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.
Highlights
Irregular Gaiotto states emerge in Liouville field theory in the colliding limit, relevant to extensions of the AGT conjecture to Argyres-Douglas type of gauge theories with asymptotic freedom [1,3,6,7,8,14,19]
In this work we address this problem by using the conformal transformations that maps operators for coherent states into regular vertex operators, expressing the interactions of the irregular states in terms of regular correlators in Liouville theory
We analyzed interactions of the irregular states in Liouville theory by using conformal transformations that map the irregular vertex operators into regular. This allows to express three-point functions of irregulars of arbitrary rank in terms of Liouville structure constants given by DOZZ (Dorn-Otto-Zamolodchikov-Zamolodchikov) formula
Summary
Irregular (coherent) Gaiotto states emerge in Liouville field theory in the colliding limit, relevant to extensions of the AGT conjecture to Argyres-Douglas type of gauge theories with asymptotic freedom [1,3,6,7,8,14,19]. Computing correlation functions describing interactions of the irregular states in Liouville theory is known to be a hard and tedious problem, especially beyond two-point functions and rank one case [5,6,9,14,15]. These correlation functions are important objects as they define the correlators in Argyres-Douglas gauge theories; namely, the Npoint correlators of Argyres-Douglas theories are idenfied with the M-point colliding limit of 2d CFT so that M = M1 + M2 + · · · + MN. In this work we address this problem by using the conformal transformations that maps operators for coherent states into regular vertex operators, expressing the interactions of the irregular states in terms of regular correlators in Liouville theory. In the concluding section we discuss the implications of our results
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