Explore a probabilistic construction of the Wess-Zumino-Witten (WZW) model with values in the hyperbolic 3-space H³ = SL(2, C)/SU(2) through this mathematical lecture. Learn how this construction applies to both standard and gauged cases on compact Riemann surfaces, and discover the proof of a physics conjecture demonstrating that correlation functions can be expressed in terms of Liouville correlation functions in a highly non-trivial manner. Gain insights into advanced topics in conformal field theory, probability theory, and mathematical physics through research conducted jointly with Kupiainen and Rhodes, presented in this hour-long mathematical exposition.