Spin factors and geometric phases in arbitrary dimensions

The Polyakov spin factor is discussed using Strominger's intuitive analysis of the fermion propagator in loop space. Modulo a "helicity" projection, the Polyakov-Strominger spin factor is shown to be a cumulant of infinitesimal "boosts" or Thomas precessions along the particle orbit. Alternative forms for the spin factor and the fermion propagator are also discussed. The results are extended to higher euclidean space dimensions with a natural interpretation in terms of abelian/non-abelian Berry phases. In even (2n) and odd (2n+1) space dimensions with n > 1 the induced Berry potential is U (2^n-1)-valued with an implicit monopole structure.