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

TL;DR

This paper conjectures that compact Calabi-Yau manifolds have a number of length-minimizing closed geodesics growing polynomially with length, supported by physical arguments relating to states in nonlinear sigma models.

Contribution

It proposes a new conjecture linking the geometry of Calabi-Yau manifolds with the growth of closed geodesics and their physical interpretation in sigma models.

Findings

Conjecture on the growth rate of closed geodesics as L^{D}

Physical argument connecting geodesics to winding states

Outline of the relationship between geometry and sigma model states

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.