On classical and semiclassical properties of the Liouville theory with defects

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.