Parafermion Fields Constructed by Current Algebra

In this letter, the parafermion fields constructed by current algebra are considered. It is proved that there must be a parafermion field with respect to each form of current algebra. We also obtain the corresponding representation and unitary relation of the parafermion field from any current algebra.     

Quantum Currents in the coset Space SU(2)/U(1)

We propose a rational quantum deformed nonlocal currentsin the homogeneous space SU(2)k/U(1),and in terms of it and a free boson field a representation for the Drinfeld currents of Yangian double at a general level k=c is obtained.In the classical limit h→0,the quantum nonlocal currents become SU(2)k parafermion,and the realization of Yangian double becomes the parafermion realization of SU(2)k current algebra.     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

Z-Algebra Construction of W-Algebra in the Parafermion Model

We propose a new construction — the Z-algebra construction for the W-algebra which includes the Virasoro algebra as a special case. The Z-algebra associated with the general affine Lie algebra is given. And we suggest a calculating program to make the derivation of the Z-algebra become possible. Using this technique we have derived the Virasoro algebra in general G_k parafermion case and the W-algebra in SU(2)_k parafermion case.     

The Unitarity of the W(3) Algebra from SU(3) Parafermion

We express the vacuum expectation value of the SU(3)k parafermion fields by that of two bosons and SU(3)_k current algebra. When k = 1,2,3, the SU(3)_k current algebra becomes an inner product of a unitary representation, and T(z), W⁽³⁾(z) are equivalent to "quasiself-adjoint" operators in this represen tation.     

The Structure of Parafermion Vertex Operator Algebras: General Case

The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.     

A(2)2 parafermions: a new conformal field theory

A new parafermionic algebra associated with the homogeneous space A⁽²⁾₂/U(1) and its corresponding Z-algebra have been recently proposed. In this paper, we give a free boson representation of the A⁽²⁾₂ parafermion algebra in terms of seven free fields. Free field realizations of the parafermionic energy–momentum tensor and screening currents are also obtained. A new algebraic structure is discovered, which contains a W-algebra type primary field with spin two.     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

Interpretation and extension of Green''s ansatz for paraparticles

The anomalous bilinear commutation relations satisfied by the components of the Green's ansatz for paraparticles are shown to derive from the comultiplication of the paraboson or parafermion algebra. The same provides a generalization of the ansatz, wherein paraparticles of order are constructed from r paraparticles of order p, =1, 2, …, r.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Parabosons,parafermions, and explicit representations of infinite-dimensional algebras

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labeled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. Alternatively, the basis vectors can be expressed as semi-standard Young tableaux.     

Graded parafermions

A graded generalization of the Z_k parafermionic current algebra is constructed. This symmetry is realized in the osp(1|2)/U(1) coset conformal field theory. The structure of the parafermionic highest-weight modules is analyzed and the dimensions of the fields of the theory are determined. A free field realization of the graded parafermionic system is obtained and the structure constants of the current algebra are found. Although the theory is not unitary, it presents good reducibility properties.     

On Solutions to the q-Deformed Oscillator Systems and the q-Oscillator Algebra Hq (4)

Discussions upon the q-oscillator systems and the q-oscillator algebra H_q(4) are provided,and the case of |q| = 1 ia detailed. A remarkable connection between SH_q(2) and H_q(4) is indicated and investigated. The single q-oscillator system t solved and shown to be analogous to the parafermion (PF), with the specific case analogous to the third order PF discussed in detail. The properties of the representations and a finite chain model are exploited.     

匀强磁场对刚性载流线圈磁力矩公式的一般推导

利用矢量代数和矢量场论的有关结果严谨地推证了匀强磁场中载流线圈所受合磁力矩表达式。     

Current Algebra Associated with Logarithmic Conformal Field Theories

We propose a general framework for deriving the OPEs within a logarithmic conformal field theory (LCFT). This naturally leads to the emergence of a logarithmetic partner of the energy momentum tensor within an LCFT and implies that the current algebra associated with an LCFT is expanded. We derive this algebra for a generic LCFT and discuss some of its implications. We observe that two constants arise in the OPE of the energy-momentum tensor with itself. One of these is the usual central charge.     

Free Field Construction for the ABF Models in Regime II

The Wakimoto construction for the quantum affine algebra U $_q$ ( $(\widehat{\mathfrak{s}\mathfrak{l}_2 })$ ) admits a reduction to the q-deformed parafermion algebras. We interpret the latter theory as a free field realization of the Andrews–Baxter–Forrester models in regime II. We give multi-particle form factors of some local operators on the lattice and compute their scaling limit, where the models are described by a massive field theory with $\mathbb{Z}$ $_k$ symmetric minimal scattering matrices.     

A discrete approach to topological quantum field theories

First, we describe a rather general scheme for constructing three-dimensional euclidean topological quantum field theories, whose basic building blocks are provided by the representation theory of a certain class of (bi-)algebras. Secondly, we discuss in some detail examples, where the algebra is either the function algebra of a finite group, the group algebra of a finite group or a deformation of the enveloping algebra of a classical simple Lie group.     

Representations of the current algebra in a model of quantum field theory

Current algebra in the model of ultralocal quantum field theory is considered. We generalize the class of ultralocal representations of the current algebra, discussed by Newman. It is shown that the representations of the current algebra (current group) can be constructed by the use of the concept of thermodynamic limit.     

On gl（2｜2）（2）k Current Superalgebra and Twisted Conformal Field Theory

Motivated by the recently discovered hidden symmetry of the type IIB Green-Schwarz superstring on certain background, the non-semisimple Kac-Moody twisted superalgebra gl（2｜2）（2）k is investigated by means of the vector coherent state method and boson-fermion realization. The free field realization of the twisted current superalgebra at general level k is constructed. The corresponding Conformai Field Theory （CFT） has zero central charge. According to the classification theory, this CFT is a nonunitary field theory. After projecting out a U（1） factor and an outer automorphism operator, we get the free field representation of psl（2｜2）（2）k, which is the a/gebra of gl（2｜2）（2）k modulo the Z4-outer automorphism, the CFT has central charge -2.     

Stochastic Evolutions in Superspace and Superconformal Field Theory

Some stochastic evolutions of conformal maps can be described by SLE and may be linked to conformal field theory via stochastic differential equations and singular vectors in highest-weight modules of the Virasoro algebra. Here we discuss how this may be extended to superconformal maps of N=1 superspace with links to superconformal field theory and singular vectors of the N=1 superconformal algebra in the Neveu–Schwarz sector.