Rationality and classification of weak modules at boundary admissible levels

Establish, for every basic classical Lie superalgebra, that the simple affine vertex operator superalgebra at a boundary admissible level is rational in the category of affine Lie-superalgebra modules in category \(\mathcal{O}\), and prove that its irreducible weak modules in category \(\mathcal{O}\) are exactly the admissible modules of the corresponding level.

Background

The paper recalls a conjecture formulated in earlier work concerning the representation theory of simple affine vertex operator superalgebras associated with basic classical Lie superalgebras at boundary admissible levels. The conjecture has two parts: rationality in category O\mathcal{O}, and an identification of all irreducible weak modules in that category with admissible affine modules.

The present paper proves finiteness, semisimplicity, and tensor-categorical properties for the category of ordinary modules, under the stated exclusions, but the quoted conjecture concerns the broader class of irreducible weak modules in category O\mathcal{O}. The paper does not present the quoted conjecture as fully resolved in its generality.

References

More specifically, we proved that $L_{\widehat{sl(2|1)}(-\frac{1}{2},0)}$ is rational in the category $\mathcal{O}$ at the boundary admissible level $-\frac{1}{2}$, while at the non-boundary admissible level $\frac{1}{2}$ we classified its infinitely many irreducible weak modules in the category $\mathcal{O}$, and we gave a conjecture that for any basic classical Lie superalgebra $\mathfrak{g}$, $L_{\widehat{\mathfrak{g}}(k,0)}$ is rational in the category $\mathcal{O}$ at boundary admissible level $k$ and the irreducible weak modules in the category $\mathcal{O}$ are exactly the admissible modules of level $k$ for $\widehat{\mathfrak{g}}$.

— Ordinary modules for affine vertex operator superalgebras  (2610.01774 - Li et al., 1 Oct 2026) in Section 1, Introduction