Local supersymmetry in one-loop quantum cosmology

The contribution of physical degrees of freedom to the one-loop amplitudes of Euclidean supergravity is here evaluated in the case of flat Euclidean backgrounds bounded by a three-sphere, recently considered in perturbative quantum cosmology. In Euclidean supergravity, the spin-3/2 potential has the pair of independent spatial components (psi_{i}^{A}, {widetilde psi}_{i}^{A'}). Massless gravitinos are here subject to the following local boundary conditions on S**3: sqrt{2} ; {_{e}n_{A}^{; ; A'}} psi_{i}^{A}=pm {widetilde psi}_{i}^{A'}, where {_{e}n_{A}^{; ; A'}} is the Euclidean normal to the three-sphere boundary. The physical degrees of freedom (denoted by PDF) are picked out imposing the supersymmetry constraints and choosing the gauge condition e_{AA'}^{; ; ; ; ; i}psi_{i}^{A}=0, e_{AA'}^{; ; ; ; ; i}{widetilde psi}_{i}^{A'}=0. These local boundary conditions are then found to imply the eigenvalue condition {[J_{n+2}(E)]}^{2}-{[J_{n+3}(E)]}^{2}=0, ; forall n geq 0, with degeneracy (n+4)(n+1). One can thus apply again a zeta-function technique previously used for massless spin-${1over 2}$ fields. The PDF contribution to the full zeta(0) value is found to be =-289/360. Remarkably, for the massless gravitino field the PDF method and local boundary conditions lead to a result for zeta(0) which is equal to the PDF value one obtains using spectral boundary conditions on S**3.