Models Of Conformal Field Theory Research Articles

Quantum group symmetry for the Φ12-perturbed and Φ21-perturbed minimal models of conformal field theory

In this paper we show the quantum group symmetry of the Zhiber-Mikhailov-Shabat model based on the non-local conserved currents that exist in the model. The S-matrix of the model is completely characterized by this symmetry. It is then used to study the RSOS version of the model which describes the Φ 21-perturbed and Φ 21-perturbed minimal models of conformal field theory as pointed out by Smirnov. The soliton and kink fields of the model are constructed explicitly.

Chiral gauged Wess-Zumino-Witten theories and coset models in conformal field theory.

The Wess-Zumino-Witten (WZW) theory has a global symmetry denoted by $G_L\otimes G_R$. In the standard gauged WZW theory, vector gauge fields (\ie\ with vector gauge couplings) are in the adjoint representation of the subgroup $H \subset G$. In this paper, we show that, in the conformal limit in two dimensions, there is a gauged WZW theory where the gauge fields are chiral and belong to the subgroups $H_L$ and $H_R$ where $H_L$ and $H_R$ can be different groups. In the special case where $H_L=H_R$, the theory is equivalent to vector gauged WZW theory. For general groups $H_L$ and $H_R$, an examination of the correlation functions (or more precisely, conformal blocks) shows that the chiral gauged WZW theory is equivalent to $(G/H)_L\otimes (G/H)_R$ coset models in conformal field theory. The equivalence of the vector gauged WZW theory and the corresponding $G/H$ coset theory then follows.

Generalized hypergeometric functions on the torus and the adjoint representation ofU q (sl 2)

We study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures. We define an action of the quantum enveloping algebraUq(sl2) on a space of relative cycles, extending results obtained previously for the sphere. Absolute cycles are identified with singular vectors. In the case of one puncture, we find that the resulting topological representation is essentially the adjoint representation.

Infinite symmetry group in D-dimensional conformal quantum field theory

The diffeomorphism group of the D-dimensional space as the symmetry group for conformal quantum field theory models is proposed. The problem of calculation of critical exponents turns out to be reduced to the study of special diffeomorphism group representations. The general properties of these representations on the space of symmetric traceless tensors are investigated.

Monte Carlo simulations of 2-dimensional quantum gravity coupled to c = 1 matter

We presents results of a high precision Monte Carlo simulation of dynamically triagulated random surfaces (up to ≈ 34,000 triangles) coupled to one scalar field ( c = 1). The mean square extent has been measured for different actions to test the universality of the leading term as a function of the size of the surfaces. Furthermore, the integrated 2-point correlation function for vertex operators is compared with conformal field theory and matrix model predictions.

Geometry and integrability of topological-antitopological fusion

Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing ground state metric on given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.

Level-rank duality of WZW models in conformal field theory

We consider the decomposition of the conformal blocks under the conformal embeddings. The case\(\widehat{gl}\left( {lr} \right)_1 \supset \widehat{sl}\left( l \right)_r \times \widehat{sl}\left( r \right)_l \times \hat a\) (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of\(\widehat{gl}\left( {lr} \right)_1 \)-modules induce a pairing between the spaces of conformal blocks of\(\widehat{sl}\left( l \right)_r \) and\(\widehat{sl}\left( r \right)_l \) Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.

(T * G) t : A toy model for conformal field theory

We study a chiral operator algebra of conformal field theory and quantum deformation of the finite-dimensional Lie group to obtain the definition of (T * G) t and its representation. The closeness of the Kac-Moody algebras, constituting the chiral operator algebra of a typical (and generic) conformal field theory model, namely the WZNW model, and quantum deformation of corresponding finite-dimensional Lie groupG has become more and more evident in recent years [1–5]. This in particular prompts further investigation of the differential geometry of such deformations. The notion of tangent and cotangent bundles is basic in classical differential geometry. It is only natural that the quantum deformations ofTG andT * G are to be introduced alongside those forG itself. Physical ideas could be useful for this goal. Indeed, theT * G can be interpreted as a phase space for a kind of a top, generalizing the usual top associated withG=SO(3). The classical mechanics is a natural language to describe differential geometry, whereas the usual quantization is nothing but the representation theory. In this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident, given the experience in this domain. As a result we get the definition of (T * G) t and its representation (t is the deformation parameter). To make the exposition most simple and formulas transparent we shall work on an example ofG=sl(2) and present results in such a way that the generalizations become evident. We shall stick to generic complex versions, real and especially compact forms requiring some additional consideration, not all of which are self-evident.

