Liouville conformal field theory (LCFT) is a family of CFTs which arise in a wide variety of contexts in the physics literature. There are two main (seemingly unrelated) approaches to LCFT in the physics literature: one in the Feynman path integral formulation and one in the conformal bootstrap approach. Recently, we constructed rigorously LCFT in the Feynman path integral formulation. In this talk, I will present recent results on the local conformal structure of LCFT in the Feynman path integral formulation (Ward and BPZ identities, operator product expansion, etc...). These results are a first step in showing that both approaches in the physics literature (Feynmam path integral and conformal bootstrap) are in fact identical. Based on joint works with F. David, A. Kupiainen and R. Rhodes.
