  We study the geometric interpretation of two dimensional rational conformal
field theories, corresponding to sigma models on Calabi-Yau manifolds. We
perform a detailed study of RCFT's corresponding to T^2 target and identify the
Cardy branes with geometric branes. The T^2's leading to RCFT's admit ``complex
multiplication'' which characterizes Cardy branes as specific D0-branes. We
propose a condition for the conformal sigma model to be RCFT for arbitrary
Calabi-Yau n-folds, which agrees with the known cases. Together with recent
conjectures by mathematicians it appears that rational conformal theories are
not dense in the space of all conformal theories, and sometimes appear to be
finite in number for Calabi-Yau n-folds for n>2. RCFT's on K3 may be dense. We
speculate about the meaning of these special points in the moduli spaces of
Calabi-Yau n-folds in connection with freezing geometric moduli.
