An equivalence of fusion categories
By \textit{G. Moore} and \textit{N. Seiberg} [Classical and conformal field theory, Commun. Math. Phys. 123, 177-254 (1989; Zbl 0694.53074)] it is known that the category \(\widetilde{\mathcal O}_\kappa\) of integrable representations of fixed (integral) positive level for the universal central extension of a loop algebra has the structure of a rigid braided tensor category. From \textit{D. Kazhdan} and \textit{G. Lusztig} [Tensor structures arising from affine Lie algebras, I--IV, J. Am. Math. Soc. 6, 905-947, 949-1011 (1993; Zbl 0786.17017); J. Am. Math. Soc. 7, 335-381 (1994; Zbl 0802.17007); 383-453 (1994; Zbl 0802.17008)] in conjunction with \textit{S. Gelfand} and \textit{D. Kazhdan} [Invent. Math. 109, 595-617 (1992; Zbl 0784.18003)] and \textit{H. Andersen} [Isr. Math. Conf. Proc. 7, 1-12 (1993; Zbl 0845.17018)], it is known how to put a structure of the same type on a suitable subquotient \(\widetilde{\mathcal O}_{-\kappa}\) of the category of highest weight modules of the same Lie algebra in fixed (integral) negative level. In this work, the author proves that these two rigid braided tensor categories are isomorphic. The idea of proof is well explained by the author in the introduction: ``It is known [see \textit{P. Deligne}, Une description de catégorie tressée (inspire par Drinfeld) (unpublished) and \textit{D. Kazhdan} and \textit{G. Lusztig} (loc. cit.)] that the data of a tensor category is equivalent to the data of a) a certain collection of local systems on the moduli spaces of configurations on a projective line; b) a certain collection of isomorphisms between these local systems and their specializations along the boundary of these moduli spaces. The local systems arising from the category \(\widetilde{\mathcal O}_\kappa\) first appeared in the work of \textit{V. Knizhnik} and \textit{A. Zamolodchikov} [Nucl. Phys. B 247, 83-103 (1984; Zbl 0661.17020)]. Their explicit description was given in [\textit{B. Feigin}, \textit{V. Schechtman} and \textit{A. Varchenko}, Commun. Math. Phys. 170, 219-247 (1995; Zbl 0842.17043)]. It turns out that exactly the same description fits the local systems arising from \(\widetilde{\mathcal O}_{-\kappa}\).''
scientific article; zbMATH DE number 912453
scientific article