MECÂNICA GRACELI GERAL - QUÂNTICA DIMENSIONAL TENSORIAL DE CAMPOS [610]

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

[

$\psi ^{*}$

$TeoriaInteraçãomediadorMagnitude relativaComportamentoFaixaCromodinâmicaForça nuclear forteGlúon10411/r71,4 × 10-15 mEletrodinâmicaForça eletromagnéticaFóton10391/r2infinitoFlavordinâmicaForça nuclear fracaBósons W e Z10291/r5 até 1/r710-18 mGeometrodinâmicaForça gravitacionalgráviton101/r2infinito$

Teoria	Interação	mediador	Magnitude relativa	Comportamento	Faixa
Cromodinâmica	Força nuclear forte	Glúon	1041	1/r7	1,4 × 10-15 m
Eletrodinâmica	Força eletromagnética	Fóton	1039	1/r2	infinito
Flavordinâmica	Força nuclear fraca	Bósons W e Z	1029	1/r5 até 1/r7	10-18 m
Geometrodinâmica	Força gravitacional	gráviton	10	1/r2	infinito

$G* = OPERADOR DE DIMENSÕES DE GRACELI.$

DIMENSÕES DE GRACELI SÃO TODA FORMA DE TENSORES, ESTRUTURAS, ENERGIAS, ACOPLAMENTOS, , INTERAÇÕES E CAMPOS, DISTRIBUIÇÕES ELETRÔNICAS, E OUTROS.

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

c\in (1,+\infty )

and its classical limit is

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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[edit]

Liouville theory describes the dynamics of a field $\phi$ 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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

V(\phi )=e^{2b\phi }\ ,

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

Reflection of an energy eigenvector with momentum

\alpha

off Liouville theory's exponential potential

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\alpha \to Q-\alpha \ ,

where the background charge is

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\Delta =\alpha (Q-\alpha )\ .

The central charge and conformal dimensions are invariant under the duality

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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[edit]

Spectrum[edit]

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

corresponds to

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\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 $c\in (1,+\infty )$ . 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[edit]

In Liouville theory, primary fields are usually parametrized by their momentum rather than their conformal dimension, and denoted $V_{\alpha }(z)$ . Both fields $V_{\alpha }(z)$ and $V_{Q-\alpha }(z)$ correspond to the primary state of the representation ${\mathcal {V}}_{\Delta }\otimes {\bar {\mathcal {V}}}_{\Delta }$ , and are related by the reflection relation

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

where the reflection coefficient is^[1]

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

R(\alpha )=\pm \lambda ^{Q-2\alpha }{\frac {\Gamma (b(2\alpha -Q))\Gamma ({\frac {1}{b}}(2\alpha -Q))}{\Gamma (b(Q-2\alpha ))\Gamma ({\frac {1}{b}}(Q-2\alpha ))}}\ .

(The sign is $+1$ if $c\in (-\infty ,1)$ and $-1$ otherwise, and the normalization parameter $\lambda$ is arbitrary.)

Correlation functions and DOZZ formula[edit]

For $c\notin (-\infty ,1)$ , the three-point structure constant is given by the DOZZ formula (for Dorn–Otto^[2] and Zamolodchikov–Zamolodchikov^[3]),

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

C_{\alpha _{1},\alpha _{2},\alpha _{3}}={\frac {\left[b^{{\frac {2}{b}}-2b}\lambda \right]^{Q-\alpha _{1}-\alpha _{2}-\alpha _{3}}\Upsilon _{b}'(0)\Upsilon _{b}(2\alpha _{1})\Upsilon _{b}(2\alpha _{2})\Upsilon _{b}(2\alpha _{3})}{\Upsilon _{b}(\alpha _{1}+\alpha _{2}+\alpha _{3}-Q)\Upsilon _{b}(\alpha _{1}+\alpha _{2}-\alpha _{3})\Upsilon _{b}(\alpha _{2}+\alpha _{3}-\alpha _{1})\Upsilon _{b}(\alpha _{3}+\alpha _{1}-\alpha _{2})}}\ ,

where the special function $\Upsilon _{b}$ is a kind of multiple gamma function.

For $c\in (-\infty ,1)$ , the three-point structure constant is^[1]

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

{\hat {C}}_{\alpha _{1},\alpha _{2},\alpha _{3}}={\frac {\left[(ib)^{{\frac {2}{b}}-2b}\lambda \right]^{Q-\alpha _{1}-\alpha _{2}-\alpha _{3}}{\hat {\Upsilon }}_{b}(0){\hat {\Upsilon }}_{b}(2\alpha _{1}){\hat {\Upsilon }}_{b}(2\alpha _{2}){\hat {\Upsilon }}_{b}(2\alpha _{3})}{{\hat {\Upsilon }}_{b}(\alpha _{1}+\alpha _{2}+\alpha _{3}-Q){\hat {\Upsilon }}_{b}(\alpha _{1}+\alpha _{2}-\alpha _{3}){\hat {\Upsilon }}_{b}(\alpha _{2}+\alpha _{3}-\alpha _{1}){\hat {\Upsilon }}_{b}(\alpha _{3}+\alpha _{1}-\alpha _{2})}}\ ,

where

{\hat {\Upsilon }}_{b}(x)={\frac {1}{\Upsilon _{ib}(-ix+ib)}}\ .

