Knizhnik-Zamolodchikov Equations Research Articles

The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem

It is shown that the Knizhnik-Zamolodchikov (KZ) equation (and corresponding vector bundle) can be viewed as a quantization of the isomonodromy problem for differential equations with several singular points.

Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration

We study the limit of asymptotically free massive integrable models in which the algebra of nonlocal charges turns into affine algebra. The form factors of fields in that limit are described by KZ equations on level 0. We show the limit to be connected with finite-gap integration of classical integrable equations.

Reduction of the Knizhnik-Zamolodchikov equation. A way of producing Virasoro singular vectors

We prove that for (half-)integer isospins the sl(2, C ) Knizhnik-Zamolodchikov equation reduces to the decoupling equation coming from Benoit-Saint-Aubin singular vectors. In the general case an algorithm is suggested which transforms, via the Knizhnik-Zamolodchikov equation, a Kac-Moody singular vector into a Virasoro one.

D-E classfiication of the local extensions ofsu 2 current algebras

A method is developed for constructing single valued rational 4-point functions of primary fields for su2 conformal current algebra satisfying the Knizhnik-Zamolodchikov equation. For integer conformal dimensions Δ these rational solutions are proven to be in one-to-one correspondence with non-diagonal modular invariant partition functions of the D-even and E-even series of the ADE classification.

DYNAMICAL SYMMETRIES OF MASSIVE INTEGRABLE MODELS 2: SPACE OF STATES OF MASSIVE MODELS AS SPACE OF OPERATORS

International Journal of Modern Physics AVol. 07, No. supp01b, pp. 839-858 (1992) No AccessDYNAMICAL SYMMETRIES OF MASSIVE INTEGRABLE MODELS 2: SPACE OF STATES OF MASSIVE MODELS AS SPACE OF OPERATORSFEDOR A. SMIRNOVFEDOR A. SMIRNOVRIMS, Kyoto University, Kyoto 606, JapanLeningrad Branch of Steklov Mathematical Institute, Leningrad, USSR, Russiahttps://doi.org/10.1142/S0217751X92004075Cited by:17 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail You currently do not have access to the full text article. Recommend the journal to your library today! FiguresReferencesRelatedDetailsCited By 17On the calculation of the correlation functions of the $ \newcommand{\slt}{\mathfrak{sl}_2} \newcommand{\sltr}{\mathfrak{sl}_3} \sltr$ -model by means of the reduced qKZ equationH Boos, A Hutsalyuk and Kh S Nirov9 October 2018 | Journal of Physics A: Mathematical and Theoretical, Vol. 51, No. 44On form factors in nested Bethe Ansatz systemsBalázs Pozsgay, Willem-Victor van Gerven Oei and Márton Kormos31 October 2012 | Journal of Physics A: Mathematical and Theoretical, Vol. 45, No. 46The elliptic quantum algebra U_{q\hbox{\scriptsize,}\, p}(\widehat{sl_N}) and its vertex operatorsWen-jing Chang and Xiang-mao Ding30 September 2009 | Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 42On the vertex operators of the elliptic quantum algebra Uq,p(sl2̂)kWen-Jing Chang and Xiang-Mao Ding1 Apr 2008 | Journal of Mathematical Physics, Vol. 49, No. 4Izergin–Korepin model with a boundaryWen-Li Yang and Yao-Zhong Zhang1 Mar 2001 | Nuclear Physics B, Vol. 596, No. 3Difference equations for correlation functions of Belavin's ℤ n -symmetric model with boundary reflectionYas-Hiro Quano9 November 2000 | Journal of Physics A: Mathematical and General, Vol. 33, No. 46The algebra ħ,η(ĝ) and infinite Hopf family of algebrasBo-Yu Hou, Liu Zhao and Xiang-Mao Ding1 Sep 1998 | Journal of Geometry and Physics, Vol. 27, No. 3-4Free boson representation of DYℏ(glN)k and DYℏ(slN)kX. M. Ding, B. Y. Hou, B. Yuan Hou and L. Zhao1 Apr 1998 | Journal of Mathematical Physics, Vol. 39, No. 4ℏ (YANGIAN) DEFORMATION OF THE MIURA MAP AND VIRASORO ALGEBRAXIANG-MAO DING, BO-YU HOU, and LIU ZHAO25 January 2012 | International Journal of Modern Physics A, Vol. 13, No. 07An -deformation of the W algebra and its vertex operatorsBo-yu Hou and Wen-li Yang1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 30, No. 17Form factors of the sausage model obtained with bootstrap fusion from sine-Gordon theoryZalán Horváth and Gábor Takács15 March 1996 | Physical Review D, Vol. 53, No. 6Annihilation poles of a Smirnov-type integral formula for solutions to the quantum Knizhnik-Zamolodchikov equationT Kojima, K Miki and Yas-Hiro Quano1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 28, No. 12An algebraic approach to form factorsMax R. Niedermaier1 May 1995 | Nuclear Physics B, Vol. 440, No. 3Free field representation for the O(3) nonlinear σ model and bootstrap fusionZalán Horváth and Gábor Takács15 March 1995 | Physical Review D, Vol. 51, No. 6Quantum Knizhnik-Zamolodchikov equation for U q (sl n ) and integral formulaT Kojima and Y -H Quano1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 27, No. 20Smirnov's integrals and the quantum Knizhnik-Zamolodchikov equation of level 0M Jimbo, T Kojima, T Miwa and Y -H Quano1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 27, No. 9The structure of n-variable polynomial rings as Hecke algebra modulesP Martin1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 26, No. 24 Recommended Vol. 07, No. supp01b Metrics History Received 1 October 1991 PDF download

