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Abstract
The Standard Model is based on the gauge invariance principle with gauge group U ( 1 ) × SU ( 2 ) × SU ( 3 ) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four-dimensional continuum by a finite noncommutative geometry F . The raison d’être for F is to correct the K -theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K -theoretic dimension six and show that the dimension (per generation) is a square of an integer k . Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k = 4 ) with the correct quantum numbers for all fields. The spectral action applied to the product M × F delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input).
Published Version
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