Non-Associative Geometry and the Spectral Action Principle

Chamseddine and Connes have shown how 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 $G_2$ gauge theory, with 8 Dirac fermions (which transform as a singlet and a septuplet under $G_2$). We use this example to illustrate how non-associative geometries may be naturally linked to ordinary (associative) geometries by a certain twisting procedure. This is just the simplest example: in a forthcoming paper we show how to construct realistic models that include Higgs fields, spontaneous symmetry breaking and fermion masses.