The effect of boundaries in one-loop quantum cosmology

The problem of boundary conditions in a supersymmetric theory of quantum cosmology is studied, with application to the one-loop prefactor in the quantum amplitude.Our background cosmological model is flat Euclidean space bounded by a three-sphere, and our calculations are based on the generalized Riemann ζ-function. One possible set of supersymmetric local boundary conditions involves field strengths for spins 1, 3/ 2 and 2, the undifferentiated spin-1/2 field, and a mixture of Dirichlet and Neumann conditions for spin 0. In this case the results we can obtain are ζ(0) =7/ 45for a complex scalar field, ζ(0) =11/360 for spin 1/2, ζ(0) =−77/ 180 (magnetic) and 13/180 (electric) for spin 1, and ζ(0) =112/45 for pure gravity when the linearized magnetic curvature is vanishing on S3. The ζ(0) values for gauge fields have been obtained by working only with physical degrees of freedom. An alternative set of boundary conditions can be motivated by studying transformation properties under local supersymmetry; these involve Dirichlet conditions for the spin-2 and spin-1 fields, a mixture of Dirichlet and Neumann conditions for spin-0, and local boundary conditions for the spin-1/2 field and the spin-3/2 potential. For the latter one finds ζ(0) =−289/360. The full ζ(0) does not vanish in extended supergravity theories,indicating that supersymmetry is one-loop divergent in the presence of boundaries.