Title : Covariant compactification: Matter and its interactions as geometry
Abstract:
The geometry of spacetime is encoded in the symmetry properties of fields under transformations. In this talk, I show how the properties of fermion fields and the standard model gauge symmetries point towards a universe with one non-compact time dimension and nine compact spatial dimensions. At any moment in cosmic time, the nine spatial dimensions form a space homeomorphic to a product of a three-sphere and a six-sphere. Each generation of the known fermions can be identified as a multiplet of harmonics of a single-component field on this space. We take this field to be the square root of the absolute value of the metric determinant. This choice has the following consequence. We take the form of the energy-momentum density tensor for a monochromatic electromagnetic wave in flat spacetime, and adapt it to Fourier modes of this field in Riemann normal coordinates. This leads to the Einstein field equation. The multiplet of fermion states transforms as the defining representations of SU (4) and an extended SU (2) isospin (Pati-Salam) group. The states include a right-handed neutrino, which does not interaction with the gauge groups. This is taken to be a background field; perturbations of the factor spaces are realised as perturbations of this background field, which are manifested as other leptons and quarks. These states are related by the gauge transformations of the standard model. The SU (4) gauge potential, which contains the gluon fields and part of the weak hypercharge field, describes the mixing of harmonics of the six-sphere under rotations of the sphere along Wilson lines. Quantisation is not taken to be a mathematical procedure or applied as a starting postulate. Instead, the quantisation of variables, and uncertainty relations between them, arise naturally due to the spacetime topology and the Lie algebras of the transformation groups.
