Time-Dependent Backgrounds in 2D String Theory and the $S$-matrix Generating Functional

TL;DR

This paper explores time-dependent backgrounds in 2D string theory using the $S$-matrix generating functional, clarifying calculation methods and applying the formalism to the sine-Gordon model, revealing new scaling behaviors.

Contribution

It develops a general formalism for analyzing time-dependent backgrounds in 2D string theory and connects different $S$-matrix calculation approaches, with detailed application to the sine-Gordon model.

Findings

Reproduces conformal field theory results for the sine-Gordon model.

Shows the tree-level partition function scales as a $c=0$ model when $p o 0$.

Clarifies the relation between Feynman rule and classical solution methods for $S$-matrix calculations.

Abstract

We study the time-dependent tachyon backgrounds of the string collective field theory using the formalism of the $S$-matrix generating functional. In the process we clarify the connection between two ways of calculating the $S$-matrix, the one using the Feynman rule and the other using the classical solution to the nonlinear equation of motion. We develop the formalism for general backgrounds and apply it to the gravitational sine-Gordon model in detail. We reproduce the conformal field theory calculation which was based on expanding around the static $c=1$ theory. Furthermore, we prove that the tree- level partition function of this model shows the scaling behavior corresponding to $c=0$ model in the limit $p \to 0$ of sine-Gordon `momentum' $p$.