A Conjecture of Warnaar-Zudilin from Deformations of Lie Superalgebras

TL;DR

The paper addresses Warnaar–Zudilin's conjectured $q$-series identities by embedding the problem in the representation theory of affine Lie superalgebras. It develops a deformation framework that deforms the simple affine VOA $L_k(\mathfrak{osp}_{1|2n})$ into a principal nilpotent subalgebra of $L_k(\mathfrak{sl}_{1|2n+1})$, enabling a comparison of their (super)characters. The main result shows that the supercharacter of $L_k(\mathfrak{osp}_{1|2n})$ matches both the supercharacter of the principal subspace of $L_k(\mathfrak{sl}_{1|2n+1})$ and the (ordinary) character of the principal subspace of $L_k(\mathfrak{sl}_{2n})$, thereby proving the identities and linking them to VOA theory and modularity questions. This work extends Stoyanovsky’s deformation program to the superalgebra setting and provides a rigorous bridge between $q$-series identities and principal subspace characters in the affine-VOA framework, with practical implications for understanding modularity phenomena in these contexts.

Abstract

We prove a collection of $q$-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k(\mathfrak{osp}_{1|2n})$ into the principal subsuperspace of $L_k(\mathfrak{sl}_{1|2n+1})$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers $N$, $k$ the character of the principal subspace of type $A_N$ at level $k$ can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.

Theorems & Definitions (38)

• Theorem 1.0.1: Main result, Corollary \ref{['cor:characters']}
• Theorem 1.0.2: Theorem \ref{['thm:characters']}
• Example 2.1.1
• Lemma 2.1.2
• proof
• Remark 2.1.3
• Lemma 2.1.4
• proof
• Remark 2.1.5
• Lemma 2.2.1