A Space-Time Orbifold: A Toy Model for a Cosmological Singularity
Vijay Balasubramanian, S. F. Hassan, Esko Keski-Vakkuri, Asad Naqvi
TL;DR
This work studies bosonic and Type II superstrings on space-time orbifolds $R^{1,d}/Z_2$—identifications that reverse time and reflect spatial coordinates—to model cosmological singularities. Using BRST formalism, it shows no negative-norm physical states for suitable $d$ and identifies a twisted sector localized at the singular locus, with momentum constraints that depend on $d$. The one-loop partition function is modular invariant, and the Type II case yields a vanishing zero-momentum dilaton tadpole at one loop, yet negative-norm virtual states fail to cancel in loops, implying no ghost-free gauge globally. The paper discusses unitarity, the meaning of S-matrix elements when twisted states are localized in time, and the potential for twisted-sector condensates to resolve singularities, providing general lessons for quantizing strings in time-dependent backgrounds.
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.