Non-associative geometry and the spectral action principle

Abstract

Chamseddine and Connes have argued that the action for Einstein gravity, coupled to the SU(3) × SU(2) × U(1) standard model of particle physics, may be elegantly recast as the “spectral action” on a certain “non-commutative geometry.” In this paper, we show how this formalism may be extended to “non-associative geometries,” and explain the motivations for doing so. As a guiding illustration, we present the simplest non-associative geometry (based on the octonions) and evaluate its spectral action: it describes Einstein gravity coupled to a G2 gauge theory, with 8 Dirac fermions (which transform as a singlet and a septuplet under G2). This is just the simplest example: in a forthcoming paper we show how to construct more realistic models that include Higgs fields, spontaneous symmetry breaking and fermion masses.