  We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yau
manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4,
has locally length minimizing closed geodesics, and that the number of these
with length less than L grows asymptotically as L^{D}. We also outline the
physical arguments behind this conjecture, which involve the claim that all
states in a nonlinear sigma model can be identified as "momentum" and "winding"
states in the large volume limit.
