Three-point correlation functions in the $\mathfrak{sl}_3$ Toda theory II: the Fateev-Litvinov formula

Toda Conformal Field Theories (CFTs) form a family of two-dimensional CFTs indexed by semisimple and complex Lie algebras. One of their remarkable features is that they are natural generalizations of Liouville CFT that enjoy an enhanced level of symmetry, prescribed by $W$-algebras. They likewise admit a probabilistic formulation in terms of Gaussian Multiplicative Chaos. Based on this probabilistic framework, this second article in a two-part series is dedicated to providing a first step towards integrability of these theories. In this perspective we prove the Fateev-Litvinov formula for a family of three-point correlation functions associated to the $\mathfrak{sl}_3$ Toda CFT. This result is the analog of the celebrated DOZZ formula in Liouville CFT. Our method of proof features techniques inspired by the physics literature together with probabilistic ones that naturally arise within the setting of Toda theories.