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 ${\displaystyle c}$ of its Virasoro symmetry algebra, but it is unitary only if

${\displaystyle c\in (1,+\infty ),}$

and its classical limit is

${\displaystyle c\to +\infty .}$

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 ${\displaystyle \varphi }$ 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

${\displaystyle V(\varphi )=e^{2b\varphi }\ ,}$

where the parameter ${\displaystyle b}$ is called the coupling constant. In a free field theory, the energy eigenvectors ${\displaystyle e^{2\alpha \varphi }}$ are linearly independent, and the momentum ${\displaystyle \alpha }$ is conserved in interactions. In Liouville theory, momentum is not conserved.

Moreover, the potential reflects the energy eigenvectors before they reach ${\displaystyle \varphi =+\infty }$, and two eigenvectors are linearly dependent if their momenta are related by the reflection

${\displaystyle \alpha \to Q-\alpha \ ,}$

where the background charge is

${\displaystyle Q=b+{\frac {1}{b}}\ .}$

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

${\displaystyle c=1+6Q^{2}\ .}$

Under conformal transformations, an energy eigenvector with momentum ${\displaystyle \alpha }$ transforms as a primary field with the conformal dimension ${\displaystyle \Delta }$ by

${\displaystyle \Delta =\alpha (Q-\alpha )\ .}$

The central charge and conformal dimensions are invariant under the duality

${\displaystyle b\to {\frac {1}{b}}\ ,}$

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 ${\displaystyle {\mathcal {S}}}$ of Liouville theory is a diagonal combination of Verma modules of the Virasoro algebra,

${\displaystyle {\mathcal {S}}=\int _{{\frac {c-1}{24}}+\mathbb {R} _{+}}d\Delta \ {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }\ ,}$

where ${\displaystyle {\mathcal {V}}_{\Delta }}$ and ${\displaystyle {\bar {\mathcal {V}}}_{\Delta }}$ denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively. In terms of momenta,

${\displaystyle \Delta \in {\frac {c-1}{24}}+\mathbb {R} _{+}}$

corresponds to

${\displaystyle \alpha \in {\frac {Q}{2}}+i\mathbb {R} _{+}.}$

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 ${\displaystyle c\in (1,+\infty )}$. 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 ${\displaystyle V_{\alpha }(z)}$. Both fields ${\displaystyle V_{\alpha }(z)}$ and ${\displaystyle V_{Q-\alpha }(z)}$ correspond to the primary state of the representation ${\displaystyle {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }}$, and are related by the reflection relation

${\displaystyle V_{\alpha }(z)=R(\alpha )V_{Q-\alpha }(z)\ ,}$

where the reflection coefficient is^[1]

${\displaystyle R(\mathrm{\alpha <!--\; \alpha \; -->})=\pm <!--\; \pm \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}^{Q-<!--\; -\; -->2\mathrm{\alpha <!--\; \alpha \; -->}}\frac{\mathrm{\Gamma <!--\; \Gamma \; -->}(}{}}$Zdroj:https://en.wikipedia.org?pojem=Liouville_gravity
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