Generalized Knizhnik-Zamolodchikov Equations Research Articles

Isomonodromic deformations, generalized Knizhnik–Zamolodchikov equations and non-skew-symmetric classical r-matrices

Using non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters, we generalize Fuchsian systems and Schlesinger equations (isomonodromy equations) viewing the last as the nonautonomous Hamiltonian equations corresponding to the generalized Gaudin Hamiltonians. Quantizing these systems we construct generalization of Knizhnik–Zamolodchikov equations corresponding to the non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters. In the case of an ordinary skew-symmetric classical r-matrix r12(u−v) depending on the difference of the spectral parameters, we reobtain, as special partial cases, a standard generalization of Fuchsian systems, Schlesinger, and Knizhnik–Zamolodchikov equations.

Generalized Knizhnik–Zamolodchikov equations, off-shell Bethe ansatz and non-skew-symmetric classical r-matrices

Using non-skew-symmetric sl ( 2 ) -valued classical r-matrices with spectral parameters we construct a generalization of the Knizhnik–Zamolodchikov (KZ) equations. We obtain integral solutions of the constructed KZ-type equations using the “off-shell” Bethe ansatz technique. We consider several examples of the obtained generalized KZ equations and their integral solutions that correspond to the “K-twisted” non-skew-symmetric classical r-matrices parametrized by arbitrary complex parameter ξ.

Many-Particle Models with Two Constants of Interaction

We discuss integrable models of quantum field theory and statistical mechanics. The dynamics and kinematics of these models are defined by Hamiltonians with symmetries determined by Lie algebras. The paper is devoted to the characterization of models such that the root vectors of their symmetry algebras run two orbits under the action of the Weyl group. Such properties possess root systems of the type B N , C N , and G 2. The main focus is on models with the symmetry of algebras B N . In this case the main characteristics of the process are obtained from the system of Yang-Baxter equation and the reflection equations.We consider the Calogero-Moser and Calogero-Sutherland models and also the formalisms of the Lax and Dunkl operators. The connection between these formalisms and method of describing these models in terms of the generalized Knizhnik-Zamolodchikov equations with the system of roots of the type B N by the example of the Gaudin model with reflection are discussed. Examples of many-particle systems that interact with each other with reflections are presented.

Isomonodromic and Bispectral Properties of the Generalized Knizhnik-Zamolodchikov Equations

Isomonodromic and Bispectral Properties of the Generalized Knizhnik-Zamolodchikov Equations

Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping

Canonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik–Zamolodchikov equations. One can solve these equations by the off-shell Bethe ansatz method provided the Knizhnik–Zamolodchikov equations are related with the highest weight representations of the corresponding Lie algebras: These solutions can be written in terms of multi-variable generalizations of special functions of the hypergeometric type. In this work, we consider a realization of the above scheme for the Maxwell–Bloch system with pumping: quantum states for this system are found in terms of the multi-variable confluent hypergeometric function.

Nonabelian statistics for calogero-sutherland systems of particles

The mathematical aspects of the theory of nonabelian statistics and anyons are discussed. The dynamics and kinematics of these objects are defined by Hamiltonians which are the sums of terms associated with different roots of a given root system. The equations of Fuchsian type which are the so-called generalized Knizhnik-Zamolodchikov equations associated with the root systemsA n ,B n , andC n , are presented. Some results the generalized pure braid groups and their application for defining the generalized anyon are discussed. The factorization of the spin Calogero-Sutherland operators and their connection with the theory of nonabelian statistics and anyons are given.

SEMICLASSICAL BLOCKS AND CORRELATORS IN RATIONAL AND IRRATIONAL CONFORMAL FIELD THEORY

The generalized Knizhnik–Zamolodchikov equations of irrational conformal field theory provide a uniform description of rational and irrational conformal field theory. Starting from the known high-level solution of these equations, we first construct the high-level conformal blocks and correlators of all the affine-Sugawara and coset constructions on simple g. Using intuition gained from these cases, we then identify a simple class of irrational processes whose high-level blocks and correlators we are also able to construct.

Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians.

Generalized Knizhnik-Zamolodchikov equations and the factorization of their solutions

Generalized Knizhnik-Zamolodchikov equations and the factorization of their solutions

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.