The Number of States of Two Dimensional Critical String Theory

TL;DR

This paper investigates the state count and vacuum structure of two-dimensional critical string theory, suggesting a finite number of states and arguing against the likelihood of only two large spacetime dimensions in string theory.

Contribution

It provides a new analysis of the state degeneracy and phase structure of two-dimensional string vacua, highlighting the role of strong coupling and quantum effects.

Findings

Naive state counting is invalid due to strong coupling singularities.

Classical solutions suggest only finite states in bounded regions.

Quantum effects likely remove vacuum degeneracy, indicating a Kosterlitz-Thouless phase.

Abstract

We discuss string theory vacua which have the wrong number of spacetime dimensions, and give a crude argument that vacua with more than four large dimensions are improbable. We then turn to two dimensional vacua, which naively appear to violate Bekenstein's entropy principle. A classical analysis shows that the naive perturbative counting of states is unjustified. All excited states of the system have strong coupling singularities which prevent us from concluding that they really exist. A speculative interpretation of the classical solutions suggests only a finite number of states will be found in regions bounded by a finite area. We also argue that the vacuum degeneracy of two dimensional classical string theory is removed in quantum mechanics. The system appears to be in a Kosterlitz-Thouless phase. This leads to the conclusion that it is also improbable to have only two large spacetime dimensions in string theory. However, we note that, unlike our argument for high dimensions, our conclusions about the ground state have neglected two dimensional quantum gravitational effects, and are at best incomplete.