Classical Liouville Three-point Functions from Riemann-Hilbert Analysis

We study semiclassical correlation functions in Liouville field theory on a two-sphere when all operators have large conformal dimensions. In the usual approach, such computation involves solving the classical Liouville equation, which is known to be extremely difficult for higher-point functions. To overcome this difficulty, we propose a new method based on the Riemann-Hilbert analysis, which is applied recently to the holographic calculation of correlation functions in AdS/CFT. The method allows us to directly compute the correlation functions without solving the Liouville equation explicitly. To demonstrate its utility, we apply it to three-point functions, which are known to be solvable, and confirm that it correctly reproduces the classical limit of the DOZZ formula for quantum three-point functions. This provides good evidence for the validity of this method.