Two and three-point functions in Liouville theory

TL;DR

The paper tackles the problem of constructing Liouville correlation functions for arbitrary real powers of the Liouville exponential, a key ingredient for gravitational dressing in noncritical string theory. It adopts a Goulian–Li continuation from integer to real values within a functional-integral framework and computes the three- and two-point functions using Dotsenko–Fateev integrals, yielding meromorphic expressions in terms of Gamma functions and a companion function $f(a,b|s)$. A nontrivial consistency check shows the Liouville equation of motion is satisfied at the level of three-point functions, lending strong support to the continuation method. The work further analyzes the analytic structure (pole-zero spectrum) and proposes a mass-shell condition for noncritical strings based on zeros of the two-point dressing factor, while highlighting unresolved issues that require higher-point functions for robust validation.

Abstract

Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.