Closed Strings in Misner Space: Stringy Fuzziness with a Twist

TL;DR

This work analyzes string dynamics in Misner space, a Lorentzian orbifold modeling cosmological singularities, by computing tree-level amplitudes involving twisted and untwisted string states. A winding-dependent non-locality emerges through the operator $\exp(\Delta(\nu) \partial_+ \partial_-)$, smearing untwisted sources over a scale $\sqrt{\Delta(\nu)}$ with $\Delta(\nu) \sim \log w$ at large winding. The authors obtain finite three-point amplitudes with two twisted strings and one untwisted state, but find divergences in certain four-point channels arising from intermediate winding with vanishing boost momentum, signaling strong back-reaction near the singularity. For configurations with three or more twists, amplitudes are derived via analytic continuation from the Nappi-Witten plane-wave model, revealing a non-local interaction kernel $\Xi(\nu_1,\nu_2)$ that can diverge for specific kinematics, underscoring the potential for winding-string condensation to back-react on the geometry. Together, these results elucidate how cosmological singularities in Misner space couple to twisted strings and hint at stringy mechanisms for resolving or reinterpreting such singularities.

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 $\exp(Δ(ν) \partial_+ \partial_-)$ reminiscent of the Moyal-product of non-commutative geometry, which, since $Δ(ν)$ 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.