Liouville conformal field theory lcft hereafter, introduced by polyakov in his 1981 seminal work quantum geometry of bosonic strings, can be. Liouville integrability in field theory gaetano vilasi universita deglistudi disalerno, italy istituto nazionale difisica nucleare, italy history commutative. The primary field in the theory takes the form e and which has the conformal weight 7, 8 4 h. Proof of the dozz formulakrv 2017 proof of the fyodorovbouchaud formular 2017integrability program for gmc and liouville theory guillaume remy ens the fyodorovbouchaud formula february 1, 2018 3 39. Pdf we study fourpoint correlation functions with logarithmic behaviour in liouville field theory on a sphere, which consist of one kind of the local. Zamolodchikov, conference on liouville field theory, montpellier january 1998, unpublished. In sturm liouville theory, we say that the multiplicity of an eigenvalue of a sturm liouville problem l. We mostly deal with the general 2ndorder ode in selfadjoint form. Simple and universal reaction of conformal theories to the vari. The fyodorovbouchaud formula and liouville conformal field.
Notes on quantum liouville theory and quantum gravity 1. Ist and poisson bracket structure g p jorjadzet, a k pogrebkov, m c polivanov and s v talalov steklov mathematical institute, moscow, ussr received 12 march 1985 abstract. Liouville theory while v and v are the same comp onen ts for the free theory the next step of the reduction pro cedure is a calculation of the canon ical form z x dx p x dq d on the constrained surface this surface can b e represen ted in the form. Dangerous liouville wave exactly wznw model from liouville.
Lft on hyperelliptic surfaces in the past few years essential progress has been achieved in the investigations of liouville field theory. Pdf an exact operator solution of the quantum liouville. Liouville field theory and the partition function for two. In other words, suppose that two particles, a and b, follow the same trajectory, except that particle a leads particle b by a finite time or equivalently, there is a finite distance in xp space between the two particles. An infinite set of the operatorvalued relations in the liouville field theory is established. Lipschitz geometry of curves and surfaces definable in ominimal structures birbrair, lev, illinois journal of. Liouville s theorem does not imply that every point along a given path has the same density. Probabilistic liouville field theory on the twodimensional. Sturmliouville theory the proof shows that if px 6qx then z b a z2p. Related content dangerous liouville wave exactly marginal but nonconformal deformation chiu man ho and yu nakayamaliouville field theory with heavy charges, i.
Pdf quantum liouville theory and btz black hole entropy. Pdf logarithmic correlation functions in liouville field. A new analysis of the intrinsic structure of liouville field theory lft is presented. Liouville field theorya decade after the revolution. Advanced classical mechanicsliouvilles theorem wikiversity. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. Sturmliouville problems, such as those involving bessels equation. There is a family of conformal boundary conditions parameterized by the boundary. I show that the grand partition function for a system of particles interacting via gravitational force can be related to the vacuumtovacuum amplitu. Dozens of papers are published on sturm liouville problems slp every year. Ist and poisson bracket structure article pdf available in journal of physics a general physics 191. The main dynamical variable is the components of the metric tensor gabx. Pdf quantum liouville field theory as solution of a flow. We prove that lft displays a zero mode if its laplacian is defined in terms of the square of the corresponding dirac operator.
In hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure locally, this measure is the 6ndimensional lebesgue measure. A probabilistic approach to liouville eld theory vincent vargas 1 ens paris 1based on a series of works with. On quantization of liouville theory and related conformal. I show that there are states that decouple from these reducible verma modules in a similar. In physics, liouville field theory or simply liouville theory is a twodimensional conformal field. Despite a lot of effort in the last years, it is still hard to draw physically relevant con clusions from these. Liouville conformal field theory is considered with conformal boundary. 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. Zamolodchikov department of physics and astronomy rutgers university p. In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as liouvilles theorem. Structure constants and conformal bootstrap in liouville field theory a. Logarithmic correlation functions in liouville field theory.
If a conformal field theory fits into the framework introduced in bpz, it is essentially fully characterized by its spectrum of primary fields and the full set of three point. These relations are enumerated by a pair of positive integers m, n, the. Aleksidze 1, 380093, tbilisi, georgia bdesy zeuthen, platanenallee 6, d15738 zeuthen, germany abstract we quantise canonical free eld zero modes p, q on a halfplane p0, as a. Bataillon, 34095 montpellier, france abstract liouville conformal. Structure constants and conformal bootstrap in liouville. Ponsot, recent progresses on liouville field theory, hepth0301193. Hosomichi, bulkboundary propagator in liouville theory on a disc, jhep 0111 044 2001 hepth0108093. For example in three dimensions, there are three spatial coordinates and three conjugate momenta. If not, then just as before we could derive by dividing by yx and the boundary term in would vanish since yb 0, and we would obtain z b a yz0. The classic references for 2d conformal field theory is 22, chap. Keywords liouville quantum field theory gaussian multiplicative chaos kpz formula polyakov formula punctures cusp singularity uniformization theorem rights creative commons attribution 4. Introduction to the gaussian free field and liouville quantum. Semiclassical limit of liouville field theory hubert lacoin 1, r emi rhodes, vincent vargas 23 abstract liouville field theory lft for short is a two dimensional model of random surfaces, which is for instance involved in 2dstring theory or in the description of the uctuations of metrics in 2dliouville quantum gravity.
