  In the one-loop approximation for Euclidean quantum gravity, the boundary
conditions which are completely invariant under gauge transformations of metric
perturbations involve both normal and tangential derivatives of the metric
perturbations $h_{00}$ and $h_{0i}$, while the $h_{ij}$ perturbations and the
whole ghost one-form are set to zero at the boundary. The corresponding
one-loop divergency for pure gravity has been recently evaluated by means of
analytic techniques. It now remains to compute the contribution of all
perturbative modes of gauge fields and gravitation to the one-loop effective
action for problems with boundaries. The functional determinant has a non-local
nature, independently of boundary conditions. Moreover, the analysis of
one-loop divergences for supergravity with non-local boundary conditions has
not yet been completed and is still under active investigation.
