W-Algebras in Conformal Field Theory (original) (raw)

W W-Algebras in Conformal Field Theory

QuantumW-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions ofW-algebras using associativity re- quirements. With this approach one explicitly obtains the first members of series of W-algebras, including in particular 'Casimir algebras' (related to simple Lie alge- bras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations ofW-algebras. In some cases consistency of the corresponding quantum field the- ory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on W- algebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level ofW-algebras exotic phenomena occur.

Representations of W-algebras with two generators and new rational models

Nuclear Physics B, 1992

In conformal field theory we investigate the representations of recently discovered W-algebras with a single generator in addition to the Virasoro field. We show that many of these W-algebras have only a finite number of highest weight representations. We describe methods for the classification and give complete lists. In a sporadic case we determine characters and fusion rules. Different methods are used for W-algebras with continously variable and for those with fixed central extension.

W-Algebras, New Rational Models and

1992

Two series of W-algebras with two generators are constructed from chiral vertex operators of a free field representation. If c = 1 − 24k, there exists a W(2, 3k) algebra for k ∈ Z Z + /2 and a W(2, 8k) algebra for k ∈ Z Z + /4. All possible lowestweight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central charge c eff = 1. The results are generalized to the case of extended supersymmetric conformal algebras.

Characters and representations of new fermionic W-algebras

Physics Letters B, 1992

W-algebras are extensions of the Virasoro algebra describing chiral subalgebras of conformal quantum field theories. Careful analysis of the four-point functions and consideration of the invariant spaces under a subgroup of the modular group SL(2, Z) allow one to find all representations of new classes of fermionic W-algebras constructed recently. For each of these W-algebras there exists only a finite number of representations. The corresponding fusion rules are calculated.

Operator algebras and conformal field theory

Communications in Mathematical Physics, 1993

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "background-independent" formulation of conformal field theories. Contents

W-Algebras with Two and Three Generators

W-Symmetry, 1995

We construct all W-algebras of chiral fields which in addition to the energy-momentum density have a single generator of conformal dimension up to 8. Some of them were unexpected, which indicates that present conjectures concerning the classification of conformally invariant quantum field theories in two dimensions are rather incomplete. We also explicitly construct the WA 3-algebra with generators of dimensions 2, 3, 4.

Conformal Linearization Versus Nonlinearity of WWW-Algebras

We review the new approach to the theory of nonlinear WWW-algebras which is developed recently and called {\it conformal linearization}. In this approach WWW-algebras are embedded as subalgebras into some {\it linear conformal} algebras with a finite set of currents and most of their properties could be understood in a much simpler way by studing their linear counterpart. The general construction is illustrated by the examples of u(N)u(N)u(N)-superconformal, W(sl(N),sl(2))W(sl(N),sl(2))W(sl(N),sl(2)), W(sl(N),sl(N))W(sl(N),sl(N))W(sl(N),sl(N)) as well as W(sl(N),sl(3))W(sl(N),sl(3))W(sl(N),sl(3)) algebras. Applications to the construction of realizations (included modulo null fields realizations) as well as central charge spectrum for minimal models of nonlinear algebras are discussed. (To appear in ``Geometry and Integrable Models'', Eds.: P.N.Pyatov & S.N.Solodukhin, World Scientific Publ. Co. (in press)).

The quantum group structure associated with non-linearly extended Virasoro algebras

Communications in Mathematical Physics, 1990

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universal R-matrix of SL(2)q, has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebras AN, N > 1, is considered along the same line. Contrary to the previous case (At) the standard R-matrix of SL(N + 1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.

The quantum symmetry of rational conformal field theories

Nuclear Physics B, 1991

The quantum group symmetry of the c < 1 Rational Conformal Field Theory. in its Coulomb gas version, is formulated in terms of a new type of screened vertex operators, which define the representation spaces of a quantum group Q. The conformal properties of these operators show a deep interplay between the quantum group Q and the Virasoro algebra.