DEGENERATE AFFINE HECKE ALGEBRAS AND TWO-DIMENSIONAL PARTICLES

We demonstrate that the quantization of momenta in different two dimensional theories of elementary particles with factorizable scattering amplitudes gives the so-called degenerate affine Hecke algebras and their versions. Some connections with quantum groups(Yangians), the two-dimensional conformal field theory and representation theory am discussed. In particular, an interpretation and generalizations of the difference counterpart of the Knizhnik-Zamolodchikov equation are found by means of the particles on a segment.

Topological invariants for 3-manifolds using representations of mapping class groups I

LET A4 be a closed orientable 3-manifold. The purpose of this paper is to define topological invariants 4,(M) parametrized by K = 1,2,. . . using representations of the mapping class group of an orientable surface. In [27], Witten introduced new topological invariants for 3manifolds based on quantum field theory and motivated by Witten’s Dehn surgery formula, Reshetikhin and Turaev [I93 gave a formula expressed by the Jones polynomial and its relatives, by means of representations of the quantized universal enveloping algebra in the case q is a root of unity. Our approach using a Heegaard decomposition of a 3-manifold is difTerent from theirs. Let Xr denote a closed oricntablc surface of genus y. First, WC consider a pants decomposition of C, and WC dcnotc by Y its dual graph, which is a trivalent graph. Given a positive intcgcr K called a Icvel, WC introduce a finite dimensional complex vector space Z,(y) whose basis is in one-to-one correspondence with the set of admissible weights /: edge(y) + (0, l/2, * . . , K/2} satisfying the Clebsch-Gordan condition and the algebraic constraintj(c,) +/(cL) +/(cJ) IS K for any edges c,, c2 and c, meeting at a vertex. This vector space appears as a combinatorial description of the space of conformal blocks for SU(2) Wess-Zumino-Witten model at level K (see [23]). Let yr and Y2 be trivalent graphs associated with markings of C,. By means of the fusing matrices describing the holonomy of the Knizhnik-Zamolodchikov equation, we obtain a canonical isomorphism Z,(Yr) z Z,(y2). A detailed study of the monodromy of the Knizhnik-Zamolodchikov equation was pursued by Tsuchiya and Kanie [22]. It is known that the representations of the braid groups appearing as the monodromy of the Knizhnik-Zamolodchikov equation provide the Jones polynomial and its relatives (see [13], [IS] and [22]). More recently, Drinfel’d established a relation between the monodromy and the quantized universal enveloping algebras in a general situation (see [6]). using the notion of quasi-Hopf algebras. In [is], Moore and Seiberg wrote down a series of polynomial equations among fusing matrices, braiding matrices and so-called switching operators expected from the viewpoint of the consistency conditions in conformal field theory. The holonomy of the Knizhnik-Zamolodchikov equation gives solutions to these polynomial equations, and as a consequence we obtain projectively linear representations of the mapping class group

N = 3 AND N = 4 SUPERCONFORMAL WZNW SIGMA MODELS IN SUPERSPACE II: THE N = 4 CASE

The issues related to the U (1) × O (4) N = 4 superconformal WZNW sigma models [with the bosonic target spaces U (1) × SU (2) and U (1) × U (1) × O (4)] are investigated in the framework of a 2D N = 4 superspace. We define the corresponding N = 4 supercurrents, both on classical and quantum levels, in terms of the basic primary N = 4 WZNW superfields and show that the generalized Sugawara form for the dimension 3/2 and 2 component currents directly follows from the constraints on the basic superfields. The N = 4 superfield analog of the Knizhnik–Zamolodchikov equation for conformal blocks in WZNW sigma models is derived. We also analyze the N = 4 WZNW superfields from the standpoint of two SU(2) N = 4 SCAs entering as subalgebras into the underlying U (1) × O (4) N = 4 SCA. We demonstrate that in the linearizing limit the U (1) × SU (2) WZNW model reproduces the recently discussed SU(2) N = 4 superconformal system of free chiral superfields.

