Liouville field theory ($S_L=\frac{1}{4\pi }\int \sqrt{h}[(\nabla \varphi {)}^2+QR\varphi +4\pi \mu e^{2b\varphi }]$)

Liouville field theory is a two-dimensional scalar theory with a curvature background charge and an exponential interaction. In the usual spacelike normalization, quantum conformal invariance gives $Q=b+b^{-1}$ and central charge $c_L=1+6Q^2$. When a noncritical string worldsheet conformal factor is retained as a dynamical field, its effective action has this form and its central charge combines with the matter and worldsheet ghost field charges. It is closely related to a background-charge scalar field, but its exponential potential supplies an additional interaction.