New dimensions for wound strings: The modular transformation of geometry to topology

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in [1]. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.