Ludmil Hadjiivanov

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 MM−1 = 1 (in order to recover the monodromy invariance of the local 2D group valued field g = uu) 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.

Quantum matrix algebra for the SU(n) WZNW model

The zero modes of the chiral SU (n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a = (a i α ), i, α = 1,....,n (with noncommuting entries) and by rational functions of n commuting elements q P 1 satisfying Π n i = 1 q p i = 1, q p i a j α = a j α q p i + δ j i - . We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U q = U q (sl n ), with each irreducible representation entering F with multiplicity 1. For an integer su (n) height h (= k + n ≥ n) the complex parameter q is an even root of unity, q h = -1, and the algebra A has an ideal I h such that the factor algebra A n = A/I h is finite dimensional. All physical Uq modules-of shifted weights satisfying p 1 n ≡ p 1 - p n < h-appear in the Fock representation of A h .

Quantum su(n)k monodromy matrices

The canonical quantization of the chiral Wess–Zumino–Novikov–Witten (WZNW) monodromy matrices, both the diagonal and the general one, requires additional numerical factors that can be attributed to renormalization. We discuss the field-theoretic and algebraic aspects of this phenomenon for the SU(n) WZNW model and show that these quantum renormalization factors are compatible with the natural definitions for the determinants of the involved matrices with non-commuting entries.

Quantum deformation of the ladder representations of U(1,1) for ‖q‖=1

Uq(u(1,1)) covariant creation and annihilation operators for q on the unit circle are constructed and the corresponding ladder and singleton representations are considered.

On the 2D Zero Modes’ Algebra of the SU(n) WZNW Model

A quantum group covariant extension of the chiral parts of the Wess-Zumino-Novikov-Witten (WZNW) model on a compact Lie group G gives rise to two matrix algebras with non-commutative entries. These are generated by “chiral zero modes” \(a_{\alpha }^{i}\,,\bar{a}_{j}^{\beta }\) which combine, in the 2D model, into \(Q_{j}^{i} = a_{\alpha }^{i} \otimes \bar{ a}_{j}^{\alpha }\). The Q-operators provide important information about the internal symmetry and the fusion ring. Here we review earlier results about the SU(n) WZNW Q-algebra and its Fock representation for n = 2 and make the first steps towards their generalization to n ≥ 3.

Zero modes of the SU(2)k Wess–Zumino–Novikov–Witten model in Euler angles parametrization

We derive the Poisson brackets of the SU(2)k Wess–Zumino–Novikov–Witten chiral zero modes directly, using Euler angles parametrization.

Indecomposable U q (sl n ) modules for q h = −1 and BRS intertwiners

A class of indecomposable representations of Uq(sln) is considered for q, an even root of unity (qh = -1) exhibiting a similar structure as (height h) indecomposable lowest weight Kac-Moody modules associated with chiral conformal field theory. In particular, Uq(sln) counterparts of the Bernard-Felder BRS operators are constructed for n = 2,3. For n = 2 a pair of dual d2(h) = h-dimensional Uq(sl2) modules gives rise to a 2h-dimensional indecomposable representation including those studied earlier in the context of tensor-product expansions of irreducible representations. For n = 3 the interplay between the Poincare-Birkhoff-Witt and (Lusztig) canonical bases is exploited in the study of d3(h) = (1/6)h(h + 1)(2h + 1)-dimensional indecomposable modules and of the corresponding intertwiners.