  In extra-dimensional models which are compactified on an n-torus (n>1) there
exist moduli associated with the torus volume (which sets the fundamental
Planck scale), the ratios of the torus radii, and the angle(s) of periodicity.
We consider a model with gravity in the bulk of n=2 large extra dimensions with
a fixed volume, taking all Standard Model fields to be confined to a "thick"
and supersymmetric 3-brane. The Casimir energy of fields in the bulk of the
2-torus accounts for the present dark energy density while the shape moduli
begin rolling at late times (z ~ 1) and induce a shift in the Kaluza-Klein
masses of the Standard Model fields. The low energy value of the fine-structure
constant is sensitive at loop level to this shift. For reasonable cosmological
initial conditions on the shape moduli we obtain a redshift dependence of the
fine-structure constant similar to that reported by Webb et al., which is
roughly compatible with Oklo and meteorite bounds. Constraints from coincident
variation in the QCD scale are also briefly discussed.
