Eta invariants, charge fractionalization and anomalies

Starting with the observation that the fermionic number operator of a charge fractionalization problem is the (inverse) Mellin transform of the eta invariant of Atiyah, Patodi and Singer, we calculate the eta invariant for odd-dimensional unit spheres. We obtain an interesting interpretation for both the Goldstone-Wilczek mass relation and the eta invariant (in this case, a generalization of the classical conformal invariant of Blaschke et. al). We briefly discuss the possible anomalies of the implied theory by connecting the eta invariant with the spin-index theorem. We show that the known relationship between Abelian and nonabelian anomalies is an interplay between the spin-index theorem and Bott periodicity. Some related questions are also discussed. (orig.)