Double affine hecke algebras, knizhnik-zamolodchikov equations, and macdonald’s operators

Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators Get access Ivan Cherednik Ivan Cherednik Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1992, Issue 9, 1992, Pages 171–180, https://doi.org/10.1155/S1073792892000199 Published: 01 May 1992 Article history Published: 01 May 1992 Received: 28 July 1992

Fock space representations of affine lie algebras and integral representations in the Wess-Zumino-Witten models

Fock space representations of affine Lie algebras are studied. Explicit forms of correction terms adding to the currentsF i (z) are determined. It is proved that the Sugawara energy-momentum tensor on the Fock spaces is quadratic in free bosons. Furthermore, screening operators are constructed. This implies the existence of generalized hypergeometric integrals satisfying the Knizhnik-Zamolodchikov equation.

Monodromy Representations for Generalized Knizhnik-Zamolodchikov Equations and Hecke Algebras

In the paper generalizations of the Knizhnik-Zamolodchikov equation with symmetries in Weyl groups for arbitrary classical r-matrices are studied. Their monodromy representations are calculated explicitly in terms of Hecke algebras and affine Hecke algebras for basic examples.

Integral Solutions of Trigonometric Knlzhnik-Zamolodchikov Equations and Kac-Moody Algebras

We use Kac-Moody algebras to get some integral solutions for trigonometric r -matrix Knizhnik-Zamolodchikov equations of type Xl (1) generalizing those in the case of Yang's r -matrix, where our construction gives a new interpretation and a short proof of Schechtman, Varchenko theorem.

Reduction of the rational spin sl(2, [formula omitted]) WZNW conformal theory

We construct explicitly the correlation functions on the plane of the rational spin and level A 1( 1) WZNW conformal model. The corresponding Virasoro minimal correlations are recovered in the limit when isospin sl(2, C ) variables x i approach the coordinates z i . The Knizhnik-Zamolodchikov equation reduces to a differential equation for the minimal conformal blocks, reflecting the decoupling of the Virasoro null vectors.

Path integral representation of the Artin braid group

Using Feynman kernels, a representation of the Artin braid group is explicitly constructed. The Schrödinger equations associated to the kernels turn out to be intimately related to the Knizhnik-Zamolodchikov equations. The representation space includes the space of correlation functions of the Wess-Zumino-Witten models.

An application of Aomoto-Gelfand hypergeometric functions to theSU(n) Knizhnik-Zamolodchikov equation

Solutions to the Knizhnik-Zamolodchikov equation for Verma modules of the Lie algebra\(\mathfrak{s}\mathfrak{l}(n + 1,\mathbb{C})\) are explicitly given by certain integrals called Aomoto-Gelfand hypergeometric functions.

Hypergeometric solutions of Knizhnik-Zamolodchikov equations

Explicit integral formulas are presented for the solutions of Knizhnik-Zamolodchikov equations associated with an arbitrary Kac-Moody Lie algebra.

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.

THE COULOMB GAS REPRESENTATION FOR SU(2) WESS-ZUMINO-WITTEN MODEL IN SUPERSPACE

We give a Coulomb gas representation for level k N = 1 supersymmetric SU(2) Kac-Moody algebra in terms of three free scalar superfields. It is clarified how this representation reduces to a Coulomb gas representation for the corresponding bosonic SU(2) Kac-Moody algebra and the free fermionic algebra. The primary superfields and the correlation functions, which satisfy the supersymmetric Knizhnik-Zamolodchikov equation, are also discussed.

BRAIDING MATRICES OF CONFORMAL BLOCKS AND COSET MODELS

We explain, from the Knizhnik-Zamolodchikov equation, the coincidence of the braiding matrices of conformal field theories having current algebras with face Boltzmann weights (at infinite spectral parameter) of the corresponding generalized Toda system (GTS). A vertex-height correspondence is introduced in the WZW theory. Braiding matrices of coset models are found to factorize into those of the WZW theories and, as an example, we evaluate those of the Ising model.

Dynamics of the coset conformal field theory. The BPZ equation from the KZ equation in the GKO construction

We have shown that the dynamics of the coset conformal field theory can be realized from the dynamics of their isometry and isotropy groups. The coset primary operators are represented by the primary or their descendent null operators of the isometry and isotropy groups.