Liouville field theory is a two-dimensional conformal field theory (CFT) that plays a significant role in both mathematical and theoretical physics, particularly in string theory, statistical mechanics, and quantum gravity. It is named after the French mathematician Joseph Liouville, who studied the properties of certain types of differential equations, and its origins are connected to the study of surfaces with curvature.
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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 and central charge . 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.