In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion resembles the Joseph Liouville's non-linear second order differential equation that appears in the classical geometrical problem of uniformizing Riemann surfaces.
The field theory is defined by the local action
where
The equation of motion associated to this action is
where
Liouville field theory is a conformal field theory that incarnates Weyl symmetry in a very special way. Its central charge
Liouville field theory is one of the best understood examples of what is called a non-rational conformal field theory, for which some observables have been computed explicitly. Such is the case of two-point and three-point correlation functions of primary operators on the topology of the sphere. Explicit expressions for observables of the theory defined on other topologies, like the partition function on the torus and the one-point function on the disk, were also calculated in the recent years.
Liouville theory is also closely related to other problems in physics and mathematics, like two-dimensional quantum gravity, two-dimensional string theory, three-dimensional general relativity in negatively curved spaces, four-dimensional superconformal gauge theories, the uniformization problem of Riemann surfaces, and other problems in conformal mapping. It is also connected to other two-dimensional non-rational conformal field theories with affine symmetry, like the Wess–Zumino–Novikov–Witten theory for the group