  Based on some recent work of the authors, we focus on the relationship
between the Casimir energy of a Majorana spinor field for a Euclidean Einstein
universe $S^4\times R$ and for a Euclidean de Sitter brane ($S^4$) embedded in
AdS(5). This is for a conformally coupled massless field. Interestingly, the
one brane effective potential is zero and the results are equivalent, as for
the scalar case, when evaluated on the conformally related cylinder. However,
using the actual metric this equivalence no longer holds because a non-trivial
contribution from the path integral measure (known as the cocycle function) is
non-zero.
