Canonical approach to the WZNW model

Abstract

The chiral Wess-Zumino-Novikov-Witten (WZNW) model provides the simplest class of rational conformal field theories which exhibit a non-abelian braid-group statistics and an associated "quantum symmetry". The canonical derivation of the Poisson-Lie symmetry of the classical chiral WZNW theory (originally studied by Faddeev, Alekseev, Shatashvili and Gawȩdzki, among others) is reviewed along with subsequent work on a covariant quantization of the theory which displays its quantum group symmetry.

“Spread” Restricted Young Diagrams from a 2D WZNW Dynamical Quantum Group

“…This work contains a brief exposition of new results based on ideas and techniques, some parts of which have been already made public in [23,24,21]. The latter relied, in turn, on the notion of quantum matrix algebras generated by the chiral zero modes of the SU (n) k Wess-Zumino-Novikov-Witten (WZNW) model introduced in [22] (see also [26,17,19]).…”

“…The relation of such algebraic objects with quantum groups [7,27] has been anticipated in [1,12]. For generic values of the deformation parameter q the Fock representation of the chiral zero modes' algebra is a model space of U q (s (n)) [4,17,21]. In the most interesting applications the deformation parameter q is a root of unity (in our case we take q = e −i π h , h = k + n).…”

“…We will recall here the basic assumptions about the chiral and 2D WZNW "zero modes" and their Fock representation, justified by the consistent application of the principles of canonical quantization (see e.g. [24,21]).…”

“…This has been done in a completely satisfactory way for n = 2 (in [24,21]; see also [18]) and the emerging picture is easy to describe. It turns out that in this case the diagonal elements of the matrix Q = (Q i j ) commute with the off-diagonal ones and both generate two copies of the (finite dimensional) restricted quantum group U q (s (2)) of [13,14,20].…”

“…(The spread condition imposes a stronger restriction, except for n = 2.) In the WZNW setting the zero modes are coupled to elementary chiral vertex operators (CVO) with similar intertwining properties [21]; having this in mind, we note that the integrable representations of the affine algebra su(n) k (or the fusion sectors of the unitary model) are labeled by all su(n) Young diagrams that fit into the "narrower" rectangle of size (n−1)×k . Extending the analogy with the n = 2 case, cf.…”