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 ).