Why the Standard Model

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).