KZ equations are an essential tool in the study of conformal field theories with affine algebra symmetry. They are satisfied by all correlation functions of affine primary fields. However, string theory in AdS3 forces us to consider fields which are not affine primaries. I will explain that some generalizations of the KZ equations nevertheless still hold. I will discuss the implications of this finding for the relation with Liouville theory and the operator product expansion in the H3 model.

I will introduce the Gaudin Model. This model is described by a system of commuting Hamiltonians. I will explain how the eigenvalue equations for these Hamiltonians arise as the critical level limit of the Knizhnik-Zamolodchikov equations. In particular, some eigenvectors can be built from H3 correlators. Then I will use the H3-Liouville relation to relate these correlators to Liouville theory correlators. The critical level limit is interpreted in Liouville theory as a geometrical limit. This leads to the construction of Gaudin eigenvalues from the accessory parameters which arise in the uniformization of certain Riemann surfaces.

I explain how arbitrary correlators of string theory in the Euclidean AdS3 can be computed from the correlators of Liouville theory. This makes use of the KZ-BPZ correspondence, which relates on the one hand the Knizhnik-Zamolodchikov equations reflecting the group symmetry in AdS3, on the other hand the Belavin-Polyakov-Zamolodchikov equations reflecting the presence of degenerate fields in the Liouville correlators.