Chiral Vertex Research Articles

A LECTURE ON THE LIOUVILLE VERTEX OPERATORS (REVIEW)

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov.

Critical RSOS and minimal models: fermionic paths, Virasoro algebra and fields

A framework is presented to extend the finitized characters and recursion methods of (off-critical) corner transfer matrices (CTMs), in a self-consistent fashion, to the calculation of CFT characters and conformal partition functions. More specifically, in this paper we consider sℓ(2) minimal conformal field theories on a cylinder from a lattice perspective. We argue that a general energy-preserving bijection exists between the one-dimensional configuration paths of the A L restricted solid-on-solid (RSOS) lattice models and the eigenstates of their double row transfer matrices and exhibit this bijection for the critical and tricritical Ising models in the vacuum sector. To each allowed one-dimensional configuration path we associate a physical state and a monomial in a finite fermionic algebra. The orthonormal states produced by the action of these monomials on the primary states | h〉 generate finite Virasoro modules with dimensions given by the finitized Virasoro characters χ ( N) h ( q). These finitized characters are the generating functions for the double row transfer matrix spectra of the critical RSOS models. We also propose a general level-by-level algorithm to build matrix representations of the Virasoro generators and chiral vertex operators (CVOs). The algorithm employs a distinguished basis which we call the L 1-basis. Our results extend to Z L−1 parafermion models by duality.

On a q-Analogue of the McKay Correspondence and the ADE Classification of [formula omitted]2 Conformal Field Theories

The goal of this paper is to give a category theory based definition and classification of “finite subgroups in U q ( s l 2)” where q= e πi/ l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ( s l 2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to s l ̂ 2 at level k= l−2. We show that “finite subgroups in U q ( s l 2)” are classified by Dynkin diagrams of types A n , D 2 n , E 6, E 8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( s l ̂ 2) k conformal field theory. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory.

Non-renormalization theorems of supersymmetric QED in the Wess–Zumino gauge

The non-renormalization theorem of chiral vertices and the generalized non-renormalization theorem of the photon self energy are derived in SQED on the basis of algebraic renormalization. For this purpose the gauge coupling is extended to an external superfield. This extension already provides detailed insight into the divergence structure. Moreover, using the local supercoupling together with an additional external vector multiplet that couples to the axial current, the model becomes complete in the sense of multiplicative renormalization, with two important implications. First, a Slavnov–Taylor identity describing supersymmetry, gauge symmetry, and axial symmetry including the axial anomaly can be established to all orders. Second, from this Slavnov–Taylor identity we can infer a Callan–Symanzik equation expressing all aspects of the non-renormalization theorems. In particular, the gauge β-function appears explicitly in the closed form.

The many faces of Ocneanu cells

We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.

Negative screenings in conformal field theory and 2D gravity: the braiding matrix

We consider an extension of the Coulomb gas picture which is motivated by Liouville theory and contains negative powers of screening operators on the same footing as positive ones. The braiding problem for chiral vertex operators in this extended framework is analyzed. We propose explicit expressions for the R-matrix with general integer screening numbers, which are given in terms of 4F3q-hypergeometric functions through natural analytic continuations of the well-known expression for positive integer screenings. These proposals are subsequently verified using a subset of the Moore–Seiberg equations that is obtained by simple manipulations in the operator approach. Interesting new relations for q-hypergeometric functions (particularly of type 4F3) arise on the way.

Non-renormalization theorems without supergraphs: the Wess–Zumino model

The non-renormalization theorems of chiral vertex functions are derived on the basis of an algebraic analysis. The property, that the interaction vertex is a second supersymmetry variation of a lower dimensional field monomial, is used to relate chiral Green functions to superficially convergent Green functions by extracting the two supersymmetry variations from an internal vertex and transforming them to derivatives acting on external legs. The analysis is valid in the massive as well as in the massless model and can be performed irrespective of properties of the superpotential at vanishing momentum.

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U q (sl n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.

MULTIPLE EDGE PARTITION FUNCTIONS FOR FRACTIONAL QUANTUM HALL STATES

We consider the multiple edge states of the Laughlin state and the Pfaffian state. These edge states are globally constrained through the operator algebra of conformal field theory in the bulk. We analyze these constraints by introducing an expression of quantum Hall state by chiral vertex operators and obtain the multiple edge partition functions by using the Verlinde formula.

Coulomb gas representation of quantum Hall effect on Riemann surfaces

Using the correlation function of chiral vertex operators of the Coulomb gas model, we find the Laughlin wavefunctions of quantum Hall effect, with filling factor $\nu =1/m$, on Riemann sufaces with Poincare metric. The same is done for quasihole wavefunctions. We also discuss their plasma analogy.

A non-Abelian square root of Abelian vertex operators

The quantum field theoretical formulation of Kadanoff’s “correlations along a line” in the critical two-dimensional Ising model is given in terms of p-products of Wightman fields. It provides a quadratic factorization, in the sense of operator-valued distributions, of Abelian chiral vertex operators into non-Abelian exchange fields. This basic result implies p-polynomial relations between fields from various conformal models in two dimensions; notably the two-dimensional local fields with chiral SU(2) symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of p-products in general.

