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arXiv:hep-th/9305127 · International Journal of Modern Physics A

MATRIX ELEMENTS OF Uq[ su (2)k] VERTEX OPERATORS VIA BOSONIZATION

A. Hamid Bougourzi, Robert Weston

Published Oct 10, 1994• 9 citations

Abstract

We construct bosonized vertex operators (VO's) and conjugate vertex operators (CVO's) of Uq[ su (2)k] for arbitrary level k and representation j ≤ k/2. Both are obtained directly as two solutions of the defining condition of vertex operators — namely that they intertwine Uq[ su (2)k] modules. We construct the screening charge and present a formula for the n-point function. Specializing to j = 1/2 we construct all VO's and CVO's explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as k = 1; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected k = 1 and q = 1 limits.

DOI: 10.1142/s0217751x9400176x

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We extend the concept of conjugate vertex operators, first introduced by Dotsenko in the case of the bosonization of the su(2) conformal field theory, to the bosonization of the dynamical vertex operators (type II in the classification of the Kyoto school) of the higher spin XXZ model. We show that the introduction of the conjugate vertex operators leads to simpler expressions for the N-point matrix elements of the dynamical vertex operators, that is, without redundant Jackson integrals that arise from the insertion of screening charges. In particular, the two-point matrix element can be represented without any integral.

We extend the concept of conjugate vertex operators, first introduced by Dotsenko (1990) in the case of the bosonization of the SU(2) conformal field theory, to the bosonization of the dynamical vertex operators (type II in the classification of the Kyoto school) of the higher spin XXZ model. We show that the introduction of the conjugate vertex operators leads to simpler expressions for the N-point matrix elements of the dynamical vertex operators, that is, without redundant Jackson integrals that arise from the insertion of screening charges. In particular, the two-point matrix element can be represented without any integral.

LEVEL 2 IRREDUCIBLE REPRESENTATIONS OF

U_q (\widehat{{\rm sl}}_2)

, VERTEX OPERATORS, AND THEIR CORRELATIONS

In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.

We extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra

U_q(su(2)_2)

of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of

U_q(su(2)_1)

.