Liouville field theory

Liouville theory describes the dynamics of a field $\backslash varphi$ 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(\varphi) = e^{2b\varphi}\ ,

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

File:Liouville reflection.svg

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

:$$

\alpha \to Q-\alpha\ ,

where the background charge is

:$$

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 \ .

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

:$$

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

The central charge and conformal dimensions are invariant under the duality

:$$

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.

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

:$$

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

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

:$\backslash Delta\; \backslash in\; \backslash frac\{c-1\}\{24\}\; +\; \backslash mathbb\{R\}\_+$

corresponds to

:$\backslash alpha\backslash in\; \backslash frac\{Q\}\{2\}+i\backslash mathbb\{R\}\_+.$

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 $c\backslash in\; (1,+\backslash infty)$. 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.

In Liouville theory, primary fields are usually parametrized by their momentum rather than their conformal dimension, and denoted $V\_\backslash alpha(z)$.

Both fields $V\_\backslash alpha(z)$ and $V\_\{Q-\backslash alpha\}(z)$ correspond to the primary state of the representation $\backslash mathcal\{V\}\_\backslash Delta\; \backslash otimes\; \backslash bar\{\backslash mathcal\{V\}\}\_\backslash Delta$, and are related by the reflection relation

:$V\_\backslash alpha(z)\; =\; R(\backslash alpha)\; V\_\{Q-\backslash alpha\}(z)\backslash \; ,$

where the reflection coefficient is{{cite arXiv |eprint=1406.4290|last1=Ribault|first1=Sylvain|title=Conformal field theory on the plane|class=hep-th|year=2014}}

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

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

For $c\backslash notin\; (-\backslash infty,\; 1)$, the three-point structure constant is given by the DOZZ formula (for Dorn–Otto{{cite journal|doi=10.1016/0550-3213(94)00352-1|arxiv=hep-th/9403141|title=Two and three point functions in Liouville theory|date=1994|last1=Dorn|first1=H.|last2=Otto|first2=H.-J.|journal=Nucl. Phys. B|volume=429|pages=375–388|bibcode = 1994NuPhB......375D |s2cid=15413971}} and Zamolodchikov–Zamolodchikov{{cite journal|doi= 10.1016/0550-3213(96)00351-3|arxiv=hep-th/9506136|title= Conformal bootstrap in Liouville field theory|date= 1996|last1= Zamolodchikov|first1= A.|last2= Zamolodchikov|first2= Al.|journal= Nuclear Physics B|volume= 477|issue= 2|pages= 577–605|bibcode = 1996NuPhB.477..577Z |s2cid=204929527}}),

:$$

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 $\backslash Upsilon\_b$ is a kind of multiple gamma function.

For $c\backslash in\; (-\backslash infty,\; 1)$, the three-point structure constant is

:$$

\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)}\ .

$N$-point functions on the sphere can be expressed in terms of three-point structure constants, and conformal blocks. An $N$-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.{{Cite journal|arxiv=hep-th/0303150|last1=Teschner|first1=J|title=A lecture on the Liouville vertex operators|journal=International Journal of Modern Physics A|volume=19|issue=2|pages=436–458|year=2003|doi=10.1142/S0217751X04020567|bibcode=2004IJMPA..19S.436T|s2cid=14792780}}{{cite arXiv|eprint=2005.11530|last1=Guillarmou|first1=C|title=Conformal Bootstrap in Liouville Theory|last2=Kupiainen|first2=A|last3=Rhodes|first3=R|first4=Vargas|last4=V|year=2020|class=math.PR }}

Liouville theory exists not only on the sphere, but also on any Riemann surface of genus $g\backslash geq\; 1$. 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.{{Cite journal|arxiv=0911.4296|last1=Hadasz|first1=Leszek|title=Modular bootstrap in Liouville field theory|journal=Physics Letters B|volume=685|issue=1|pages=79–85|last2=Jaskolski|first2=Zbigniew|last3=Suchanek|first3=Paulina|year=2010|doi=10.1016/j.physletb.2010.01.036|bibcode=2010PhLB..685...79H|s2cid=118625083}}{{Cite journal|arxiv=1503.02067|last1=Ribault|first1=Sylvain|title=Liouville theory with a central charge less than one|journal=Journal of High Energy Physics|volume=2015|issue=8|pages=109|last2=Santachiara|first2=Raoul|year=2015|doi=10.1007/JHEP08(2015)109|bibcode=2015JHEP...08..109R|s2cid=54193340}}

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 $b$ and the momenta,
• degenerate fields exist.

