  We compute (as functions of the shape and Wilson-line moduli) the one-loop
Casimir energy induced by higher-dimensional supergravities compactified from
6D to 4D on 2-tori, and on some of their Z_N orbifolds. Detailed calculations
are given for a 6D scalar field having an arbitrary 6D mass m, and we show how
to extend these results to higher-spin fields for supersymmetric 6D theories.
Particular attention is paid to regularization issues and to the identification
of the divergences of the potential, as well as the dependence of the result on
m, including limits for which m^2 A<< 1 and m^2 A>> 1 where A is the volume of
the internal 2 dimensions. Our calculation extends those in the literature to
very general boundary conditions for fields about the various cycles of these
geometries. The results have potential applications towards Supersymmetric
Large Extra Dimensions (SLED) as a theory of the Dark Energy. First, they
provide an explicit calculation within which to follow the dependence of the
result on the mass of the bulk states which travel within the loop, and for
heavy masses these results bear out the more general analysis of the
UV-sensitivity obtained using heat-kernel methods. Second, because the
potentials we find describe the dynamics of the classical flat directions of
these compactifications, within SLED they would describe the present-day
dynamics of the Dark Energy.
