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.
Introduction
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.
Spectrum and correlation functions
Spectrum
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. In terms of momenta,
corresponds to
. The reflection relation is responsible for the momentum taking values on a half-line, instead of a full line for a free theory.
Liouville theory is unitary if and only if . The spectrum of Liouville theory does not include a vacuum state. A vacuum state can be defined, but it does not contribute to operator product expansions.
Fields and reflection relation
In Liouville theory, primary fields are usually parametrized by their momentum rather than their conformal dimension, and denoted . Both fields and correspond to the primary state of the representation , and are related by the reflection relation
where the reflection coefficient is
(The sign is if and otherwise, and the normalization parameter is arbitrary.)
Correlation functions and DOZZ formula
For , the three-point structure constant is given by the DOZZ formula (for Dorn–Otto and Zamolodchikov–Zamolodchikov),
where the special function is a kind of multiple gamma function.
For , the three-point structure constant is
where
-point functions on the sphere can be expressed in terms of three-point structure constants, and conformal blocks. An -point function may have several different expressions: that they agree is equivalent to crossing symmetry of the four-point function, which has been checked numerically and proved analytically.
Liouville theory exists not only on the sphere, but also on any Riemann surface of genus . Technically, this is equivalent to the modular invariance of the torus one-point function. Due to remarkable identities of conformal blocks and structure constants, this modular invariance property can be deduced from crossing symmetry of the sphere four-point function.
Uniqueness of Liouville theory
Using the conformal bootstrap approach, Liouville theory can be shown to be the unique conformal field theory such that the spectrum is a continuum, with no multiplicities higher than one, the correlation functions depend analytically on and the momenta, degenerate fields exist.
Lagrangian formulation
Action and equation of motion
Liouville theory is defined by the local action
where is the metric of the two-dimensional space on which the theory is formulated, is the Ricci scalar of that space, and is the Liouville field. The parameter , which is sometimes called the cosmological constant, is related to the parameter that appears in correlation functions by .