WZNW Theory Research Articles

Correspondences among CFTs with different W-algebra symmetry

W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra g and an sl(2)-embedding into g. We derive correspondences among correlation functions of theories having different W-algebras as symmetry algebras. These W-algebras are associated to the same g but distinct sl(2)-embeddings.For this purpose, we first explore different free field realizations of W-algebras and then generalize previous works on the path integral derivation of correspondences of correlation functions. For g=sl(3), there is only one non-standard (non-regular) W-algebra known as the Bershadsky-Polyakov algebra. We examine its free field realizations and derive correlator correspondences involving the WZNW theory of sl(3), the Bershadsky-Polyakov algebra and the principal W3-algebra. There are three non-regular W-algebras associated to g=sl(4). We show that the methods developed for g=sl(3) can be applied straightforwardly. We briefly comment on extensions of our techniques to general g.

Superpolynomials for torus knots from evolution induced by cut-and-join operators

The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist.

The string theory on [formula omitted] as a marginal deformation of a linear dilaton background

We investigate N-point string scattering amplitudes in AdS 3 space. Based on recent observations on the solutions of KZ and BPZ-type differential equations, we discuss how to describe the string theory in AdS 3 as a marginal deformation of a (flat) linear dilaton background. This representation resembles the called “discrete light-cone Liouville” realization as well as the FZZ dual description in terms of the sine-Liouville field theory. Consequently, the connection and differences between those and this realization are discussed. The free field representation presented here permits to understand the relation between correlators in both Liouville and WZNW theories in a very simple way. Within this framework, we discuss the spectrum and interactions of strings in Lorentzian AdS 3 .

On spectral flow symmetry and Knizhnik–Zamolodchikov equation

It is well known that five-point function in Liouville field theory provides a representation of solutions of the SL(2,R)k Knizhnik–Zamolodchikov equation at the level of four-point function. Here, we make use of such representation to study some aspects of the spectral flow symmetry of slˆ(2)k affine algebra and its action on the observables of the WZNW theory. To illustrate the usefulness of this method we rederive the three-point function that violates the winding number in SL(2,R) in a very succinct way. In addition, we prove several identities holding between exact solutions of the Knizhnik–Zamolodchikov equation.

Topics in quantum integrable systems

Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.

Quantisation of gauged \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$SL(2,\mathbb R)$\end{document} WZNW theories

AbstractCanonical quantisation of the free‐field zero modes q, p on a half‐line p > 0 provides for WZNW coset theories self‐adjoint vertex operators on account of hidden symmetries generated by an S‐matrix.

CANONICAL APPROACH TO 2D SUPERSYMMETRIC WZNW MODEL COUPLED TO SUPERGRAVITY

Starting from the known representation of the Kac–Moody algebra in terms of the coordinates and momenta, we extend it to the representation of the super Kac–Moody and super Virasoro algebras. Then we use general canonical method to construct an action invariant under local gauge symmetries, where components of the super energy–momentum tensor L± and G± play the role of the diffeomorphisms and supersymmetry generators respectively. We obtain covariant extension of WZNW theory with respect to local supersymmetry as well as explicit expressions for gauge transformations.

Characters of sl(4)k fusion algebra at the non-rational level

We construct the fusion ring of a quasi-rational $\hat{sl}(4)_k$ WZNW theory at generic level $k \not\in Q$. It is generated by commutative elements in the group ring $Z[\tilde{W}]$ of the affine Weyl group $\tilde{W}$ which extend polynomially the formal characters of finite dimensional representations of $sl(4)$.

Poisson structure and Moyal quantisation of the Liouville theory

The symplectic and Poisson structures of the Liouville theory are derived from the SL(2, R) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of local fields are constructed and their ∗-products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given.

OPERATOR QUANTIZATION OF CONSTRAINED WZNW THEORIES AND COSET CONSTRUCTIONS

Using two WZNW theories for Lie algebras g and h, h ⊂ g, we construct the associative quotient algebra which includes a class of g/h coset primary fields and currents.

Twisted Kac–Moody algebras and the entropy of AdS 3 black hole

We show that an SL(2, R) L × SL(2, R) R Chern–Simons theory coupled to a source on a manifold with the topology of a disk correctly describes the entropy of the AdS 3 black hole. The resulting boundary WZNW theory leads to two copies of a twisted affine Kac–Moody algebra, for which the respective Virasoro algebras have the same central charge c as the corresponding untwisted theory. But the eigenvalues of the respective L 0 operators are shifted. We show that the asymptotic density of states for this theory is, up to logarithmic corrections, the same as that obtained by Strominger using the asymptotic symmetry of Brown and Henneaux.

Integration of the SL(2,R) / U(1) gauged WZNW theory by reduction and quantum parafermions

Using a gauge-invariant reduction, we directly integrate the SL(2,ℝ/U(1) WZNW theory. We prove that the conserved parafermions in this theory are coset currents. The parafermion algebra, the energy–momentum tensor, and “auxiliary” parafermions are quantized in a self-consistent manner.

