Level-rank duality of WZW models in conformal field theory
The decomposition of the conformal blocks under the conformal embedding is considered. In addition to the known mutually commutative embedding of affine Lie algebras \(\widehat {sl}(l)_ r\times \widehat{sl}(r)_ l\subset\widehat{gl}(lr)_ 1\) (Frenkel, 1982), the case of \(\widehat {sl}(l)_ r\times\widehat{sl}(r)_ l\times \hat a\subset gl(lr)_ 1\) where \(\hat a\) is an affine extension of the abelian subalgebra of central elements of \(gl(lr)\), is studied in detail. It is shown that a pairing between the spaces of conformal blocks of \(\widehat{sl}(l)_ r\) and \(\widehat{sl}(r)_ l\) Wess-Zumino-Witten models on the Riemann sphere (Witten, 1984; Knizhnik-Zamolodchikov, 1984) which is induced by the reciprocal decompositions of \(\widehat{gl}(lr)_ 1\)-modules and satisfies the completeness condition, defines the level-rank duality [\textit{S. G. Naculich} and \textit{H. J. Schnitzer}, Phys. Lett. 244, No. 2, 235-240 (1990); Nucl. Phys. B 347, No. 3, 687-742 (1990)]\ between two spaces in conformal field theory under consideration.
scientific article; zbMATH DE number 54750
scientific article