Non-associative geometry and the spectral action principle

Shane Farnsworth, Latham Boyle
2015 Journal of High Energy Physics  
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
more » ... n: 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 ). 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.
doi:10.1007/jhep07(2015)023 fatcat:7a632r2wjbbshgowuhtxfw373y