Dimensional Mutation and Spacelike Singularities

TL;DR

This paper explores how string theory compactified on a Riemann surface transitions to a higher-dimensional phase at small volume, revealing increased degrees of freedom and a new perspective on spacelike singularities.

Contribution

It demonstrates that compactification on a Riemann surface leads to a higher-dimensional phase at small volume, supported by density of states and RG flow analyses.

Findings

Effective dimensionality increases at small volume

Density of states grows exponentially with winding modes

Spacelike singularities can be replaced by higher-dimensional phases

Abstract

I argue that string theory compactified on a Riemann surface crosses over at small volume to a higher dimensional background of supercritical string theory. Several concrete measures of the count of degrees of freedom of the theory yield the consistent result that at finite volume, the effective dimensionality is increased by an amount of order $2h/V$ for a surface of genus $h$ and volume $V$ in string units. This arises in part from an exponentially growing density of states of winding modes supported by the fundamental group, and passes an interesting test of modular invariance. Further evidence for a plethora of examples with the spacelike singularity replaced by a higher dimensional phase arises from the fact that the sigma model on a Riemann surface can be naturally completed by many gauged linear sigma models, whose RG flows approximate time evolution in the full string backgrounds arising from this in the limit of large dimensionality. In recent examples of spacelike singularity resolution by tachyon condensation, the singularity is ultimately replaced by a phase with all modes becoming heavy and decoupling. In the present case, the opposite behavior ensues: more light degrees of freedom arise in the small radius regime. I comment on the emerging zoology of cosmological singularities that results.