  This paper studies the 1-loop approximation for a massless spin-1/2 field on
a flat four-dimensional Euclidean background bounded by two concentric
3-spheres, when non-local boundary conditions of the spectral type are imposed.
The use of $\zeta$-function regularization shows that the conformal anomaly
vanishes, as in the case when the same field is subject to local boundary
conditions involving projectors. A similar analysis of non-local boundary
conditions can be performed for massless supergravity models on manifolds with
boundary, to study their 1-loop properties.
