In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge of its Virasoro symmetry algebra, but it is unitary only if and its classical limit is Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically. Liouville theory describes the dynamics of a field called the Liouville field, which is defined on a two-dimensional space. This field is not a free field due to the presence of an exponential potential where the parameter is called the coupling constant. In a free field theory, the energy eigenvectors are linearly independent, and the momentum is conserved in interactions. In Liouville theory, momentum is not conserved. Moreover, the potential reflects the energy eigenvectors before they reach , and two eigenvectors are linearly dependent if their momenta are related by the reflection where the background charge is While the exponential potential breaks momentum conservation, it does not break conformal symmetry, and Liouville theory is a conformal field theory with the central charge Under conformal transformations, an energy eigenvector with momentum transforms as a primary field with the conformal dimension by The central charge and conformal dimensions are invariant under the duality The correlation functions of Liouville theory are covariant under this duality, and under reflections of the momenta. These quantum symmetries of Liouville theory are however not manifest in the Lagrangian formulation, in particular the exponential potential is not invariant under the duality. The spectrum of Liouville theory is a diagonal combination of Verma modules of the Virasoro algebra, where and denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively.

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Related concepts (4)
Two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.
Operator product expansion
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories. Whether this result can be extended to QFT in general, thus resolving many of the difficulties of a perturbative approach, remains an open research question.
Conformal bootstrap
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of resolution. Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients.
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