arXiv:hep-th/0109219v3 24 Oct 2001 EPJ manuscript No. (will be inserted by the editor) On the rational solutions of the  su(2) k Knizhnik-Zamolodchikov equation Ludmil Hadjiivanov and Todor Popov Theoretical Physics Division, Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria Received: date / Revised version: date Abstract. We present some new results on the rational solutions of the Knizhnik-Zamolodchikov (KZ) equation for the four-point conformal blocks of isospin I primary fields in the SU (2) k Wess-Zumino- Novikov-Witten (WZNW) model. The rational solutions corresponding to integrable representations of the affine algebra  su(2) k have been classified in [1], [2]; provided that the conformal dimension is an integer, they are in one-to-one correspondence with the local extensions of the chiral algebra. Here we give another description of these solutions as specific braid-invariant combinations of the so called regular basis introduced in [3] and display a new series of rational solutions for isospins I = k +1 ,k ∈ N corresponding to non-integrable representations of  su(2) k . PACS. 11.25.H Conformal field theory – 02.20.U Quantum groups 1 Introduction The 2D nonlinear σ-model with a suitably normalized WZ term, known as WZNW model [4], is a conformally invari- ant (and therefore integrable) field theory with a huge internal symmetry, beautiful geometric structure at the classical level and rich algebraic content in the quantized case. The model describes a closed (respectively, open, for the so called boundary model) string moving freely on a Lie group manifold G. After choosing the group, the only parameter left which fixes the theory is a positive integer k playing the role of WZ term coupling constant. Here we will only consider the case of the compact group G = SU (2) . To solve the model, one can use different approaches in both the classical and the quantum cases. In the axiomatic approach to the quantized model one constructs the space of states as a direct sum of superselection sectors (tensor products of integrable representations of the correspond- ing left and right current algebras, both of which appear to be affine algebras of the type ˆ G k where G is the Lie algebra of G and k is the level). Each sector is generated from the vacuum by a primary field. The interplay of affine and conformal invariance leads to linear systems of partial differential equations for the correlation functions of pri- mary fields (conformal blocks), one for each set of chiral variables. These are the famous KZ equations [5], [6] de- termining, in principle, the chiral structure of the theory; the correlation functions of the 2D theory are recovered by combining the left and the right conformal blocks (which are, typically, multivalued) in such a way that 2D locality is restored [7]. In some cases this can be done in different ways constrained by modular invariance of the WZNW partition function; for G = SU (2) this leads to the ADE classification of [8]. The canonical quantization of the chirally split WZNW model [9]-[15] leads to a description in terms of chiral fields revealing the quantum group (QG) invariance of their ex- change algebras which is the quantum counterpart of the Poisson-Lie invariance of the underlying classical theory (see [16] for a recent comprehensive exposition of the clas- sical situation and [17], focused on the boundary WZNW model). The fact that the monodromies of the chiral cor- relation functions are related to U q (G)6j -symbols (for q an even root of unity, q = e ±i π k+2 for G = SU (2) ) has been known for a long time [18]. On the other hand, it is clear that the true ”internal” symmetry of the model is much more involved (see e.g. [19] and references therein for a recent analysis of the relation between weak C * -Hopf algebras and rational conformal field theories). A plausi- ble way out of this apparent contradiction would be the assumption [20] that U q (G) plays the role of a generalized ”gauge” group on the extended (chirally split) WZNW so that one is facing an alternative analogous to choosing unitary or covariant gauges in gauge theories, the latter necessarily including unphysical states. In the case at hand this means that we have to consider indecomposable rep- resentations of both the conformal current algebra and of the quantum group. For the SU (2) k WZNW model the minimal exten- sion should involve primary fields with isospins covering at least twice the range of the unitarizable representations, 0 ≤ 2I ≤ 2k +2 , since the indecomposable QG counter-