Liouville theory is defined by the local action

:$$

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

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

:$\backslash lambda\text{'}\; =4\; \backslash frac\{\backslash Gamma(1-b^2)\}\{\backslash Gamma(b^2)\}\; \backslash lambda^b.$

The equation of motion associated to this action is

:$$

\Delta \varphi(x) = \frac {1}{2} Q R(x) + \lambda' b e^{2b\varphi (x)} \ ,

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

:$$

\left(\frac{\partial ^2}{\partial x_1^2} + \frac{\partial ^2}{\partial x_2^2} \right) \varphi (x_1,x_2) = \lambda' b e^{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.{{cite journal |last1=Andrei Ioan|first1=Dogaru|last2=Campos Delgado|first2=Ruben |title=Cylinder quantum field theories at small coupling |journal=J. High Energy Phys.|year=2022 |volume=2022|issue=10 |page=110 |doi=10.1007/JHEP10(2022)110|doi-access=free|arxiv=2205.07363}}

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

:$g\_\{\backslash mu\; \backslash nu\}dx^\backslash mu\; dx^\backslash nu=\; dzd\backslash bar\{z\},$

the energy–momentum tensor's components obey

:$$

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_{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

:$c\; =\; 1\; +\; 6Q^2.$

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

:$\backslash Delta\; =\; \backslash alpha(Q-\backslash alpha).$

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

:$\backslash Delta(b)\; =\; 1.$

This leads to the relation

:$Q\; =\; b+\backslash frac\{1\}\{b\}$

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

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

:$$

\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 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 $b\backslash to\; b^\{-1\}$ 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{{cite arXiv |eprint=1707.08785|last1=Kupiainen|first1=Antti|title=Integrability of Liouville theory: Proof of the DOZZ Formula|last2=Rhodes|first2=Rémi|last3=Vargas|first3=Vincent|class=math.PR|year=2017}} and the conformal bootstrap.

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.{{Cite journal|arxiv=hep-th/0306026|last1=Schomerus|first1=Volker|title=Rolling Tachyons from Liouville theory|journal=Journal of High Energy Physics|volume=2003|issue=11|pages=043|year=2003|doi=10.1088/1126-6708/2003/11/043|bibcode=2003JHEP...11..043S|s2cid=15608105}} 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 $SL\_2(\backslash mathbb\{R\})$ Wess–Zumino–Witten model by a quantum Drinfeld–Sokolov reduction. Moreover, correlation functions of the $H\_3^+$ model (the Euclidean version of the $SL\_2(\backslash mathbb\{R\})$ WZW model) can be expressed in terms of correlation functions of Liouville theory.{{Cite journal|arxiv=hep-th/0502048|last1=Ribault|first1=Sylvain|title=H(3)+ correlators from Liouville theory|journal=Journal of High Energy Physics|volume=2005|issue=6|pages=014|last2=Teschner|first2=Joerg|year=2005|doi=10.1088/1126-6708/2005/06/014|bibcode=2005JHEP...06..014R|s2cid=119441269}}{{Cite journal|arxiv=0706.1030|last1=Hikida|first1=Yasuaki|title=H^+_3 WZNW model from Liouville field theory|journal=Journal of High Energy Physics|volume=2007|issue=10|pages=064|last2=Schomerus|first2=Volker|year=2007|doi=10.1088/1126-6708/2007/10/064|bibcode=2007JHEP...10..064H|s2cid=1807250}} This is also true of correlation functions of the 2d black hole $SL\_2/U\_1$ coset model. Moreover, there exist theories that continuously interpolate between Liouville theory and the $H\_3^+$ model.{{Cite journal|arxiv=0803.2099|last1=Ribault|first1=Sylvain|title=A family of solvable non-rational conformal field theories|journal=Journal of High Energy Physics|volume=2008|issue=5|pages=073|year=2008|doi=10.1088/1126-6708/2008/05/073|bibcode=2008JHEP...05..073R|s2cid=2591498}}

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 $\backslash varphi$, and whose symmetry algebras are W-algebras rather than the Virasoro algebra.

Liouville theory admits two different supersymmetric extensions called $\backslash mathcal\{N\}=1$ supersymmetric Liouville theory and $\backslash mathcal\{N\}=2$ supersymmetric Liouville theory.{{cite journal|doi=10.1142/S0217751X04019500|arxiv=hep-th/0402009|title=Liouville Field Theory: A Decade After the Revolution|date=2004|last1=Nakayama|first1=Yu|journal=International Journal of Modern Physics A|volume=19|issue=17n18|pages=2771–2930|bibcode = 2004IJMPA..19.2771N |citeseerx=10.1.1.266.6964|s2cid=119519820}}

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

:$$

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

In two dimensions, Liouville theory can be used to build a quantum theory of gravity called Liouville gravity. It should not be confused{{cite journal | last1 = Grumiller | first1 = Daniel | authorlink1 = Daniel Grumiller | last2 = Kummer | first2 = Wolfgang | last3 = Vassilevich | first3 = Dmitri | authorlink3 = Dmitri Vassilevich | date = October 2002 | title = Dilaton Gravity in Two Dimensions | journal = Physics Reports | volume = 369 | issue = 4 | pages = 327–430 | doi = 10.1016/S0370-1573(02)00267-3 |arxiv = hep-th/0204253 |bibcode = 2002PhR...369..327G | s2cid = 119497628 | url = https://cds.cern.ch/record/549532 | type = Submitted manuscript }}