In the boundary liouville field theory i investigate the reducible verma modules in the elliptic sector, which correspond to certain irreducible representations of the quantum algebra. It asserts that the phasespace distribution function is constant along the trajectories of the systemthat is that the density of system points in the vicinity of a given system point traveling through phasespace is constant with time. In the physics literature, the quantum liouville theory was. In the context of quantum field theory, liouville theory is the name of a certain type of 2d cft properties relation to 3d quantum gravity. In physics, liouville field theory or simply liouville theory is a twodimensional conformal field theory whose classical equation of motion is a generalization of liouville s equation. Aleksidze 1, 380093, tbilisi, georgia bdesy zeuthen, platanenallee 6, d15738 zeuthen, germany abstract we quantise canonical free eld zero modes p, q on a halfplane p0, as a prerequisite for the calculation of correlation functions in periodic liouville theory. In 1910 hermann weyl published one of the most widely quoted papers in analysis 607. Adkins master of science graduate department of mathematics university of toronto 2014 a basic introduction into sturmliouville theory. Applying this result in the examples above we immediately recover familiar orthogonality statements.
The eigenvalues of a sturm liouville problem are all of multiplicity one. An exact operator solution of the quantum liouville field theory. An example of the theoretical utility of the hamiltonian formalism is liouvilles theorem. Pdf stochastic quantization of liouville conformal field. Notes on 2d quantum gravity and liouville theory harold erbin.
Quantum liouville field theory as solution of a flow equation. Yasuaki hikida and volker schomerus jhep102007064 liouville theories view the article online for updates and enhancements. Another important aspect of the liouville field theory is the exponential of the gaussian free field. Since einstein the term gravity means the dynamic theory of the spacetime metric structure.
Liouville theory raises a number of hard technical problems, which in most of the physi cally relevant cases are not yet resolved. The second motivation for studying liouville theory is that it is an interesting example of an integrable conformal field theory in two dimensions. An argument via chernsimons gravity, but see the caveats there that 3d quantum gravity with negative cosmological constant has as boundary field theory 2d liouville theory is due to coussaerthenneauxvandriel 95. Essential building block of the theory is the gaussian free field, which can be thought of as a random gaussian field with the covariance operator given by the inverse of the laplacian. Fermilab on quantization of liouville theory and related. In physics, liouvilles theorem, named after the french mathematician joseph liouville, is a key theorem in classical statistical and hamiltonian mechanics. Further, by interpreting the spacetime asso2,1so1, 1 analogous toso3, 2so3, 1 we present, arguments which support the nontrivial approximativesmatrix. The eigenvalues of a sturmliouville problem are all of multiplicity one. Notes on 2d quantum gravity and liouville theory lpthe. Landau institute for theoretical physics kosygina 2, 117334, moscow, russia al. Keywords gaussian free field gaussian multiplicative chaos boundary liouville field theory conformal field theory bpz equations. Just as the 183637 papers of sturm and liouville started the study of regular slp, the 1910 paper of weyl initiated the investigation of singular slp.
Wznw model from liouville field theory to cite this article. Liouville field theory a decade after the revolution. In physics, liouville field theory or simply liouville theory is a twodimensional 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. Mukhi, topological matrix models, liouville matrix model and c 1 string theory, hepth0310287. The constants of this field are just the complex numbers c. Vincent vargas 14 liouville conformal field theory and. This normalization is very convenient for liouville theory since we will be dealing with a spinless. In this paper we consider the integrable field theories which can be formulated as sine. Many of the important subsequent developments are omitted. In classical mechanics, the complete state of a particle can be given by its coordinates and momenta. The general solution for the liouville equation with any given number n of singularitiv is considered. Twodimensional field theories, conformal field theories, etc. Liouville field theory on a hyperelliptic surface sciencedirect. Zero mode problem of liouville field theory george jorjadzea 1 and gerhard weigtb 2 arazmadze mathematical institute, m.
The fyodorovbouchaud formula and liouville conformal. As an example, the field cx of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. Gaussian multiplicative chaos,liouville correlations consequences. The eigenvalues of a sturmliouville problem are all. Integrable deformations of sineliouville conformal field theory. The partition function of the liouville field theory on a hyperelliptic surface is expressed as a correlation function of the liouville vertex operators on a sphere and the twisted fields. This dynamics may be either classical classical gravity or quantum, in which case we talk about quantum gravity. Gaussian multiplicative chaos, liouville correlations consequences. A proof of liouville s theorem can be found in section 12.
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