Structure Constants and Conformal Bootstrap in Liouville Field Theory

An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.Comment: 31 pages, 2 Postscript figures. Important note about existing (but unfortunately previously unknown to us) paper which has significant overlap with this work is adde

N=1 SUSY Conformal Block Recursive Relations

We present explicit recursive relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. The results are compared with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursive relations are an efficient tool for numerically studying the four-point correlation function in Super Conformal Field Theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry.Comment: 12 pages, typos corrected, reference adde

Massless Flows I: the sine-Gordon and O(n) models

The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coefficient of the cosine potential is imaginary. This flow is studied, partly numerically, from three different points of view. First we work out the expansion close to the Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge going up and down in between the UV and IR values of $c=1$. Next we analytically continue the Casimir energy of the massive flow (i.e. with real cosine term). Finally we consider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature phases. A detailed discussion of the case $n=0$ is then given using the underlying N=2 supersymmetry, which is spontaneously broken in the low-temperature phase. The ``index'' \tr F(-1)^F follows from the Painleve III differential equation, and is shown to have simple poles in this phase. These poles are interpreted as occuring from level crossing (one-dimensional phase transitions for polymers). As an application, new exact results for the connectivity constants of polymer graphs on cylinders are obtained.Comment: 39 pages, 7 uuencoded figures, BUHEP-93-5, USC-93/003, LPM-93-0

Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed

Conformal blocks related to the R-R states in the \hat c =1 SCFT

We derive an explicit form of a family of four-point Neveu-Schwarz blocks with $\hat c =1,$ external weights $\Delta_i = 1/8$ and arbitrary intermediate weight. The derivation is based on a set of identities obeyed in the free superscalar theory by correlation functions of fields satisfying Ramond condition with respect to the bosonic (dimension 1) and the fermionic (dimension 1/2) currents.Comment: 15 pages, no figure

Conserved charges in the chiral 3-state Potts model

We consider the perturbations of the 3-state Potts conformal field theory introduced by Cardy as a description of the chiral 3-state Potts model. By generalising Zamolodchikov's counting argument and by explicit calculation we find new inhomogeneous conserved currents for this theory. We conjecture the existence of an infinite set of conserved currents of this form and discuss their relevance to the description of the chiral Potts models

On the Yang-Lee and Langer singularities in the O(n) loop model

We use the method of `coupling to 2d QG' to study the analytic properties of the universal specific free energy of the O(n) loop model in complex magnetic field. We compute the specific free energy on a dynamical lattice using the correspondence with a matrix model. The free energy has a pair of Yang-Lee edges on the high-temperature sheet and a Langer type branch cut on the low-temperature sheet. Our result confirms a conjecture by A. and Al. Zamolodchikov about the decay rate of the metastable vacuum in presence of Liouville gravity and gives strong evidence about the existence of a weakly metastable state and a Langer branch cut in the O(n) loop model on a flat lattice. Our results are compatible with the Fonseca-Zamolodchikov conjecture that the Yang-Lee edge appears as the nearest singularity under the Langer cut.Comment: 38 pages, 16 figure

Ising Field Theory on a Pseudosphere

We show how the symmetries of the Ising field theory on a pseudosphere can be exploited to derive the form factors of the spin fields as well as the non-linear differential equations satisfied by the corresponding two-point correlation functions. The latter are studied in detail and, in particular, we present a solution to the so-called connection problem relating two of the singular points of the associated Painleve VI equation. A brief discussion of the thermodynamic properties is also presented.Comment: 39 pages, 6 eps figures, uses harvma

Mutual Information and Boson Radius in c=1 Critical Systems in One Dimension

We study the generic scaling properties of the mutual information between two disjoint intervals, in a class of one-dimensional quantum critical systems described by the c=1 bosonic field theory. A numerical analysis of a spin-chain model reveals that the mutual information is scale-invariant and depends directly on the boson radius. We interpret the results in terms of correlation functions of branch-point twist fields. The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge.Comment: 4.1 pages, 5 figure

The long delayed solution of the Bukhvostov Lipatov model

In this paper I complete the solution of the Bukhvostov Lipatov model by computing the physical excitations and their factorized S matrix. I also explain the paradoxes which led in recent years to the suspicion that the model may not be integrable.Comment: 9 page