Closed Strings in Misner Space: Stringy Fuzziness with a Twist

TL;DR

This paper investigates string interactions in Misner space, revealing that twisted strings become 'fuzzy' at large winding numbers and analyzing the finiteness and divergence of various scattering amplitudes, with implications for cosmological singularities.

Contribution

It introduces a detailed computation of scattering amplitudes involving twisted states in Misner space, highlighting stringy fuzziness and non-local effects with a novel operator resembling non-commutative geometry.

Findings

Twisted strings with large winding are fuzzy on a scale ew log w.

Certain scattering amplitudes are finite, others diverge due to winding string propagation.

The results enable potential calculations of back-reaction from winding string condensates.

Abstract

Misner space, also known as the Lorentzian orbifold $R^{1,1}/boost$, is the simplest tree-level solution of string theory with a cosmological singularity. We compute tree-level scattering amplitudes involving twisted states, using operator and current algebra techniques. We find that, due to zero-point quantum fluctuations of the excited modes, twisted strings with a large winding number $w$ are fuzzy on a scale $\sqrt{\log w}$, which can be much larger than the string scale. Wave functions are smeared by an operator $\exp(\Delta(\nu) \partial_+ \partial_-)$ reminiscent of the Moyal-product of non-commutative geometry, which, since $\Delta(\nu)$ is real, modulates the amplitude rather than the phase of the wave function, and is purely gravitational in its origin. We compute the scattering amplitude of two twisted states and one tachyon or graviton, and find a finite result. The scattering amplitude of two twisted and two untwisted states is found to diverge, due to the propagation of intermediate winding strings with vanishing boost momentum. The scattering amplitude of three twisted fields is computed by analytic continuation from three-point amplitudes of states with non-zero $p^+$ in the Nappi-Witten plane wave, and the non-locality of the three-point vertex is found to diverge for certain kinematical configurations. Our results for the three-point amplitudes allow in principle to compute, to leading order, the back-reaction on the metric due to a condensation of coherent winding strings.