A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.q-alg/9507034

We study the generic scaling properties of the mutual information between two disjoint intervals, in a class of one-dimensional quantum critical systems described by the c = 1 bosonic field theory. A numerical analysis of a spin-chain model reveals that the mutual information is scale-invariant and depends directly on the boson radius. We interpret the results in terms of correlation functions of branch-point twist fields. The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge. Given a microscopic model, an important and often nontrivial issue is how to obtain the effective field theory controlling its long-distance behavior. The notion of quantum entanglement, or more specifically, the entanglement entropy, has been extensively applied as a new way to address this basic matter. From a quantum ground state |Ψ , one constructs the reduced density matrix ρ A := TrĀ |Ψ Ψ| on a subsystem A by tracing out the exteriorĀ. The entanglement entropy is defined as S A := −Tr ρ A log ρ A . In 1D quantum critical systems, the entanglement entropy for an interval A = [x 1 , x 2 ] embedded in a chain exhibits a universal scaling [6,7,8,9,10,11]:where c is the central charge of the CFT and s 1 is a nonuniversal constant related to the ultra-violet (UV) cutoff. This scaling allows to determine the universal number c as a representative of the ground state structure, without having to worry about the precise correspondence between the microscopic model and the field theory.As it is well known, the central charge is not the only important number specifying a CFT. In the bosonic field theory with c = 1, the boson compactification radius R (or equivalently, the TLL parameter K = 1/(4πR 2 )) is a dimensionless parameter which changes continuously in a phase and controls the power-law behavior of various physical quantities. It is natural to ask how to identify the boson radius as a generic structure of the ground state. In this Letter, we demonstrate that the entanglement entropy can achieve this task if we consider two disjoint intervals, A = [x 1 , x 2 ] and B = [x 3 , x 4 ]. We analyze the scaling of the mutual information defined asThis measures the amount of information shared by two subsystems [12,13]. A numerical analysis of a spin-chain model reveals a robust relation between I A:B and R, irrespective of microscopic details. We compare the result with the general prediction of Calabrese and Cardy (CC) [9], and find a relevant correction to their result. Roughly speaking, the mutual information (2) may be regarded as a region-region correlator. It is known that I A:B is non-negative, and becomes zero iff ρ A∪B = ρ A ⊗ ρ B , i.e., in a situation of no correlation [14]. A motivation to consider I A:B comes from that microscopic details at short-range scales, which are often obstacles when analyzing point-point correlators, can be smoothed out over regions. As we enlar...

We derive a quantum deformation of the W N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.q-alg/9508011

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [1], Felder [2]). Frønsdal [3,4] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U q (g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U q (g). We also prove the shifted cocycle condition for the twistors, thereby completing Frønsdal's findings.This construction entails that, for generic values of the deformation parameters, representation theory for U q (g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra A q,p ( sl 2 ).

Abstract. We establish the equivalence between the refined topological vertex of IqbalKozcaz-Vafa and a certain representation theory of the quantum algebra of type W 1+∞ introduced by Miki. Our construction involves trivalent intertwining operators Φ and Φ * associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors ∈ Z 2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Φ and Φ * give the refined topological vertex C λµν (t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C λµ ν (q, t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Φ and Φ * . The spectral parameters attached to Fock spaces play the role of the Kähler parameters.

Abstract. We introduce a unital associative algebra A over degenerate CP 1 . We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii [FO]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors [S2]. It is found that the Ding-Iohara algebra [DI]

On the basis of the collective field method, we analyze the Calogero-Sutherland model (CSM) and the Selberg-Aomoto integral, which defines, in particular case, the partition function of the matrix models. Vertex operator realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Virasoro constraint for the generalized matrix models and indicate relations with the CSM operators. Similar results are presented for the q-deformed case (the Macdonald operator and polynomials), which gives the generating functional of infinitely many conserved charges in the CSM. * JSPS fellow

We investigate the structure of the elliptic algebra U q,p ( sl 2 ) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U q ( sl 2 ), which are elliptic analogs of the Drinfeld currents. They enable us to identify U q,p ( sl 2 ) with the tensor product of U q ( sl 2 ) and a Heisenberg algebra generated by P, Q with [Q, P ] = 1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element c. The vertex operators of Lukyanov and Pugai arise as 'intertwiners' of U q,p ( sl 2 ) for the level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.The principle of infinite dimensional symmetry has seen an impressive success in conformal field theory (CFT). With the aim of understanding non-critical lattice models in the same spirit, the method of algebraic analysis [1, 2, 3] has been developed. In this approach, a central role is played by the notion of vertex operators (VO's). There are two kinds of VO's with distinct physical significance: the type I VO, which describes the operation of adding one lattice site, and the type II VO, which plays the role of particle creation/annihilation operators. In the most typical example of the XXZ spin chain, these VO's have a clear mathematical meaning as intertwiners [4] of certain modules over the quantum affine algebra U q ( sl 2 ).An important class of CFT is the minimal unitary series [5]. Their lattice counterpart are the solvable models of Andrews-Baxter-Forrester (ABF) [6]. These are 'solid-on-solid' (SOS, or 'face') models whose Boltzmann weights are expressed by elliptic functions. Their Lie theoretic generalizations have also been studied extensively [7,8,9]. The vertex operator approach to the ABF models and their fusion hierarchy was formulated in [10] by a coset-type construction. In [10], U q ( sl 2 ) was used only as an auxiliary tool to define the VO's, and its role as a symmetry algebra was somewhat indirect. In [11], Lukyanov and Pugai constructed a free boson realization of type I VO's for the ABF models. (The formulas for type II VO's can be found in [12].) They have shown further that these VO's commute with the action of the deformed Virasoro algebra (DVA) [13], making clear the parallelism with CFT. However, unlike the case of CFT, the VO's did not allow for direct interpretation as intertwiners, because DVA lacks a coproduct 1) . It has remained an open problem to understand the conceptual meaning of VO's.In [14], one of the authors introduced an elliptic algebra U q,p ( sl 2 ) and proposed it as an algebra of screening currents of conjectural extended DVA associated with the fusion SOS models. The aim of the present article is to continue the study of U q,p ( sl 2 ), and to show that it offers a characterization of the VO's for SOS models in close analogy with the XXZ model.