Liouville field theory

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 ${\displaystyle \partial _{\mu }=\partial /\partial x^{\mu },\ g_{\mu \nu }}$ is the metric of the two-dimensional space on which the field theory is formulated, ${\displaystyle R}$ is the Ricci scalar of such space, and ${\displaystyle b}$ is a real coupling constant. The field ${\displaystyle \phi }$ is consequently dubbed the Liouville field.

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