Chiral Algebra Research Articles

TOPOLOGICAL QUANTUM FIELD THEORIES FROM GENERALIZED 6J-SYMBOLS

Given an associative algebra with a distinguished finite set of representations that is closed under a (deformed) tensor product, and satisfies some technical assumptions, we define generalized 6j-symbols, and show that they can be associated, in a natural way, with certain labeled tetrahedra. Given a 3-dimensional compact oriented manifold M with boundary ∂M = Σ we choose an arbitrary triangulation [Formula: see text] of M and exploit the above correspondence between 6j-symbols and labeled tetrahedra to construct a vectorspace UΣ and a vector Z(M) ∈ UΣ, independent of [Formula: see text], and fulfilling the axioms of a topological quantum field theory as formulated by Atiyah [11]. Examples covered by our approach are quantum groups corresponding to the classical simple Lie algebras as well as, expectedly, chiral algebras of 2-dimensional rational conformal field theories.

On modular invariant partition functions for tensor products of conformal field theories

We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type SU(2) K A ⊗SU(2) K B , finding all consistent theories for K A , K B odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and antiholomorphic theories do not share the same chiral algebra. Some explicit examples are given.

Surface critical exponents and cylindrical partition functions for the CSOS model

Fixed boundary condition cylindrical partition functions of the CSOS model are determined by using their connection with certain sums evaluated in corner transfer matrix calculations. The details of this connection are also discussed. In identifying the appropriate chiral algebra we are led to consider the p p′ algebras as well as the U(1) Kac-Moody algebras. Surface critical exponents are calculated, and related to those of physical operators of the model in a bounded geometry, which we have obtained by the Coulomb gas approach.

Remarks on the physical states and the chiral algebra of 2D gravity coupled toC?1 matter

Some elaboration is given to the structure of physical states in 2D gravity coupled to C ≤ 1 matter, and to the chiral algebra (ω ∞) of C M = 1 theory which has been found recently, in the continuum approach, by Witten and by Klebanov and Polyakov. It is shown then that the chiral algebra is being realized as well in the minimal models of gravity (C M < 1), so that it stands as a general symmetry of 2D gravity theories

QUANTUM TODA THEORY AND THE CASIMIR ALGEBRA OF B2 AND C2

We consider W-algebras and their relation to quantum Toda field theory. We show that the chiral algebra is given as the commutant of the screening charges. The algebra WBC2, which is the Casimir algebra of B2 and C2, is then constructed using a free field representation. From this we are able to deduce the determinant in the Verma module and consequently some of the representation theory for the algebra.

Integrable systems in topological field theory

Integrability of the system of PDE for dependence on coupling parameters of the (tree-level) primary partition function in massive topological field theories, being imposed by the associativity of the perturbed primary chiral algebra, is proved. In the conformal case it is shown that all the topological field theories are classified as solutions of a universal high-order Painlevé-type equation. Another integrable hierarchy (of systems of hydrodynamic type) is shown to describe coupling to gravity of the matter sector of any topological field theory. Different multicritical models with the given structure of primary correlators are identified with particular self-similar solutions of the hierarchy. The partition function of any of the models is calculated as the corresponding τ-function of the hierarchy.

Level 1 WZW superselection sectors

The superselection structure of the Wess-Zumino-Witten theory based on the affine Lie algebra\(\widehat{so}(N)\) at level one is investigated for arbitraryN. By making use of the free fermion representation of the affine algebra, the endomorphisms which represent the superselection sectors on the observable algebra can be constructed as endomorphisms of the underlying Majorana algebra. These endomorphisms do not close on the chiral algebra of the theory, but we are able to obtain a larger algebra on which the endomorphisms close. The composition of equivalence classes of the endomorphisms reproduces the WZW fusion rules.

Results of the classification of superconformal algebras in two dimensions

A list of superconformal chiral operator product expansion algebras with quadratic nonlinearity in two dimensions is completed on the basis of the known classification of little conformal Lie superalgebras. In addition to the previously known cases and the in our previous paper constructed exceptional N=8 superalgebra associated with F(4), a novel exceptional N=7 superconformal algebra associated with G(3) is found, as well as a whole family of superalgebras containing affine su 2⊕ usp 2N . A classification scheme for quasisuperconformal algebras is also outlined.

THE ANNIHILATING IDEALS OF MINIMAL MODELS

We describe several aspects of the annihilating ideals and reduced chiral algebras of conformal field theories, especially, minimal models of Wn algebras. The structure of the annihilating ideal and a vanishing condition is given. Using the annihilating ideal, the structure of quasi-finite models of the Virasoro (2,q) minimal models are studied, and their intimate relation to the Gordon identities are discussed. We also show the examples in which the reduced algebras of Wn and Wℓ algebras at the same central charge are isomorphic to each other.

