In 1981, Polyakov presented in a pioneer article a canonical way of defining the notion of random surface in the setting of 2d quantum gravity, usually called Liouville conformal field theory. A few years later Belavin, Polyakov and Zamolodchikov (BPZ) presented in their 1984 seminal work a systematic procedure to « solve » more generally a certain class of models, the so-called 2d CFTs, which like Liouville theory possess certain conformal symmetries.
The main input of their method was to translate in a representation theoretical language the constraints imposed by conformal symmetry and study the consequences of these constraints from an algebraic viewpoint. The main drawback of this machinery being that in order to go from representation theory to actual models (such as Liouville theory), physicists usually rely on certain axioms and often lack mathematically rigorous formulation.
In this talk we will investigate how to implement the method of BPZ to a mathematically rigorous model : the probabilistic construction of Liouville theory. If time permits we will discuss the generalisation of this method to Toda theories, which are natural extensions of Liouville theories.