  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
zeta-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: zeta(0)=7/45 for a complex scalar field,
zeta(0)=11/360 for spin 1/2, zeta(0)= -77/180 (magnetic) and 13/180 (electric)
for spin 1, and zeta(0)=112/45 for pure gravity when the linearized magnetic
curvature is vanishing on $S^3$. The zeta(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: zeta(0)=-289/360. The full zeta(0) does
not vanish in extended supergravity theories, indicating that supersymmetry is
one-loop divergent in the presence of boundaries.
