Stabilization of Extra Dimensions at Tree Level

TL;DR

The paper addresses why string-theoretic extra dimensions remain compact by analyzing winding and momentum modes on a time-dependent background at weak coupling and tree-level in $\alpha'$, showing stabilization of six internal dimensions at the self-dual radius $R=1$. It derives and uses T-duality to relate winding/momentum contributions to the background evolution, demonstrating a robust stabilization mechanism that drives the internal space to $R=1$ while the three large dimensions expand. The results hinge on balancing negative pressure from winding and positive pressure from momentum modes, with the dilaton playing a crucial role in maintaining a weak-coupling regime. The work aligns with earlier qualitative arguments and highlights the need for a dilaton potential and higher-order corrections to bridge to late-time FRW cosmology.

Abstract

By considering the effects of string winding and momentum modes on a time dependent background, we find a method by which six compact dimensions become stabilized naturally at the self-dual radius while three dimensions grow large.

Figures (2)

• Figure 1: Here we take the six small dimensions to be at the self-dual radius and with vanishing expansion rate initially, $b_{0}=1$ and $\dot{\nu}_{0}=0$. We find that the dimensions remain stable at the self-dual point regardless of the behavior of the dilaton and of the three large dimensions. These are shown in the figure for the initial values $\lambda_{0}=3$, $\dot{\lambda}_{0}=0.5$, and $\phi_{0}=-3$. However, this result holds for generic initial values as long as we respect the weak coupling limit (i.e. $g_{s}<<1$). Note that we are using Planck units.
• Figure 2: We consider initial values of $b(t)$ away from the self-dual radius and find oscillations about the self-dual radius which are damped by the dilaton and by the evolution of the large dimensions. This is again a generic result. Here we present the evolution for the following initial data: $\dot{\lambda}_{0} = .500, \phi_{0} = -3.00, \dot{\phi}_{0} = .380, \lambda_{0} = 3.00, \dot{\nu}_{0} = 0, \nu_{0} = 0$$\dot{\lambda}_{0} = .500, \phi_{0} = -3.00, \dot{\phi}_{0} = -.020, \lambda_{0} = 3.00, \dot{\nu}_{0} = -.100, \nu_{0} = .010$$\dot{\lambda}_{0} = .500, \phi_{0} = -3.00, \dot{\phi}_{0} = -.008, \lambda_{0} = 3.00, \dot{\nu}_{0} = -.100, \nu_{0} = .500$