# Wess-Zumino-Witten models and path integrals

# Wess-Zumino-Witten models and path integrals

The Wess-Zumino-Witten (WZW) model is a 2 dimensional conformal field theory (CFT) where the field takes values in a Lie group G or its coset space. For a compact G this CFT is rational and its cosets G/H include for instance all unitary rational CFTs (e.g. the Ising model). WZW model has a formal path integral representation whose rigorous construction has remained elusive and in fact most of its conjectured properties have been discussed using the representation theory of affine Lie algebras. In this talk I will review the basic facts about the path integral formulation of WZW models and then discuss the coset theory SL(2,C)/SU(2). This theory can be formulated in terms of field taking values in the 3-dimensional hyperbolic space and by the work of Ribault, Teschner, Hikida and Schomerus it has been argued to have a mapping to the Liouville CFT. It has interesting connections to the 3d Chern-Simons topological QFT as well as a “quantum” deformation of the geometric and analytic Langlands correspondence. I will explain briefly how this theory can be constructed probabilistically using the theory of Gaussian Multiplicative Chaos and how on a general Riemann surface the correlation functions of its primary fields can be mapped to those of the Liouville CFT.