Liouville theory, AdS2 string, and three-point functions
This is a write-up of the lectures given in Young Researchers Integrability School 2017. The main goal is to explain the connection between the ODE/IM correspondence and the classical integrability of strings in AdS. As a warm up, we first discuss the classical three-point function of the Liouville theory. Our starting point is the well-known fact that the classical solutions to the Liouville equation can be constructed by solving a certain Schrödinger-like differential equation. We then convert it into a set of functional equations using the idea akin to the ODE/IM correspondence. The classical three-point functions can be computed directly from these functional equations and the result matches with the classical limit of the celebrated DOZZ formula. We then proceed to discuss the semi-classical three-point function of strings in AdS2 and show that one can apply a similar idea by making use of the classical integrability of the string sigma model on AdS2.