Strings in Discrete and Continuous Target Spaces: A Comparison

TL;DR

This paper compares two approaches to two-dimensional string theory, showing their observables coincide under certain conditions and revealing that discrete target spaces simplify amplitude calculations while preserving key correlation functions.

Contribution

It establishes the precise relationship between loop gas and matrix quantum mechanics methods, highlighting the effects of target space discretization on string theory observables.

Findings

Target space loop correlators coincide in both models.

Discrete target space leads to factorization of amplitudes.

Correlation functions of tachyon operators are calculable in the discrete setting.

Abstract

We find the precise relationship between the loop gas method and the matrix quantum mechanics approach to two-dimensional string theory. The two systems are distinguished by different target spaces ($\Z$ and $\R$, respectively) as far as {\it observables} are concerned. We argue that target space loop correlators should coincide in the two models and demonstrate this for a number of examples. As a consequence some interesting generic observations about the structure of two-dimensional string theory may be made: Restricting to a discrete target space leads to {\it factorization} of amplitudes and thus to very simple sewing rules. It is also demonstrated that the restriction to the discrete target space still allows to calculate the correlation functions of tachyon operators in the unrestricted theory.