Interactions of Irregular Gaiotto States in Liouville Theory

TL;DR

This paper develops a method to compute correlation functions of irregular Gaiotto states in Liouville theory by transforming them into regular vertex operators, expressing complex correlators in terms of primary fields and regularized Schwarzians.

Contribution

It introduces a novel regularization approach for irregular states in Liouville theory, enabling explicit calculation of their correlation functions using conformal transformations.

Findings

Correlation functions expressed via primary fields and Schwarzians

Explicit formulas for three-point functions involving DOZZ constants

Regularization examples provided for ranks one and two

Abstract

We compute the correlation functions of irregular Gaiotto states appearing in the colliding limit of the Liouville theory by using "regularizing" conformal transformations mapping the irregular (coherent) states to regular vertex operators in the Liouville theory. The $N$-point correlation functions of the irregular vertex operators of arbitrary ranks are expressed in terms of $N$-point correlators of primary fields times the factor that involves regularized higher-rank Schwarzians of the above conformal transformation. In particular, in the case of three-point functions the general answer is expressed in terms of DOZZ (Dorn-Otto-Zamolodchikov-Zamolodchikov) structure constants times exponents of regularized higher-derivative Schwarzians. The explicit examples of the regularization are given for the ranks one and two.