  We find a class of four dimensional deformed conformal field theories which
appear extra dimensional when their gauge symmetries are spontaneously broken.
The theories are supersymmetric moose models which flow to interacting
conformal fixed points at low energies, deformed by superpotentials. Using
a-maximization we give strong nonperturbative evidence that the hopping terms
in the resulting latticized action are relevant deformations of the fixed point
theories. These theories have an intricate structure of RG flows between
conformal fixed points. Our results suggest that at the stable fixed points
each of the bulk gauge couplings and superpotential hopping terms is turned on,
in favor of the extra dimensional interpretation of the theory. However, we
argue that the higher dimensional gauge coupling is generically small compared
to the size of the extra dimension. In the presence of a brane the topology of
the extra dimension is determined dynamically and depends on the numbers of
colors and bulk and brane flavors, which suggests phenomenological
applications. The RG flows between fixed points in these theories provide a
class of tests of Cardy's conjectured a-theorem.