$�$ -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 $g\geq 1$ . 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.^[7]^[4]

Uniqueness of Liouville theory[edit]

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 formulation[edit]

Action and equation of motion[edit]

Liouville theory is defined by the local action

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

S[\phi ]={\frac {1}{4\pi }}\int d^{2}x{\sqrt {g}}(g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi +QR\phi +\lambda 'e^{2b\phi })\ ,

where $g_{\mu \nu }$ is the metric of the two-dimensional space on which the theory is formulated, $�$ is the Ricci scalar of that space, and $\phi$ is the Liouville field. The parameter $\lambda '$ , which is sometimes called the cosmological constant, is related to the parameter $\lambda$ that appears in correlation functions by

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\lambda '=4{\frac {\Gamma (1-b^{2})}{\Gamma (b^{2})}}\lambda ^{b}

The equation of motion associated to this action is

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\Delta \phi (x)={\frac {1}{2}}QR(x)+\lambda 'be^{2b\phi (x)}\ ,

where $\Delta =|g|^{-1/2}\partial _{\mu }(|g|^{1/2}g^{\mu \nu }\partial _{\nu })$ is the Laplace–Beltrami operator. If $g_{\mu \nu }$ is the Euclidean metric, this equation reduces to

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\left({\frac {\partial ^{2}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}}{\partial x_{2}^{2}}}\right)\phi (x_{1},x_{2})=\lambda 'be^{2b\phi (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.^[8]

Conformal symmetry[edit]

Using a complex coordinate system $�$ and a Euclidean metric

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

g_{\mu \nu }dx^{\mu }dx^{\nu }=dzd{\bar {z}}

the energy–momentum tensor's components obey

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

T=T_{zz}=(\partial _{z}\phi )^{2}+Q\partial _{z}^{2}\phi \;,\quad {\bar {T}}=T_{{\bar {z}}{\bar {z}}}=(\partial _{\bar {z}}\phi )^{2}+Q\partial _{\bar {z}}^{2}\phi \ .

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

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

c=1+6Q^{2}

For both of these Virasoro algebras, a field $e^{2\alpha \phi }$ is a primary field with the conformal dimension

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\Delta =\alpha (Q-\alpha )

For the theory to have conformal invariance, the field $e^{2b\phi }$ that appears in the action must be marginal, i.e. have the conformal dimension

\Delta (b)=1

This leads to the relation

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

Q=b+{\frac {1}{b}}

between the background charge and the coupling constant. If this relation is obeyed, then $e^{2b\phi }$ is actually exactly marginal, and the theory is conformally invariant.

Path integral[edit]

The path integral representation of an $�$ -point correlation function of primary fields is

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

\left\langle \prod _{i=1}^{N}V_{\alpha _{i}}(z_{i})\right\rangle =\int D\phi \ e^{-S[\phi ]}\prod _{i=1}^{N}e^{2\alpha _{i}\phi (z_{i})}\ .

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 $b\to b^{-1}$ 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^[9] and the conformal bootstrap.^[6]^[10]

Relations with other conformal field theories[edit]

Some limits of Liouville theory[edit]

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 $b^{2}\notin \mathbb {R} ,b^{2}\to \mathbb {Q} _{<0}$ .^[4] So, for $b^{2}\in \mathbb {Q} _{<0}$ , 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 models[edit]

Liouville theory can be obtained from the $SL_{2}(\mathbb {R} )$ Wess–Zumino–Witten model by a quantum Drinfeld–Sokolov reduction. Moreover, correlation functions of the $H_{3}^{+}$ model (the Euclidean version of the $SL_{2}(\mathbb {R} )$ 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 $SL_{2}/U_{1}$ coset model.^[12] Moreover, there exist theories that continuously interpolate between Liouville theory and the $H_{3}^{+}$ model.^[14]

Conformal Toda theory[edit]

Liouville theory is the simplest example of a Toda field theory, associated to the $A_{1}$ Cartan matrix. More general conformal Toda theories can be viewed as generalizations of Liouville theory, whose Lagrangians involve several bosons rather than one boson $\phi$ , and whose symmetry algebras are W-algebras rather than the Virasoro algebra.

Supersymmetric Liouville theory[edit]

Liouville theory admits two different supersymmetric extensions called ${\mathcal {N}}=1$ supersymmetric Liouville theory and ${\mathcal {N}}=2$ supersymmetric Liouville theory.^[15]

Relations with integrable models[edit]

Sinh-Gordon model[edit]

In flat space, the sinh-Gordon model is defined by the local action:

$equação Graceli estatística tensorial quântica de campos$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

//////

S[\phi ]={\frac {1}{4\pi }}\int d^{2}x\left(\partial ^{\mu }\phi \partial _{\mu }\phi +\lambda \cosh(2b\phi )\right)

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 $0<b<1$ , and it is formally invariant under $b\to b^{-1}$ . However, it has been argued that the model itself is not invariant.^[16]

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MECÂNICA GRACELI GERAL - QUÂNTICA DIMENSIONAL TENSORIAL DE CAMPOS [610]

Introduction[edit]

Spectrum and correlation functions[edit]

Spectrum[edit]

Fields and reflection relation[edit]

Correlation functions and DOZZ formula[edit]

Uniqueness of Liouville theory[edit]

Lagrangian formulation[edit]

Action and equation of motion[edit]

Conformal symmetry[edit]

Path integral[edit]

Relations with other conformal field theories[edit]

Some limits of Liouville theory[edit]

WZW models[edit]

Conformal Toda theory[edit]

Supersymmetric Liouville theory[edit]

Relations with integrable models[edit]

Sinh-Gordon model[edit]

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