Liouville field theory


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.

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


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 functionsEdit


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 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


where the reflection coefficient is[1]


(The sign is   if   and   otherwise, and the normalization parameter   is arbitrary.)

Correlation functions and DOZZ formulaEdit

For  , the three-point structure constant is given by the DOZZ formula (for Dorn-Otto[2] and Zamolodchikov-Zamolodchikov[3]),


where the special function   is a kind of multiple gamma function.

For  , the three-point structure constant is[1]




 -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[3][4] and proved analytically.[5][6]

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.[7][4]

Uniqueness of Liouville theoryEdit

Using the conformal bootstrap approach, Liouville theory can be shown to be the unique conformal field theory such that[1]

  • the spectrum is a continuum, with no multiplicities higher than one,
  • the correlation functions depend analytically on   and the momenta,
  • degenerate fields exist.

Lagrangian formulationEdit

Action and equation of motionEdit

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


The equation of motion associated to this action is


where   is the Laplace–Beltrami operator. If   is the Euclidean metric, this equation reduces to


which is equivalent to Liouville's equation.

Conformal symmetryEdit

Using a complex coordinate system   and a Euclidean metric


the energy–momentum tensor's components obey


The non-vanishing components are


Each one of these two components generates a Virasoro algebra with the central charge


For both of these Virasoro algebras, a field   is a primary field with the conformal dimension


For the theory to have conformal invariance, the field   that appears in the action must be marginal, i.e. have the conformal dimension


This leads to the relation


between the background charge and the coupling constant. If this relation is obeyed, then   is actually exactly marginal, and the theory is conformally invariant.

Path integralEdit

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[8] and the conformal bootstrap.[9][10]

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.[1]

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.[11] Generalizations of Runkel-Watts theory are obtained from Liouville theory by taking limits of the type  .[4] 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.

WZW modelsEdit

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.[12][13] This is also true of correlation functions of the 2d black hole   coset model.[12] Moreover, there exist theories that continuously interpolate between Liouville theory and the   model.[14]

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.[15]

Relations with integrable modelsEdit

Sinh-Gordon modelEdit

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.[16]


Liouville gravityEdit

In two dimensions, the Einstein equations reduce to Liouville's equation, so Liouville theory provides a quantum theory of gravity that is called Liouville gravity. It should not be confused[17][18] with the CGHS model or Jackiw–Teitelboim gravity.

String theoryEdit

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.[19] 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.[20] 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).

Other applicationsEdit

Liouville theory is related to other subjects in physics and mathematics, such as three-dimensional general relativity in negatively curved spaces, the uniformization problem of Riemann surfaces, and other problems in conformal mapping. It is also related to instanton partition functions in a certain four-dimensional superconformal gauge theories by the AGT correspondence.

Naming confusion for c<=1Edit

Liouville theory with   first appeared as a model of time-dependent string theory under the name timelike Liouville theory.[21] It has also been called a generalized minimal model.[22] It was first called Liouville theory when it was found to actually exist, and to be spacelike rather than timelike.[4] As of 2022, not one of these three names is universally accepted.


  1. ^ a b c d Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th].
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External linksEdit

  • Mathematicians Prove 2D Version of Quantum Gravity Really Works, Quanta Magazine article by Charlie Wood, June 2021.
  • An Introduction to Liouville Theory, Talk at Institute for Advanced Study by Antti Kupiainen, May 2018.