A Space-Time Orbifold: A Toy Model for a Cosmological Singularity

TL;DR

This paper investigates string theory in a simple time-dependent orbifold background, revealing insights into unitarity, spectrum, and quantum consistency issues relevant to cosmological singularities.

Contribution

It introduces a toy model for cosmological singularities using time-dependent orbifolds and analyzes its quantum properties, including spectrum and unitarity challenges.

Findings

No negative norm physical excitations in the model

Negative norm virtual states contribute to loops without cancellation

Partition function is modular invariant and dilaton tadpole vanishes

Abstract

We explore bosonic strings and Type II superstrings in the simplest time dependent backgrounds, namely orbifolds of Minkowski space by time reversal and some spatial reflections. We show that there are no negative norm physical excitations. However, the contributions of negative norm virtual states to quantum loops do not cancel, showing that a ghost-free gauge cannot be chosen. The spectrum includes a twisted sector, with strings confined to a ``conical'' singularity which is localized in time. Since these localized strings are not visible to asymptotic observers, interesting issues arise regarding unitarity of the S-matrix for scattering of propagating states. The partition function of our model is modular invariant, and for the superstring, the zero momentum dilaton tadpole vanishes. Many of the issues we study will be generic to time-dependent cosmological backgrounds with singularities localized in time, and we derive some general lessons about quantizing strings on such spaces.