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

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 $S{p}_{\nu }(\tilde{g},f)$ with slope $\nu =u/m$. The central proposal is that each non-empty $C^{\ast }$-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 ${}^3{D}_4$ and ${}^2{A}_3$, where our geometric counting reproduces known results for both admissible and non-admissible $W$-algebras.

51 pages, 24 tables