Regular basis and \(R\)-matrices for the \(\widehat{su}(n)_k\) Knizhnik-Zamolodchikov equation (Q5944747)

There are two different approaches to the WZNW model describing the conformally invariant dynamics of a closed string moving on a compact Lie group \(G\). One of them, the axiomatic approach, relies on the representation theory of Kac-Moody current algebras and on the Sugawara formula for the stress-energy tensor. The symmetries of the chiral conformal block solutions of the KZ equation, characterizing the correlation functions of the model, is related to the quantum groups. The second canonical approach is based on the Poisson-Lie symmetry of the WZNW action that give rise to the quantum group invariant quadratic exchange relations at the quantum level. The object of the paper is to provide a step towards the consistent operator formulation of the chiral WZNW model that would reproduce the known conformal blocks for the KZ equation. The first result is the statement of the precise correspondence between the monodromy representation of the braid group and the \(R\)-matrix exchange relations for the step operators, that is the field operators that are transformed under the representation of \(SU(n)\). To this end the four-point conformal block including two step operators is considered. The second (new) result is the extension to \(\widehat {\mathfrak {su}}(n)\) step operators of the regular basis used earlier for the case \(\widehat {\mathfrak {su}}(2)\). Then the authors extend their preceding results on the finite monodromy problem for \(\widehat {\mathfrak {su}}(2)\) to the case \(\widehat {\mathfrak {su}}(n)\). The standard notion of a chiral vertex operator and its zero modes counterpart are applied for studying the braid properties of the physical solutions of the KZ equations.