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.
Reflection of an energy eigenvector with momentum off Liouville theory's exponential potential
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
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.
where and denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively. In terms of momenta,
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 relationEdit
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
-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 theoryEdit
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,
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
The equation of motion associated to this action is
between the background charge and the coupling constant. If this relation is obeyed, then is actually exactly marginal, and the theory is conformally invariant.
The path integral representation of an -point correlation function of primary fields is
It has been difficult to define and to compute this path integral. In the path integral representation, it is not obvious that Liouville theory has exact conformal invariance, and it is not manifest that correlation functions are invariant under and obey the reflection relation. Nevertheless, the path integral representation can be used for computing the residues of correlation functions at some of their poles as Dotsenko-Fateev integrals in the Coulomb gas formalism, and this is how the DOZZ formula was first guessed in the 1990s. It is only in the 2010s that a rigorous probabilistic construction of the path integral was found, which led to a proof of the DOZZ formula and the conformal bootstrap.
Relations with other conformal field theoriesEdit
Some limits of Liouville theoryEdit
When the central charge and conformal dimensions are sent to the relevant discrete values, correlation functions of Liouville theory reduce to correlation functions of diagonal (A-series) Virasoro minimal models.
On the other hand, when the central charge is sent to one while conformal dimensions stay continuous, Liouville theory tends to Runkel-Watts theory, a nontrivial conformal field theory (CFT) with a continuous spectrum whose three-point function is not analytic as a function of the momenta. Generalizations of Runkel-Watts theory are obtained from Liouville theory by taking limits of the type . So, for , two distinct CFTs with the same spectrum are known: Liouville theory, whose three-point function is analytic, and another CFT with a non-analytic three-point function.
Liouville theory can be obtained from the Wess–Zumino–Witten model by a quantum Drinfeld-Sokolov reduction. Moreover, correlation functions of the model (the Euclidean version of the WZW model) can be expressed in terms of correlation functions of Liouville theory. This is also true of correlation functions of the 2d black hole coset model. Moreover, there exist theories that continuously interpolate between Liouville theory and the model.
Conformal Toda theoryEdit
Liouville theory is the simplest example of a Toda field theory, associated to the Cartan matrix. More general conformal Toda theories can be viewed as generalizations of Liouville theory, whose Lagrangians involve several bosons rather than one boson , and whose symmetry algebras are W-algebras rather than the Virasoro algebra.
Supersymmetric Liouville theoryEdit
Liouville theory admits two different supersymmetric extensions called supersymmetric Liouville theory and supersymmetric Liouville theory.
Relations with integrable modelsEdit
In flat space, the sinh-Gordon model is defined by the local action:
The corresponding classical equation of motion is the sinh-Gordon equation.
The model can be viewed as a perturbation of Liouville theory. The model's exact S-matrix is known in the weak coupling regime , and it is formally invariant under . However, it has been argued that the model itself is not invariant.
Liouville theory appears in the context of string theory when trying to formulate a non-critical version of the theory in the path integral formulation. Also in the string theory context, if coupled to a free bosonic field, Liouville field theory can be thought of as the theory describing string excitations in a two-dimensional space(time).
Random energy modelsEdit
There is an exact mapping between Liouville theory with , and certain log-correlated random energy models. These models describe a thermal particle in a random potential that is logarithmically correlated. In two dimensions, such potential coincides with the Gaussian free field. In that case, certain correlation functions between primary fields in the Liouville theory are mapped to correlation functions of the Gibbs measure of the particle. This has applications to extreme value statistics of the two-dimensional Gaussian free field, and allows to predict certain universal properties of the log-correlated random energy models (in two dimensions and beyond).
Liouville theory with first appeared as a model of time-dependent string theory under the name timelike Liouville theory.
It has also been called a generalized minimal model. It was first called Liouville theory when it was found to actually exist, and to be spacelike rather than timelike. As of 2022, not one of these three names is universally accepted.
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Mathematicians Prove 2D Version of Quantum Gravity Really Works, Quanta Magazine article by Charlie Wood, June 2021.
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