Relaxed highest-weight modules I: rank $1$ cases

TL;DR

This paper derives character formulas and fully describes the structure of relaxed highest-weight modules over certain affine vertex operator superalgebras, confirming several conjectures and providing new results at specific levels.

Contribution

It provides explicit character formulas and structural descriptions for relaxed highest-weight modules over admissible-level affine superalgebras, including new results for $ ext{osp}(1|2)$.

Findings

Proved character formulas for modules over $ ext{sl}_2$ and $ ext{osp}(1|2)$.

Confirmed several conjectures in the literature.

Established module structures at various admissible levels.

Abstract

Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$. Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for $\mathfrak{sl}_2$, at arbitrary admissible levels, and for $\mathfrak{osp}(1|2)$ at level $-\frac{5}{4}$. For other admissible levels, the $\mathfrak{osp}(1|2)$ results are believed to be new.