STRUCTURE OF IRREDUCIBLE SU(2) PARAFERMION MODULES DERIVED VIA THE FEIGIN-FUCHS CONSTRUCTION

We show how the Feigin-Fuchs Coulomb-gas construction, with two free Gaussian bosons, can be used to derive the representation theory of the SU(2) parafermion models. We identify the generators of the chiral algebra within the bosonic Fock space and derive the chiral algebra of the finitely reducible models, which correspond to the SU(2) and SU(1, 1) parafermion algebras. We then focus on the SU(2) highest-weight modules in the remainder of the paper. Unitarity of the modules requires that the states of the parafermion theory be independent of the zero modes of two fermionic vertex operators of the bosonic theory. The expressions for the Virasoro highest weights of the models are doubly degenerate in the bosonic Fock space. We formulate the correlation functions of these operators in the parafermion Hilbert space, and in particular, the fusion rules for the Virasoro highest weights are derived in an elegant way. Finally, the irreducible parafermion characters are derived. We discuss the connection between our analysis and previous work on representation theory based on BRST cohomology.

Kähler-Chern-Simons theory and symmetries of anti-self-dual gauge fields

Kähler-Chern-Simons theory, which was proposed as a generalization of ordinary Chern-Simons theory, is explored in more detail. The theory describes anti-self-dual instantons on a four-dimensional Kähler manifold. The phase space is the space of gauge potentials, whose symplectic reduction by the constraints of anti-self-duality leads to the moduli space of instantons. We show that infinitesimal Bäcklund transformations, previously related to “hidden symmetries” of instantons, are canonical transformations generated by the anti-self-duality constraints. The quantum wave functions naturally lead to a generalized Wess-Zumino-Witten action, which in turn has associated chiral current algebras. The dimensional reduction of the anti-self-duality equations leading to integrable two-dimensional theories is briefly discussed in this framework.

Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models

We show that the bi-hamiltonian structure of the averaged Gelfand-Dikii hierarchy is involved in the Landau-Ginsburg topological models (forA n -Series): the Casimirs for the first P.B. give the correct coupling parameters for the perturbed topological minimal model; the correspondence {coupling parameters}→{primary fields} is determined by the second P.B. The partition function (at the tree level) and the chiral algebra for LG models are calculated for any genusg.

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Zp,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZp factors.

N=2 SUPERCONFORMAL INTERACTING BOSONIC MODELS

Bosonic representations of N=2 superconformal algebra are studied. We show that the free energy momentum tensor decomposes into an orthogonal sum of the interacting bosonic model (IBM) and a coset-like tensors. We define the notion of flags of models and show that the central charge does not decrease along the flags. We examine the conditions for an arbitrary un-oriented graph to form an IBM. We discuss several properties of the chiral algebra of these models and examine the role of the continuous parameters by studying an example. Finally we discuss the relations between these models and the N=2 superconformal coset models.

N = 2 superconformal CP n model

We study the Feigin-Fuchs representation of the Kazama-Suzuki N = 2 coset model CP n = SU( n + 1)/SU( n) × U(1). The chiral algebra of this model is the N = 2 super W-algebra obtained from the quantum super Miura transformation associated with the Lie superalgebra A( n, n − 1). We construct screening operators, which commute with the W algebra, and investigate the null field structure of the model. We also study the chiral ring structure of the N = 2 CP n model using the free field realization.

ExtendedU(1) Conformal Field Theories and Zk-Parafermions

A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z1, g) Ψ(z2, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Zk-parafermions. The case Δ2 = 4 (Δ1 − 1), where Δv = Δk−v(Δ0 = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δv = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z2 [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ1. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.

On the chiral algebra of topological conformal field theory

A chiral algebra extending the minimal topological algebra of Eguchi and Yang is given. This algebra is found to be present in the BRST structure of a wide variety of topological CFTs and string theories, including topological minimal models, topological gravity, and the usual bosonic and NSR strings.

On a new hierarchy associated with the W 3(2) conformal algebra

In a realization of the Kac-Moody algebra A 2 (1) intermediate between the homogeneous and the principal realization, an integrable hierarchy of differential equations is constructed. The hierarchy shares features of both the AKNS and the KdV hierarchies. The same hierarchy can also be constructed using the Drinfeld-Sokolov approach in terms of zero curvature conditions, with as hamiltonian structure the W 3 (2) algebra of Polyakov and Bershadsky. It provides evidence for the conjecture that there exists a general relation between hierarchies constructed in some intermediate realization and the covariantly coupled chiral algebras of Bais, Tjin and van Driel.

CHIRAL ALGEBRAS—TWO EXAMPLES

We examine the chiral algebras of quotient superconformal theories by studying two examples: the SO(n) HSS models and the models based on the SU (n) group. We study the automorphism group which identifies fields in each model, and its fixed-point fields. We calculate the Poincare polynomial for the SU (n) example, see whether there are duality relations and find new Landau-Ginzburg scalar theories. For the second example we explain how to calculate the number of fields in the algebra and show a duality relation by giving the duality map explicitly.

On the algebraic structure ofN=2 string theory

InN=2 string theory the chiral algebra expresses the generations and anti-generations of the theory and the Yukawa couplings among them and is thus crucial to the phenomenological properties of the theory. Also the connection with complex geometry is largely through the algebras. These algebras are systematically investigated in this paper. A solution for the algebras is found in the context of rational conformal field theory based on Lie algebras. A statistical mechanics interpretation for the chiral algebra is given for a large family of theories and is used to derive a rich structure of equivalences among the theories (dihedralities). The Poincare polynomials are shown to obey a resolution series which cast these in a form which is a sum of complete intersection Poincare polynomials. It is suggested that the resolution series is the proper tool for studying allN=2 string theories and, in particular, exposing their geometrical nature.