OPERATOR ALGEBRA IN TWO-DIMENSIONAL CONFORMAL FIELD THEORY CONTAINING SPIN 4/3 “PARAFERMIONIC” CONSERVED CURRENTS

Recently Zamolodchikov and Fateev have constructed a series of models of the two-dimensional conformal field theory containing spin 4/3 nonlocal (parafermion) currents. From degenerated fields one can construct a closed operator algebra with respect to the operator product expansions. All the structure constants of this algebra are computed in this paper.

Geometry of Wess-Zumino-Witten models of conformal field theory

We discuss the geometric aspects of the SU(2) Wess-Zumino-Witten sigma model. Their relation to the quantum deformation of the SU(2) group is explained.

On algebraic equations satisfied by correlators in Wess-Zumino-Witten models

Apart from the Knizhnik-Zamolodchikov differential equations, correlation functions in Wess-Zumino-Witten models of conformal field theory satisfy a certain system of algebraic equations. We show that hypergeometric solutions of KZ equations constructed in [3] do satisfy these algebraic equations.

Reductions of the sine-Gordon model as a perturbation of minimal models of conformal field theory

It is shown that for special values of the coupling constant a reduction is possible in the sine-Gordon model which preserves the locality of certain operators. The reduced model corresponds to perturbed M 2/(2 n+1) models of conformal field theory. It is explained that for any rational coupling constant, the reduction is possible which leads to the perturbed M p/ q model.

SOLUTION OF A CLASS OF CONFORMAL QUANTUM FIELD THEORY MODELS IN D-DIMENSIONAL SPACE

A method of solving a class of conformal quantum field theory models in D-dimensional Euclidean space-time is proposed. Some of the models are determined by regularized field equations. The method allows us to obtain closed differential equations for each Green function of fundamental and composite fields, and also algebraic equations for scale dimensions of fields. Each D>2-model involves an analogue of the central charge, i.e., a special scalar field P of dimension dP=D−2. When D=2, this becomes a constant field. We also obtain a new class of D=2 models with broken infinite parameter symmetry. Closed differential equation for Green functions of these models are found.

Vacuum ward identities for higher genera

The minimal models of two-dimensional conformal field theory are considered on surfaces with nontrivial topology. Due to degeneration of the vacuum module in these models, the stress tensor components satisfy special equations of motion - the vacuum Ward identities. It is shown that these identities can be written in the form of partial differential equations on the moduli space, satisfied by the partition function of the theory. Some examples are written down explicitly in the case of torus and g = 2 surface, represented as a double-fold covering of a sphere. For the simplest minimal theory M( 2 5 ), equations are closed on hyperelliptic surface of any genus and the situation is governed by the other minimal model M( 3 10 ).

Modular covariance of minimal model correlation functions

We prove that one-point functions of all scaling fields in minimal left-right diagonal models of conformal field theory are modular covariant. This consistency condition should allow one to extend these minimal models to Riemann surfaces of arbitrary genus.

EXACT SOLUTIONS OF CONFORMAL FIELD THEORY IN TWO DIMENSIONS AND CRITICAL PHENOMENA

Modern development of conformal field theory in two dimensions and its applications to critical phenomena are briefly reviewed. The specific properties of the renormalization group in two dimensions and the fundamentals of 2-dimensional conformal field theory are presented. The properties of degenerate representations of the Virasoro algebra and other infinite dimensional algebras, "minimal" models of conformal and superconformal field theory, "parafermionic" and other symmetries are discussed. We also investigate a perturbation theory around conformal solutions of field theory.

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.

Conformal quantum field theory models in two dimensions having Z 3 symmetry

An infinite set of conformal quantum field theory models is constructed in two dimensions. The models possess an infinite-dimensional symmetry generated by a local spin-3 current in addition to the conformal symmetry, and exhibit Z 3 invariance. The simplest model of this set describes the critical point of a 2-state Potts model.

Quasiprimary composite fields and operator product expansions in two-dimensional conformal models

Renormalized composite conformal fields are constructed and applied to derive global operator product expansions (OPEs) in two-dimensional conformal quantum field theory (QFT) models.

Composite models of conformal field theory with a nonintegral number of patrons

Models are proposed, which have an explicit solution at any space-time dimension with nontrivial conformal interaction, but in general with ghost states. At two dimensions ghosts can be removed, and one obtains the Thirring model. The problem of ghosts at four dimensions remains open.