Moduli Stabilization with Long Winding Strings

TL;DR

This work tackles the problem of stabilizing all moduli arising from string compactifications by combining string gas cosmology with flux on a torus. By analyzing fully time-dependent equations in a simple ${ m R}^{1,3} \times T^2$ setup and incorporating long strings winding all internal directions, the authors show that all geometric moduli (except the dilaton, which is fixed by hand) can be dynamically stabilized; the angle modulus and flux are stabilized by the gas and exhibit harmonic fluctuations about the fixed point. The main contributions are the classical stabilization from extended winding strings and the quantum stabilization with flux, demonstrated within a fully dynamical framework rather than a static effective potential. This work highlights winding modes as the key missing ingredient in EFT approaches to moduli stabilization and points to future extensions to dilaton stabilization and more general backgrounds, potentially involving branes. $R$, $\theta$, and $b$ enter as dynamical moduli, with stability analyzed via linearized equations like $d^2\theta/dt^2 + 4(1+ b^2)K^{-1/2}e^{-2\phi}\,\theta = 0$ and $d^2 b/dt^2 + K^{-1/2} b = 0$, where $K \equiv 4 + 2 b^2 + 2N$.

Abstract

Stabilizing all of the modulus fields coming from compactifications of string theory on internal manifolds is one of the outstanding challenges for string cosmology. Here, in a simple example of toroidal compactification, we study the dynamics of the moduli fields corresponding to the size and shape of the torus along with the ambient flux and long strings winding both internal directions. It is known that a string gas containing states with non-vanishing winding and momentum number in one internal direction can stabilize the radius of this internal circle to be at self-dual radius. We show that a gas of long strings winding all internal directions can stabilize all moduli, except the dilaton which is stabilized by hand, in this simple example.