New Dimensions for Wound Strings: The Modular Transformation of Geometry to Topology

TL;DR

This paper demonstrates how string theory on negatively curved compact spaces exhibits increased effective dimensions due to exponential growth in winding modes, linking geometry and topology to dimensionality through modular invariance.

Contribution

It generalizes previous results by explicitly connecting negative curvature, winding modes, and effective dimensions in string theory using Milnor's theorem.

Findings

Exponential growth of winding modes correlates with increased effective dimensions.

Explicit agreement between effective central charge and infrared spectrum in time-dependent solutions.

Establishes a fundamental relation between geometry, topology, and dimensionality in string theory.

Abstract

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 hep-th/0510044. 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.