On classical and semiclassical properties of the Liouville theory with defects
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ELEMENTARY PARTICLES AND FIELDS Theory
On Classical and Semiclassical Properties of the Liouville Theory with Defects∗ H. Poghosyan1)** and G. Sarkissian1), 2)*** Received May 18, 2016
Abstract—The Lagrangian of the Liouville theory with topological defects is analyzed in detail and general solution of the corresponding defect equations of motion is found. We study the heavy and light semiclassical limits of the defect two-point function found before via the bootstrap program. We show that the heavy asymptotic limit is given by the exponential of the Liouville action with defects, evaluated on the solutions with two singular points. We demonstrate that the light asymptotic limit is given by the finite-dimensional path integral over solutions of the defect equations of motion with a vanishing energy– momentum tensor. DOI: 10.1134/S1063778817040214
1. INTRODUCTION Defects in two-dimensional conformal field theories can be realized as oriented lines, separating different theories. We are interested in the special class of topological defects, for which the energy– momentum tensor is continuous across the defect [1]. During the last few years topological defects in the Liouville and Toda field theories attracted some attention due to their relation to the Wilson lines in the AGT correspondence [2]. Defects in the Liouville field theory have been constructed in [3, 4]. In these papers defects were constructed as operators on the Hilbert space of Liouville theory. To obtain these operators, two-point functions in the presence of defects were calculated using the conformal bootstrap program for defects, developed in [3, 5]. It was shown in [3] that there are two families of defects: discrete, corresponding to the degenerate fields and labeled by a pair of positive integers, and continuous, labeled by one continuous parameter. The Lagrangian for the continuous family of twodimensional topological defects was suggested in [6]. It is demonstrated in [6] that topological defects are so called type-II defects, proposed in [7], allowing additional degrees of freedom associated with the defect itself. It is also shown in [6], that requiring the additional degrees of freedom to be represented by a ∗
The text was submitted by the authors in English. Yerevan Physics Institute, Yerevan, Armenia. 2) Department of Physics, Yerevan State University, Yerevan, Armenia. ** E-mail: [email protected] *** E-mail: [email protected] 1)
holomorphic field leads to the topological defects. The aim of this work is to study correspondence between the continuous family of defects realized as operators in the Hilbert space of the Liouville field theory in [3, 4] and the one-parametric family of Lagrangians with defect proposed in [6]. First we find the general solution of the defect equations of motion coming from the Lagrangian proposed in [6]. To link two-point functions in the presence of defects to the Lagrangian with defects we use two strategies: heavy and light asymptotic semiclassical limits [8–13]. Here we develop both procedures of the s
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