On the Representation Theory of Non-Admissible $W$-Algebras: Part I

Dan Xie

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arXiv · 2606.16714

On the Representation Theory of Non-Admissible $W$-Algebras: Part I

Dan Xie

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Abstract

Motivated by the mirror symmetry for circle compactified 4d $N=2$ theories, we propose a geometric framework for studying the representation theory of non-admissible $W$-algebras $W^{k}(\mathfrak g,f)$ at levels $k=-h^\vee+\frac{1}{n}\frac{m}{u}$, using the geometry of generalized affine Springer fibers $Sp_{\nu}(\tilde{g},f)$ with slope $\nu=u/m$. The central proposal is that each non-empty $C^*$-fixed locus, labeled by a double coset $\tilde w\in W_\nu\backslash\tilde{W}/W_f$, gives rise to simple modules whose highest weight is determined by the map $\tilde w\mapsto\tilde w(k\Lambda_0+\tilde\rho)-\tilde\rho$, while the dimension of the fixed variety encodes additional non-semisimple structure (logarithmic modules). We verify this correspondence in numerous examples, including $D_4$, $E_6$, $E_8$, and twisted theories of type $^3D_4$ and $^2A_3$, where our geometric counting reproduces known results for both admissible and non-admissible $W$-algebras.

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Dan Xie. 2026-06-15. On the Representation Theory of Non-Admissible $W$-Algebras: Part I. https://arxiv.org/abs/2606.16714

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