Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture
The author studies the representation theory of the \(\mathcal{W}\) algebra \(\mathcal{W}_k(\mathfrak{g},f)\) where \(\mathfrak{g}\) is a simple finite-dimensional Lie superalgebra, \(f\) is its principal nilpotent element and \(k\in \mathbb{C}\) is a level of the Kac-Moody affinization \(\hat{\mathfrak{g}}\) of \(\mathfrak{g}\). The structure theory of \(\mathcal{W}_k(\mathfrak{g},f)\) was developed by \textit{V. G. Kac, S.-S. Roan} and \textit{M. Wakimoto} in [Commun. Math. Phys. 241, 307--342 (2003; Zbl 1106.17026)]. They constructed a functor \(H\) from the category of restricted \(\hat{\mathfrak{g}}\)-modules of level \(k\) to the category of positive energy modules over \(\mathcal{W}_k(\mathfrak{g},f)\) and conjectured that if \(L(\Lambda)\) is an admissible \(\hat{\mathfrak{g}}\)-module then \(\mathcal{W}_k(\mathfrak{g},f)\)-module \(H(L(\Lambda))\) is either zero or irreducible. In the paper under review the author proves the Kac-Roan-Wakimoto conjecture. Moreover, he proves that the character of any irreducible highest weight \(\mathcal{W}_k(\mathfrak{g},f)\)-module is determined by the character of the corresponding irreducible highest weight \(\hat{\mathfrak{g}}\)-module of level \(k\).
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