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 c {\displaystyle c} of its Virasoro symmetry algebra, but it is unitary only if
c ∈ ( 1 , + ∞ ) , {\displaystyle c\in (1,+\infty ),}
and its classical limit is
c → + ∞ . {\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 φ {\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
V ( φ ) = e 2 b φ , {\displaystyle V(\varphi )=e^{2b\varphi }\ ,}
where the parameter b {\displaystyle b} is called the coupling constant. In a free field theory, the energy eigenvectors e 2 α φ {\displaystyle e^{2\alpha \varphi }} are linearly independent, and the momentum α {\displaystyle \alpha } is conserved in interactions. In Liouville theory, momentum is not conserved.
Moreover, the potential reflects the energy eigenvectors before they reach φ = + ∞ {\displaystyle \varphi =+\infty } , and two eigenvectors are linearly dependent if their momenta are related by the reflection
α → Q − α , {\displaystyle \alpha \to Q-\alpha \ ,}
where the background charge is
Q = b + 1 b . {\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
c = 1 + 6 Q 2 . {\displaystyle c=1+6Q^{2}\ .}
Under conformal transformations, an energy eigenvector with momentum α {\displaystyle \alpha } transforms as a primary field with the conformal dimension Δ {\displaystyle \Delta } by
Δ = α ( Q − α ) . {\displaystyle \Delta =\alpha (Q-\alpha )\ .}
The central charge and conformal dimensions are invariant under the duality
b → 1 b , {\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 S {\displaystyle {\mathcal {S}}} of Liouville theory is a diagonal combination of Verma modules of the Virasoro algebra,
S = ∫ c − 1 24 + R + d Δ V Δ ⊗ V ¯ Δ , {\displaystyle {\mathcal {S}}=\int _{{\frac {c-1}{24}}+\mathbb {R} _{+}}d\Delta \ {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }\ ,}
where V Δ {\displaystyle {\mathcal {V}}_{\Delta }} and V ¯ Δ {\displaystyle {\bar {\mathcal {V}}}_{\Delta }} denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively. The conformal dimension Δ {\displaystyle \Delta } takes values Δ ∈ c − 1 24 + R + {\displaystyle \Delta \in {\tfrac {c-1}{24}}+\mathbb {R} _{+}} , which in terms of the momentum α {\displaystyle \alpha } amounts to
α ∈ Q 2 + i P with P ∈ R + . {\displaystyle \alpha \in {\frac {Q}{2}}+iP\qquad {\text{with}}\qquad P\in \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 c ∈ ( 1 , + ∞ ) {\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 V P ( z ) {\displaystyle V_{P}(z)} . Both fields V P ( z ) {\displaystyle V_{P}(z)} and V − P ( z ) {\displaystyle V_{-P}(z)} correspond to the primary state of the representation V Δ ⊗ V ¯ Δ {\displaystyle {\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }} , and are related by the reflection relation
V P ( z ) = R P V − P ( z ) , {\displaystyle V_{P}(z)=R_{P}V_{-P}(z)\ ,}
where the reflection coefficient is
R P = ± λ − 2 i P Γ ( 2 i b P ) Γ ( 2 i b − 1 P ) Γ ( − 2 i b P ) Γ ( − 2 i b − 1 P ) . {\displaystyle R_{P}=\pm \lambda ^{-2iP}{\frac {\Gamma (2ibP)\Gamma (2ib^{-1}P)}{\Gamma (-2ibP)\Gamma (-2ib^{-1}P)}}\ .}
(The sign is + 1 {\displaystyle +1} if c ∈ ( − ∞ , 1 ) {\displaystyle c\in (-\infty ,1)} and − 1 {\displaystyle -1} otherwise, and the normalization parameter λ {\displaystyle \lambda } is arbitrary.)
