On Correlation Functions for Non-critical Strings with c<1 but d>1

TL;DR

This work extends non-critical string theory formulations at $c\le 1$ to target spaces with $d\ge 1$ by developing a Goulian-Li-type continuation in the number of cosmological-constant insertions. It provides explicit continuum expressions for the three-point function $A_3$ and a special four-point function, formulated through an integral representation of $f(a,b|s)$ and a structured set of pole–zero factors, while analyzing how the analytic structure depends on dimensionality $d$ and the integer continuation parameter $s$. A key finding is that, for $d>1$, the pole structure does not factorize into leg poles as in the $d=1$ case, and integer $s$ can align with a $(d+1)$-dimensional critical-string interpretation, whereas generic $s$ may yield unphysical spectra, signaling potential limits of the higher-dimensional picture and the need for Liouville-mass-shell constraints or screening. The work thus clarifies when a continuum, higher-dimensional string interpretation is valid for non-critical strings with $c\le 1$ and outlines how to test and possibly restore a consistent spectrum via screening insertions.

Abstract

We construct a Goulian-Li-type continuation in the number of insertions of the cosmological constant operator which is no longer restricted to one dimensional target space. The method is applied to the calculation of the three-point and a special four-point correlation function. Various aspects of the emerging analytical structure are discussed.