{{cite journal|last1=Grumiller |first1=Daniel |authorlink1=Daniel Grumiller |last2=Meyer |first2=Rene |authorlink2=Rene Meyer |year=2006 |title=Ramifications of Lineland |journal=Turkish Journal of Physics |volume=30 |issue=5 |pages=349–378 |url=http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=fiz&rak=0604-8 |archiveurl=https://web.archive.org/web/20110822060534/http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=fiz&rak=0604-8 |archivedate=22 August 2011 |arxiv=hep-th/0604049 |bibcode=2006TJPh...30..349G |url-status=dead }}

with the CGHS model or Jackiw–Teitelboim gravity.

In two dimensions, the Einstein-Hilbert action is topological, i.e. it is proportional to the Euler characteristic. Nevertheless, after quantization, general relativity is no longer topological, because of the Weyl anomaly: under a rescaling of the metric $g\backslash mapsto\; e^\backslash phi\; g$, while the action is invariant, the functional integration measure is not, and gives rise to a term proportional to the Liouville action for $\backslash phi$. This leads to the construction of Liouville gravity as a product of three CFTs: Liouville theory for the gravitational sector, Faddeev-Popov ghosts for Weyl invariance (viewed as a gauge symmetry), and an arbitrary CFT that describes matter. The central charges of these CFTs must sum to zero in order to cancel the Weyl anomaly, and ensure that the quantum theory is topological.

The observables of Liouville gravity are correlation numbers: correlation functions of the product CFT, integrated over the moduli. Correlation numbers can be computed explicitly in some examples, such as the Virasoro minimal string. Correlation numbers at fixed Euler characteristic are the coefficients of quantum gravity correlators, when expanded in powers of the gravitational constant.

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.{{cite journal|doi=10.1016/0370-2693(81)90743-7|title=Quantum geometry of bosonic strings|date=1981|last1=Polyakov|first1=A.M.|journal=Physics Letters B|volume=103|issue=3|pages=207–210|bibcode = 1981PhLB..103..207P }} The theory also appears as the description of bosonic string theory in two spacetime dimensions with a linear dilaton and a tachyon background. The tachyon field equation of motion in the linear dilaton background requires it to take an exponential solution. The Polyakov action in this background is then identical to Liouville field theory, with the linear dilaton being responsible for the background charge term while the tachyon contributing the exponential potential.{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume I: An Introduction to the Bosonic String|url=|doi=|location=|publisher=Cambridge University Press|chapter=9|pages=323–325|isbn=978-0143113799}}

There is an exact mapping between Liouville theory with $c\backslash geq\; 25$, 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 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.

Liouville theory with $c\backslash leq\; 1$ first appeared as a model of time-dependent string theory under the name timelike Liouville theory.{{Cite journal|arxiv=hep-th/0303221|last1=Strominger|first1=Andrew|title=Correlators in Timelike Bulk Liouville Theory|url=https://projecteuclid.org/download/pdf_1/euclid.atmp/1112627637|journal=Adv. Theor. Math. Phys.|volume=7|pages=369–379|last2=Takayanagi|first2=Tadashi|year=2003|issue=2|mr=2015169|bibcode=2003hep.th....3221S|doi=10.4310/atmp.2003.v7.n2.a6|s2cid=15080926}}

It has also been called a generalized minimal model.{{Cite journal|arxiv=hep-th/0505063|last1=Zamolodchikov|first1=Al|title=On the Three-point Function in Minimal Liouville Gravity|journal=Theoretical and Mathematical Physics|volume=142|issue=2|pages=183–196|year=2005|doi=10.1007/s11232-005-0048-3|bibcode=2005TMP...142..183Z|s2cid=55961140}} 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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{{cite journal| last1=Bernard | first1=Denis | last2=LeClair | first2=André | title=The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems | journal=Journal of High Energy Physics | date=2021-12-10 | volume=2022 | issue=5 | page=22 | doi=10.1007/JHEP05(2022)022 | arxiv=2112.05490v1| bibcode=2022JHEP...05..022B | s2cid=245117303 }}

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}}

• [https://www.quantamagazine.org/mathematicians-prove-2d-version-of-quantum-gravity-really-works-20210617/ Mathematicians Prove 2D Version of Quantum Gravity Really Works], Quanta Magazine article by Charlie Wood, June 2021.
• [https://www.youtube.com/watch?v=dLn_50Tpdf8&t=2937s An Introduction to Liouville Theory], Talk at Institute for Advanced Study by Antti Kupiainen, May 2018.

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