Some remarks on torsion in Kaluza–Klein unification

Abstract

The role of manifolds endowed with a parallelizing torsion in Kaluza–Klein theories is examined. In particular the spin connection on such manifolds is demonstrated to be a single pure gauge almost everywhere on the manifold. This follows from the Frobenius integration theorem. As a consequence of this result the computation of the representation of massless fermions follows immediately and trivially. The spectrum of massless fermions on manifolds with a parallelizing torsion is contrasted with the analogous spectrum on manifolds with nontrivial topological configurations. While the remarks are primarily of pedagogic value much of the relevant mathematics is made intuitive.