G/u(1)dparafermions from constrained WZNW theories

The conformal field theory based on the $g/u(1)^d$ coset construction is treated as the WZNW theory for the affine Lie algebra $\hat g$ with the constrained $\hat u(1)^d$ subalgebra.Using a modification of the generalized canonical quantization method generators and primary fields of an extended symmetry algebra are found for arbitrary d.

Factorization of correlation functions in coset conformal field theories

We use the conformal Ward identities to study the structure of correlation functions in coset conformal field theories. For a large class of primary fields of arbitrary g/h theory, a factorization anzatz is found. Corresponding correlation functions are explicitly expressed in terms of correlation functions of two independent WZNW theories for g and h.

Constrained current algebras and g/u(1)d conformal field theories

Operator quantization of the WZNW theory invariant with respect to an affine Lie algebra ĝ with a constrained u ̂ (1) d subalgebra is performed using Dirac's procedure. Upon quantization the initial energy-momentum tensor is replaced by the g / u (1) d coset construction. The su ̂ (2) WZNW theory with a constrained u ̂ (1) current is equivalent to the su (2)/ u (1) conformal field theory.

Hamiltonian reduction of supersymmetric WZNW models on bosonic groups and superstrings

It is shown that an alternative supersymmetric version of the Liouville equation extracted from D = 3 Green-Schwarz superstring equations naturally arises as a super-Toda model obtained from a properly constrained supersymmetric WZNW theory based on the sl(2, R) algebra. Hamiltonian reduction is performed by imposing a non-linear superfield constraint which turns out to be a mixture of a first- and second-class constraint on supercurrent components. Supersymmetry of the model is realized non-linearly and is spontaneously broken. The set of independent current fields which survive the Hamiltonian reduction contains (in the holomorphic sector) one bosonic current of spin 2 (the stress tensor of the spin-0 Liouville mode) and two fermionic fields of spin 3 2 and −1 2 . The n = 1 superconformal system thus obtained is of the same kind as one describing non-critical fermionic strings in a universal string theory. The generalization of this procedure allows one to produce from any bosonic Lie algebra super-Toda models and associated super- W algebras together with their non-standard realizations.

Closed superstrings in magnetic flux tube background

We consider a 2-parameter class of solvable closed superstring models which `interpolate' between Kaluza-Klein and dilatonic Melvin magnetic flux tube backgrounds. The spectrum of string states has similarities with Landau spectrum for a charged particle in a uniform magnetic field. The presence of spin-dependent `gyromagnetic' interaction implies breaking of supersymmetry and possible existence (for certain values of magnetic parameters) of tachyonic instabilities. We study in detail the simplest example of the Kaluza-Klein Melvin model describing a superstring moving in flat but non-trivial 10-d space containing a 3-d factor which is a `twisted' product of a 2-plane and an internal circle. We also discuss the compact version of this model constructed by `twisting' the product of the two groups in SU(2) x U(1) WZNW theory without changing the local geometry (and thus the central charge). We explain how the supersymmetry is broken by continuous `magnetic' twist parameters and comment on possible implications for internal space compactification models. (Contribution to the Proceedings of the 1995 Erice School "String Gravity and Physics at the Planck Scale")

Heterotic Toda fields

A non-left-right symmetric conformal integrable Toda field theory is constructed. It is found that the conformal algebra for this model is the product of a left chiral W r+1 algebra and a right chiral W r+1 2 algebra. The general classical solution is constructed out of the chiral vectors satisfying the so-called classical exchange algebra. In addition, we derived an explicit Wronskian type solution in relation to the constrained WZNW theory. We also showed that the A ∞ limit of this model is precisely the ( B 2, C 1) flow of the standard Toda lattice hierarchy.

FRACTIONAL QUANTUM HALL EFFECT FROM ANOMALIES IN WZNW MODEL

An approach to understand fractional quantum Hall effect (FQHE) using anomalies is studied in this letter. More specifically, this is done by looking at the anomaly in the current conservation equation of a WZNW theory describing fields living at the edge of the two-dimensional Hall sample. This WZNW theory itself comes from the non-Abelian bosonization of fermions living at the edge. By identifying the anomaly at the edge with the Hall conductivity in the bulk, it is shown that this model can describe the integer or the FQHE depending on the charge matrix used as the generator of gauge transformations at the boundary.

A conformally broken system as the constrained SL(2,R) WZNW model, its integrability and dressing symmetry

This paper is devoted to the study of the conformally broken system related to the conformally invariant SL(2,R) WZNW model. Our starting point is such an action that is similar in form to the action of gauged WZNW theories developed by Balog et al. (1990). However, the Lagrangian multipliers in our case are assumed to take their values on the mixing of upper and lower Borel subalgebras of sl(2). The system obtained turns out to be an off-conformal extension of the sinh-Gordon model. Its integrability is displayed by the Lax formalism with an arbitrary spectral parameter, the classical r-matrix, and the hidden dressing symmetry.