Liouville theory
Urs Schreiber
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In the context of quantum field theory, Liouville theory is the name of a certain type of 2d CFT.
An argument (via Chern-Simons gravity, but see the caveats there) that 3d quantum gravity with negative cosmological constant has as boundary field theory 2d Liouville theory is due to (Coussaert-Henneaux-vanDriel 95). See also at AdS3-CFT2...