On the fermion mass-spectrum of Kaluza-Klein supergravity

A general setup for the harmonic expansions of D = 11 supergravity about classical compactifying solutions is given. Mass-matrices for the fermion mass-spectrum are derived both in the case of Freund-Rubin backgrounds and of Englert type solutions. The mass-matrix is explicitly given in terms of the eigenvalues of invariant operators on the 7-manifold. In the Freund-Rubin case the mass formula is very easy, while in the Englert case it is quite complicated and requires special techniques. In the second part of the paper applications to special manifolds for the Freund-Rubin background are considered, postponing the application of the general Englert formula to future publications. In particular, the round 7-sphere with the complete fermion spectrum and the SU(3) × SU(2) × U(1) isometric M^pqr-spaces are considered. For this latter the condition for the existence of a spinor structure (p = odd number) is established, deriving the explicit spectrum and the mass-eigenvalues for one of the two sectors of which it is composed. This relies on a general distinction between longitudinal and transverse fermion particle-modes which is emphasized in the first part of the paper. In this paper the longitudinal spectrum of M^pqr cosets is derived, postponing the transverse one to future work.