Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$

TL;DR

This paper constructs and analyzes the rigid tensor category structure of modules for certain non-semisimple $W$-algebras and their superalgebra counterparts, revealing new examples of ribbon categories at non-integer levels.

Contribution

It establishes the first known ribbon tensor category structures for finitely-generated weight modules of specific non-integer level affine and $W$-algebras, including superalgebras.

Findings

Complete classification of indecomposable projective modules.

Fusion rules for simple modules in these categories.

First examples of ribbon categories at non-integral admissible levels.

Abstract

We establish rigid tensor category structure on finitely-generated weight modules for the subregular $W$-algebras of $\mathfrak{sl}_n$ at levels $ - n + \frac{n}{n+1}$ (the $\mathcal{B}_{n+1}$-algebras of Creutzig-Ridout-Wood) and at levels $- n + \frac{n+1}{n}$ (the finite cyclic orbifolds of the $\beta\gamma$-vertex algebra), as well as for their Feigin-Semikhatov dual principal $W$-superalgebras of $\mathfrak{sl}_{n|1}$. These categories are neither finite nor semisimple, and in the $W$-algebra case they contain modules with infinite-dimensional conformal weight spaces and no lower bound on conformal weights. We give complete lists of indecomposable projective modules in these tensor categories and fusion rules for simple modules. All these vertex operator (super)algebras are simple current extensions of singlet algebras tensored with a rank-one Heisenberg algebra, so we more generally study simple current extensions in direct limit completions of vertex algebraic tensor categories. Then our results for $W$-(super)algebras follow from the known ribbon category structure on modules for the singlet algebras. Our results include and generalize those of Allen-Wood on the $\beta\gamma$-vertex algebra, as well as our own on the affine vertex superalgebra of $\mathfrak{gl}_{1|1}$. Our results also include the first examples of ribbon category structure on all finitely-generated weight modules for an affine vertex algebra at a non-integral admissible level, namely for affine $\mathfrak{sl}_2$ at levels $-\frac{4}{3}$ and $-\frac{1}{2}$.