Eur. Phys. J. C (2018) 78: 507
https://doi.org/10.1140/epjc/s10052-018-5976-9

Regular Article - Theoretical Physics

Interactions of irregular Gaiotto states in Liouville theory

¹ Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu, 6100064, China
² Institute of Information Transmission Problems (IITP), Bolshoi Karetny per. 19/1, Moscow, 127994, Russia

^* e-mail: polyakov@scu.edu.cn
^** e-mail: polyakov@sogang.ac.kr

Received: 17 January 2018
Accepted: 7 June 2018
Published online: 18 June 2018

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.