem for almost-commutative spectral triples
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• On globally non-trivial almost-commutative manifolds
• Understanding truncated non-commutative geometries through computer simulations
• Finding the Standard Model of Particle Physics, A Combinatorial Problem
• The graded product of real spectral triples
• Seesaw and noncommutative geometry
• From Noncommutative Geometry to Random Matrix Theory
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Twisted reality condition for Dirac operators
• Spectral metric spaces for Gibbs measures
• Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
• Fermion masses in noncommutative geometry
• Non-commutative geometry and the standard model vacuum
• Connes-Lott model building on the two-sphere
• Renormalizability conditions for almost-commutative geometries
• On a Classification of Irreducible Almost-Commutative Geometries V
• Canonical quantization and the spectral action, a nice example
• A new algebraic structure in the standard model of particle physics
• Going beyond the Standard Model with noncommutative geometry
• A Dark Sector Extension of the Almost-Commutative Standard Model
• Massive Neutrinos in Almost-Commutative Geometry
• BV quantization of dynamical fuzzy spectral triples
• Moduli spaces of discrete gravity I: A few points
• Noncommutative geometry, topology and the standard model vacuum
• The Inverse Seesaw Mechanism in Noncommutative Geometry
• Moduli of quantum Riemannian geometries on <= 4 points
• Gauge theories with graded differential Lie algebras
• Spectral triples and differential calculi related to the Kronecker foliation
• Towards modified bimetric theories within non-product spectral geometry
• Real Spectral Triples on Crossed Products
• Gauge unification in noncommutative geometry
• Krajewski diagrams and the Standard Model
• Lifting Bratteli Diagrams between Krajewski Diagrams: Spectral Triples, Spectral Actions, and $AF$ algebras
• The loop equations for noncommutative geometries on quivers