In my programme of covariant compactification, I seek to explain all the fundamental fields of physics as geometric quantities. This is based on the work of Kaluza and Klein to extend general relativity.

In this talk, I explore how the geometry of product spacetimes determines their diffeomorphism and covariance groups, and the general linear and (pseudo-)orthogonal actions these induce on tangent spaces. I show how tensors decompose and gauge fields arise when transformation groups are non-linearly realised.

I explain how the topology of factor spaces determines their harmonics, which are manifested as matter fields. With an appropriate choice of geometry, these can be identified with the known fermions, as each harmonic can be labelled by its quantum numbers.

Unlike most post-1960 Kaluza–Klein theories, unitary gauge transformations do not act directly on the additional dimensions. Instead, orthogonal diffeomorphisms on compact factor spaces have a ‘dragging action’ on fields defined on these spaces. The induced actions on multiplets of harmonics are unitary representations of the orthogonal groups. This includes spinor representations. This allows us to identify multiplets of harmonics with spinors, on which the unitary gauge transformations act.

And in contrast to perceived wisdom about modern Kaluza–Klein theories, symmetry restoration does not take place at high energies. Instead, it occurs at the zero-curvature ‘decompactification limit’, in which all dimensions appear on the same footing.