Smeared and Unsmeared Chiral Vertex Operators

We prove unboundedness and boundedness of the unsmeared and smeared chiral vertex operators, respectively. We use elementary methods in bosonic Fock space, only. Possible applications to conformal two - dimensional quantum field theory, perturbation thereof, and to the perturbative construction of the sine-Gordon model by the Epstein-Glaser method are discussed. From another point of view the results of this paper can be looked at as a first step towards a Hilbert space interpretation of vertex operator algebras.

Chiral quantization of the WZW SU ( n) model

We quantize the SU ( n) Wess-Zumino-Witten model in terms of left and right chiral variables choosing an appropriate gauge and we compare our results with the results that have been previously obtained in the algebraic treatment of the problem. The algebra of the chiral vertex operators in the fundamental representation is verified by solving the appropriate Knizhnik-Zamolodchikov equations.

Hidden U q (sl(2)) $otimes$ U q (sl(2)) Quantum Group Symmetry in Two Dimensional Gravity

In a previous paper, we proposed a construction of $U_q(sl(2))$ quantum group symmetry generators for 2d gravity, where we took the chiral vertex operators of the theory to be the quantum group covariant ones established in earlier works. The basic idea was that the covariant fields in the spin $1/2$ representation themselves can be viewed as generators, as they act, by braiding, on the other fields exactly in the required way. Here we transform this construction to the more conventional description of 2d gravity in terms of Bloch wave/Coulomb gas vertex operators, thereby establishing for the first time its quantum group symmetry properties. A $U_q(sl(2))\otimes U_q(sl(2))$ symmetry of a novel type emerges: The two Cartan-generator eigenvalues are specified by the choice of matrix element (bra/ket Verma-modules); the two Casimir eigenvalues are equal and specified by the Virasoro weight of the vertex operator considered; the co-product is defined with a matching condition dictated by the Hilbert space structure of the operator product. This hidden symmetry possesses a novel Hopf like structure compatible with these conditions. At roots of unity it gives the right truncation. Its (non linear) connection with the $U_q(sl(2))$ previously discussed is disentangled.

Operator realization of the SU(2) WZNW model

Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables M and M . Earlier work on the subject, which traced back the quantum group symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: • - How to reconcile the necessity to set M M −1 = 1 (in order to recover the monodromy invariance of the local 2D group valued field g = u u ) with the fact the M and M obey different exchange relations? • - What is the status of the quantum symmetry in the 2D theory in which the chiral fields u( x− t) and u(x + t) commute? • - Is there a consistent operator formalism in the chiral (and the extended 2D) theory in the continuum limit? We propose a constructive affirmative answer to these questions for G = SU(2) by presenting the quantum fields u and u as sums of products of chiral vertex operators and q-Bose creation and annihilation operators.

Chiral vertex operators in off-conformal theory: Sine-Gordon example.

We study chiral vertex operators in the sine-Gordon [SG] theory, viewed as an off-conformal system. We find that these operators, which would have been primary fields in the conformal limit, have interesting and, in some ways, unexpected properties in the SG model. Some of them continue to have scale- invariant dynamics even in the presence of the non-conformal cosine interaction. For instance, it is shown that the Mandelstam operator for the bosonic representation of the Fermi field does {\it not} develop a mass term in the SG theory, contrary to what the real Fermi field in the massive Thirring model is expected to do. It is also shown that in the presence of the non-conformal interactions, some vertex operators have unique Lorentz spins, while others do not.

Status of the operator approach to liouville theory

These notes contain a review of work by J.-L. Gervais and the author, as well as recent work in progress by the author. The two main themes are the construction of local observables in Liouville theory — the Liouville exponentials — together with the underlying algebra of chiral vertex operators, and the proper definition of the three point function in the operator framework.

Quantisation of the SU (N) WZW model at level k

The quantisation of the Wess-Zumino-Witten model on a circle is discussed in the case of SU(N) at level k. The quantum commutation of the chiral vertex operators is described by an exchange relation with a braiding operator, Q. Using quantum consistency conditions, the braiding operator is found explicitly in the fundamental representation. Its matrix elements are identified as the Racah coefficients for Ut(SL(N)). From studying the monodromy of the four-point functions by solving Knizhnik-Zamolodchikov equations, the deformation parameter t is shown to be t = exp(iπ/ (k + N)) when the level k ⩾ 2. For k = 1, there are two possible types of braiding, t = exp(iπ/(1 + N)) or t = exp(iπ). In the latter case, the chiral vertex operators are constructed explicitly by extending the free field realisation à la Frenkel-Kac and Segal. This construction gives an explicit description of how to chirally factorise the SU(N)k=1 WZW model.

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi quantum group $A_{g,t}$, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The $R$ matrix describes the braiding of the chiral vertex operators and the coassociator $\Phi$ gives rise to a modification of the duality property. For generic $q$ the quasi quantum group is isomorphic to the coassociative quantum group $U_{q}(g)$ and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasi quantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case of $g=su(2)$ is worked out in detail.

A theory of tensor products for modulé categories for a vertex operator algebra, IV

This is the fourth part in a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, we establish the associativity of P( z)-tensor products for nonzero complex numbers z constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that they relate the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion.