Non-Rational 2D Quantum Gravity: I. World Sheet CFT

TL;DR

This paper investigates tachyon correlation functions in non-rational Liouville gravity with c<1, deriving recurrence relations and explicit solutions for 3- and 4-point functions in two variants of the theory, including a diagonal deformation.

Contribution

It introduces a novel diagonal deformation of Liouville gravity and computes explicit correlation functions, extending understanding of non-rational 2D quantum gravity models.

Findings

Derived recurrence relations for tachyon correlation functions.

Obtained explicit 3- and 4-point function solutions.

Provided a closed expression for 4-point functions in the diagonal theory.

Abstract

We address the problem of computing the tachyon correlation functions in Liouville gravity with generic (non-rational) matter central charge c<1. We consider two variants of the theory. The first is the conventional one in which the effective matter interaction is given by the two matter screening charges. In the second variant the interaction is defined by the Liouville dressings of the non-trivial vertex operator of zero dimension. This particular deformation, referred to as "diagonal'', is motivated by the comparison with the discrete approach, which is the subject of a subsequent paper. In both theories we determine the ground ring of ghost zero physical operators by computing its OPE action on the tachyons and derive recurrence relations for the tachyon bulk correlation functions. We find 3- and 4-point solutions to these functional equations for various matter spectra. In particular, we find a closed expression for the 4-point function of order operators in the diagonal theory.