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

Abstract

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.

minor corrections, 43 pages, to appear in frontiers in mathematical physics. Frontiers in Physics, Dec 16, 2013