Correlation functions and DOZZ formula For c ∉ ( − ∞ , 1 ) {\displaystyle c\notin (-\infty ,1)} , the three-point structure constant is given by the DOZZ formula (for Dorn–Otto and Zamolodchikov–Zamolodchikov),
C P 1 , P 2 , P 3 = [ b 2 b − 2 b λ ] − Q 2 − i ( P 1 + P 2 + P 3 ) Υ b ′ ( 0 ) ∏ j = 1 3 Υ b ( − 2 i P j ) ∏ ± , ± Υ b ( Q 2 + i P 1 ± i P 2 ± i P 3 ) , {\displaystyle C_{P_{1},P_{2},P_{3}}={\frac {\left[b^{{\frac {2}{b}}-2b}\lambda \right]^{-{\frac {Q}{2}}-i(P_{1}+P_{2}+P_{3})}\Upsilon _{b}'(0)\prod _{j=1}^{3}\Upsilon _{b}(-2iP_{j})}{\prod _{\pm ,\pm }\Upsilon _{b}({\frac {Q}{2}}+iP_{1}\pm iP_{2}\pm iP_{3})}}\ ,}
where the special function Υ b {\displaystyle \Upsilon _{b}} is a kind of multiple gamma function. For c ∈ ( − ∞ , 1 ) {\displaystyle c\in (-\infty ,1)} , the three-point structure constant is
C ^ P 1 , P 2 , P 3 = [ β 2 ( β + β − 1 ) λ − i ] β − 1 − β 2 + P 1 + P 2 + P 3 ∏ ± , ± Υ β ( β + β − 1 2 + P 1 ± P 2 ± P 3 ) Υ β ( β − 1 ) ∏ j = 1 3 Υ β ( β − 1 + 2 P j ) , {\displaystyle {\hat {C}}_{P_{1},P_{2},P_{3}}={\frac {\left[\beta ^{2(\beta +\beta ^{-1})}\lambda ^{-i}\right]^{{\frac {\beta ^{-1}-\beta }{2}}+P_{1}+P_{2}+P_{3}}\prod _{\pm ,\pm }\Upsilon _{\beta }({\frac {\beta +\beta ^{-1}}{2}}+P_{1}\pm P_{2}\pm P_{3})}{\Upsilon _{\beta }(\beta ^{-1})\prod _{j=1}^{3}\Upsilon _{\beta }(\beta ^{-1}+2P_{j})}}\ ,}
where
β = i b ∈ R . {\displaystyle \beta =ib\in \mathbb {R} \ .}
N {\displaystyle N} -point functions on the sphere can be expressed in terms of three-point structure constants, and conformal blocks. An N {\displaystyle N} -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. The sphere N-point function ⟨ ∏ j = 1 N V P j ( z j ) ⟩ {\displaystyle \left\langle \prod _{j=1}^{N}V_{P_{j}}(z_{j})\right\rangle } is a meromorphic function of P j {\displaystyle P_{j}} . For c ∉ ( − ∞ , 1 ) {\displaystyle c\notin (-\infty ,1)} its poles are at i ∑ j = 1 N ϵ j P j ∈ N − 2 2 Q + N b + N b − 1 {\displaystyle i\sum _{j=1}^{N}\epsilon _{j}P_{j}\in {\tfrac {N-2}{2}}Q+\mathbb {N} b+\mathbb {N} b^{-1}} for any ϵ j = ± {\displaystyle \epsilon _{j}=\pm } . (These poles are simple if they do not coincide.) For c ∈ ( − ∞ , 1 ) {\displaystyle c\in (-\infty ,1)} the poles are due to factors 1 Υ β ( β − 1 + 2 P j ) {\displaystyle {\frac {1}{\Upsilon _{\beta }(\beta ^{-1}+2P_{j})}}} , and can be removed by a field renormalization. The sphere 4-point function transforms covariantly under
P 1 ↦ P Σ , P j ↦ j = 2 , 3 , 4 P 1 + P j − P Σ with P Σ = 1 2 ∑ j = 1 4 P j . {\displaystyle P_{1}\mapsto P_{\Sigma }\quad ,\quad P_{j}{\underset {j=2,3,4}{\mapsto }}P_{1}+P_{j}-P_{\Sigma }\quad {\text{with}}\quad P_{\Sigma }={\frac {1}{2}}\sum _{j=1}^{4}P_{j}\ .}
Namely, under this transformation, for c ∉ ( − ∞ , 1 ) {\displaystyle c\notin (-\infty ,1)} , the following is invariant:
( ∏ j < k | z j − z k | 2 P j 2 + 2 P k 2 ) ( ∏ j = 1 4 1 Υ b ( − 2 i P j ) ) ⟨ ∏ j = 1 4 V P j ( z j ) ⟩ . {\displaystyle \left(\prod _{j<k}|z_{j}-z_{k}|^{2P_{j}^{2}+2P_{k}^{2}}\right)\left(\prod _{j=1}^{4}{\frac {1}{\Upsilon _{b}(-2iP_{j})}}\right)\left\langle \prod _{j=1}^{4}V_{P_{j}}(z_{j})\right\rangle \ .}
Liouville theory exists not only on the sphere, but also on any Riemann surface of genus g ≥ 1 {\displaystyle g\geq 1} . 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 b {\displaystyle b} and the momenta.
