148 research outputs found
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
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 . 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 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 . 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 external weights 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
Higher Equations of Motion in N = 1 SUSY Liouville Field Theory
Similarly to the ordinary bosonic Liouville field theory, in its N=1
supersymmetric version an infinite set of operator valued relations, the
``higher equations of motions'', holds. Equations are in one to one
correspondence with the singular representations of the super Virasoro algebra
and enumerated by a couple of natural numbers . We demonstrate
explicitly these equations in the classical case, where the equations of type
survive and can be interpreted directly as relations for classical
fields. General form of the higher equations of motion is established in the
quantum case, both for the Neveu-Schwarz and Ramond series.Comment: Two references adde
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