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We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yaumanifold, 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 thesewith length less than L grows asymptotically as L^{D}. We also outline thephysical arguments behind this conjecture, which involve the claim that allstates in a nonlinear sigma model can be identified as "momentum" and "winding"states in the large volume limit.

Geodesics on Calabi-Yau manifolds and winding states in nonlinear sigma models