Lagrangian formulation
Action and equation of motion Liouville theory is defined by the local action
S [ φ ] = 1 4 π ∫ d 2 x g ( g μ ν ∂ μ φ ∂ ν φ + Q R φ + λ ′ e 2 b φ ) , {\displaystyle S[\varphi ]={\frac {1}{4\pi }}\int d^{2}x\,{\sqrt {g}}(g^{\mu \nu }\partial _{\mu }\varphi \partial _{\nu }\varphi +QR\varphi +\lambda 'e^{2b\varphi })\ ,}
where g μ ν {\displaystyle g_{\mu \nu }} is the metric of the two-dimensional space on which the theory is formulated, R {\displaystyle R} is the Ricci scalar of that space, and φ {\displaystyle \varphi } is the Liouville field. The parameter λ ′ {\displaystyle \lambda '} , which is sometimes called the cosmological constant, is related to the parameter λ {\displaystyle \lambda } that appears in correlation functions by
λ ′ = 4 Γ ( 1 − b 2 ) Γ ( b 2 ) λ b . {\displaystyle \lambda '=4{\frac {\Gamma (1-b^{2})}{\Gamma (b^{2})}}\lambda ^{b}.}
The equation of motion associated to this action is
Δ φ ( x ) = 1 2 Q R ( x ) + λ ′ b e 2 b φ ( x ) , {\displaystyle \Delta \varphi (x)={\frac {1}{2}}QR(x)+\lambda 'be^{2b\varphi (x)}\ ,}
where Δ = | g | − 1 / 2 ∂ μ ( | g | 1 / 2 g μ ν ∂ ν ) {\displaystyle \Delta =|g|^{-1/2}\partial _{\mu }(|g|^{1/2}g^{\mu \nu }\partial _{\nu })} is the Laplace–Beltrami operator. If g μ ν {\displaystyle g_{\mu \nu }} is the Euclidean metric, this equation reduces to
( ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 ) φ ( x 1 , x 2 ) = λ ′ b e 2 b φ ( x 1 , x 2 ) , {\displaystyle \left({\frac {\partial ^{2}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}}{\partial x_{2}^{2}}}\right)\varphi (x_{1},x_{2})=\lambda 'be^{2b\varphi (x_{1},x_{2})}\ ,}
which is equivalent to Liouville's equation. Once compactified on a cylinder, Liouville field theory can be equivalently formulated as a worldline theory.
Conformal symmetry Using a complex coordinate system z {\displaystyle z} and a Euclidean metric
g μ ν d x μ d x ν = d z d z ¯ , {\displaystyle g_{\mu \nu }dx^{\mu }dx^{\nu }=dzd{\bar {z}},}
the energy–momentum tensor's components obey
T z z ¯ = T z ¯ z = 0 , ∂ z ¯ T z z = 0 , ∂ z T z ¯ z ¯ = 0 . {\displaystyle T_{z{\bar {z}}}=T_{{\bar {z}}z}=0\;,\quad \partial _{\bar {z}}T_{zz}=0\;,\quad \partial _{z}T_{{\bar {z}}{\bar {z}}}=0\ .}
The non-vanishing components are
T = T z z = ( ∂ z φ ) 2 + Q ∂ z 2 φ , T ¯ = T z ¯ z ¯ = ( ∂ z ¯ φ ) 2 + Q ∂ z ¯ 2 φ . {\displaystyle T=T_{zz}=(\partial _{z}\varphi )^{2}+Q\partial _{z}^{2}\varphi \;,\quad {\bar {T}}=T_{{\bar {z}}{\bar {z}}}=(\partial _{\bar {z}}\varphi )^{2}+Q\partial _{\bar {z}}^{2}\varphi \ .}
Each one of these two components generates a Virasoro algebra with the central charge
c = 1 + 6 Q 2 . {\displaystyle c=1+6Q^{2}.}
For both of these Virasoro algebras, a field e 2 α φ {\displaystyle e^{2\alpha \varphi }} is a primary field with the conformal dimension
Δ = α ( Q − α ) . {\displaystyle \Delta =\alpha (Q-\alpha ).}
For the theory to have conformal invariance, the field e 2 b φ {\displaystyle e^{2b\varphi }} that appears in the action must be marginal, i.e. have the conformal dimension
Δ ( b ) = 1. {\displaystyle \Delta (b)=1.}
This leads to the relation
Q = b + 1 b {\displaystyle Q=b+{\frac {1}{b}}}
between the background charge and the coupling constant. If this relation is obeyed, then e 2 b φ {\displaystyle e^{2b\varphi }} is actually exactly marginal, and the theory is conformally invariant.
Path integral The path integral representation of an N {\displaystyle N} -point correlation function of primary fields is
⟨ ∏ i = 1 N V α i ( z i ) ⟩ = ∫ D φ e − S [ φ ] ∏ i = 1 N e 2 α i φ ( z i ) . {\displaystyle \left\langle \prod _{i=1}^{N}V_{\alpha _{i}}(z_{i})\right\rangle =\int D\varphi \ e^{-S[\varphi ]}\prod _{i=1}^{N}e^{2\alpha _{i}\varphi (z_{i})}\ .}
It has been difficult to define and to compute this pa