The free field approach to Liouville CFT

The goal of my lectures will be to describe an older approach to the solution to Liouville theory which can supplement the powerful probabilistic approach of Kupiainen, Rhodes and Vargas by providing detailed information on the chiral factorisation and on the spectrum of Liouville conformal field theory. The main idea of this approach is to quantise the well-known representation of the solution to the Liouville equation of motion in terms of a free bosonic field. We will present the crucial ingredient in the verification that the quantised Liouville field is local, the braid relations of the chiral vertex operators representing the main building blocks of the construction. If time permits we will explain how the calculation of braid relations gives hints on connections to quantum Teichmueller theory and the harmonic analysis of Diff(S^1)