In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Liouville field theory (or simply Liouville theory) is a
two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fie ...
whose classical
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verla ...
is a generalization of
Liouville's equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
.
Liouville theory is defined for all
complex values of the
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
of its
Virasoro symmetry algebra, but it is
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
only if
:
,
and its
classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
is
:
.
Although it is an interacting theory with a
continuous spectrum
In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable ...
, Liouville theory has been solved. In particular, its three-point function on the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
has been determined analytically.
Introduction
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
In physics a free field is a field without interactions, which is described by the terms of motion and mass.
Description
In classical physics, a free field is a field whose equations of motion are given by linear partial differential equati ...
due to the presence of an exponential potential
:
where the parameter
is called the
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
. 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.
Moreover, the potential reflects the energy eigenvectors before they reach
, and two eigenvectors are linearly dependent if their momenta are related by the
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
:
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
In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the ...
with the
conformal dimension In mathematics, the conformal dimension of a metric space ''X'' is the infimum of the Hausdorff dimension over the conformal gauge of ''X'', that is, the class of all metric spaces quasisymmetric to ''X''.John M. Mackay, Jeremy T. Tyson, ''Con ...
by
:
The central charge and conformal dimensions are invariant under the
duality
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
:
The
correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
s 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
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
formulation, in particular the exponential potential is not invariant under the duality.
Spectrum and correlation functions
Spectrum
The
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of Liouville theory is a diagonal combination of
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
s of the
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
,
:
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
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
. A vacuum state can be defined, but it does not contribute to
operator product expansion
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex ...
s.
Fields and reflection relation
In Liouville theory, primary fields are usually
parametrized by their momentum rather than their
conformal dimension In mathematics, the conformal dimension of a metric space ''X'' is the infimum of the Hausdorff dimension over the conformal gauge of ''X'', that is, the class of all metric spaces quasisymmetric to ''X''.John M. Mackay, Jeremy T. Tyson, ''Con ...
, 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
:
(The sign is
if
and
otherwise, and the normalization parameter
is arbitrary.)
Correlation functions and DOZZ formula
For
, the three-point structure constant is given by the DOZZ formula (for Dorn–Otto and Zamolodchikov–Zamolodchikov
),
:
where the special function
is a kind of
multiple gamma function
In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by . At the end of this paper he mentioned the existence of multiple gamma funct ...
.
For
, the three-point structure constant is
:
where
:
-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
In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time.
Crossing states that the same formula that determines th ...
of the four-point function, which has been checked numerically
and proved analytically.
Liouville theory exists not only on the sphere, but also on any
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of genus
. Technically, this is equivalent to the
modular invariance
In theoretical physics, modular invariance is the invariance under the group such as SL(2,Z) of large diffeomorphisms of the torus. The name comes from the classical name modular group of this group, as in modular form theory.
In string theory, m ...
of the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
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
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of resolution.
Overview
U ...
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
and the momenta,
* degenerate fields exist.
Lagrangian formulation
Action and equation of motion
Liouville theory is defined by the local
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
:
where
is the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
of the
two-dimensional space
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
on which the theory is formulated,
is the
Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
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
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
. If
is the
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
, this equation reduces to
:
which is equivalent to
Liouville's equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
.
Once compactified on a cylinder, Liouville field theory can be equivalently formulated as a worldline theory.
Conformal symmetry
Using a
complex coordinate system and a
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
:
,
the
energy–momentum tensor Energy–momentum may refer to:
*Four-momentum
*Stress–energy tensor
*Energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also ...
's components obey
:
The non-vanishing components are
:
Each one of these two components generates a
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
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
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
, the field
that appears in the action must be
marginal
Marginal may refer to:
* ''Marginal'' (album), the third album of the Belgian rock band Dead Man Ray, released in 2001
* ''Marginal'' (manga)
* '' El Marginal'', Argentine TV series
* Marginal seat or marginal constituency or marginal, in polit ...
, 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 integral
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
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
, 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
residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
s of correlation functions at some of their
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
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 and the conformal bootstrap.
Relations with other conformal field theories
Some limits of Liouville theory
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. 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.
WZW models
Liouville theory can be obtained from the
Wess–Zumino–Witten model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edwa ...
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.
This is also true of correlation functions of the 2d black hole
coset model.
Moreover, there exist theories that continuously interpolate between Liouville theory and the
model.
Conformal Toda theory
Liouville theory is the simplest example of a
Toda field theory
In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Kac–Moody algebra and a specific Lagrangian.
Fixing the Kac–Mo ...
, associated to the
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the ...
. 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-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchi ...
s rather than the Virasoro algebra.
Supersymmetric Liouville theory
Liouville theory admits two different
supersymmetric
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
extensions called
supersymmetric Liouville theory and
supersymmetric Liouville theory.
Relations with integrable models
Sinh-Gordon model
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
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
is known in the weak coupling regime