On Minisuperspace Models of S-branes

TL;DR

This work studies time-dependent tachyon backgrounds through minisuperspace toy models for rolling and bouncing tachyons, showing that self-adjoint extensions with boundary conditions at infinity are essential in Lorentzian settings. It derives the spectra for the rolling case as $Spec_{\nu_0}(H_{ m rt}) = (-\infty,0] \cup \bigcup_{n=0}^\infty \{4(\nu_0+n)^2\}$ and establishes a discrete, two-branch structure for the bouncing tachyon governed by a 2-parameter boundary data, with large-energy behaviors linked to Liouville-like sectors and modified Mathieu functions. The paper also computes minisuperspace 2- and 3-point functions for rolling tachyons, revealing how boundary parameters enter correlators and how integrating over them reproduces results from Wick-rotated Euclidean models (Strominger’s framework). Together, these results clarify how time-like worldsheet theories acquire discrete conformal weights through boundary conditions and offer a roadmap for extending to full time-dependent string field theories and their amplitudes.

Abstract

In this note we reconsider the minisuperspace toy models for rolling and bouncing tachyons. We show that the theories require to choose boundary conditions at infinity since particles in an exponentially unbounded potential fall to infinity in finite world-sheet time. Using standard techniques from operator theory, we determine the possible boundary conditions and we compute the corresponding energy spectra and minisuperspace 3-point functions. Based on this analysis we argue in particular that world-sheet models of S-branes possess a discrete spectrum of conformal weights containing both positive and negative values. Finally, some suggestions are made for possible relations with previous studies of the minisuperspace theory.

Figures (3)

• Figure 1: An illustration of the spectrum of the rolling tachyon model with $\lambda =0.2$ and boundary parameter $\nu _{0}=3/4$. For $\Delta >0$ the spectrum is discrete and we plotted the wave functions $\psi ^{\nu _{0}}_{\nu _{n}}$ for $n=0,1$. The spectrum is continous for $\Delta <0$ and two wave functions $\psi ^{\nu _{0}}_{\omega }$ are shown as representatives.
• Figure 2: Energy $\Delta$ versus phase-space volume $\Gamma (0,\Delta )$. Neighbouring grid lines correspond to a spacing $\delta \Gamma =h$, i.e. in the energy interval $\delta \Delta$ between two horizontal grid lines we expect to find one quantum mechanical state.
• Figure 3: An illustration of the spectrum of the bouncing tachyon model. The boundary labels are $\nu _{+}=3/4$ and $\nu _{-}=1/4$, and we set $\lambda =0.2$. The spectrum is purely discrete, and the level spacing agrees well with the semi-classical expectations as we can see by comparing with the neighbouring figure \ref{['fig:psv']}. The right half of the drawing resembles strongly the figure of the rolling tachyon model (fig. \ref{['fig:rt']}).