Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

TL;DR

The paper develops general, categorical criteria for rigidity in braided monoidal categories with commutative algebras, linking rigidity of the ambient category $\mathcal{C}$ to rigidity of module categories $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ via exactness and Grothendieck–Verdier duality. It replaces rigidity prerequisites with exactness criteria for finite tensor categories, proving that $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ are rigid under suitable conditions, and showing that rigidity can descend from $\mathcal{C}_A^{\text{loc}}$ to $\mathcal{C}$ in a GV framework. The results have direct implications for vertex operator algebra extensions, establishing rigidity, fusion, and modularity properties for extension categories and their local representations without requiring nonzero categorical dimensions; these apply to strongly finite VOAs and their subalgebra extensions, including cosets and $W$-algebras. The framework is designed to handle non-semisimple, logarithmic VOA categories and is intended to facilitate rigidity proofs for weight-module categories of affine VOAs (e.g., $L_k(\mathfrak{sl}_2)$ at admissible levels) and related non-rational C2-cofinite theories, with several concrete examples and a pathway to broader Kazhdan–Lusztig-type correspondences. Overall, the work provides scalable, dimension-free criteria for rigidity and an organizing GV-theoretic approach to VOA-extension phenomena.

Abstract

Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ and its subcategory $\mathcal{C}_A^{\text{loc}}$ of local modules inherit rigidity from $\mathcal{C}$, and then we find conditions for $\mathcal{C}$ and $\mathcal{C}_A$ to inherit rigidity from $\mathcal{C}_A^{\text{loc}}$. First, we assume $\mathcal{C}$ is a braided finite tensor category and prove rigidity of $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ under conditions based on criteria of Etingof-Ostrik for $A$ to be an exact algebra in $\mathcal{C}$. As a corollary, we show that if $A$ is a simple $\mathbb{Z}_{\geq 0}$-graded VOA with a strongly rational vertex operator subalgebra $V$, then $A$ is strongly rational, without requiring the categorical dimension of $A$ as a $V$-module to be non-zero. Next, we assume $\mathcal{C}$ is a Grothendieck-Verdier category, i.e., $\mathcal{C}$ admits a weaker duality structure than rigidity. We first prove $\mathcal{C}_A$ is also a Grothendieck-Verdier category. Using this, we prove that if $\mathcal{C}_A^{\text{loc}}$ is rigid, then so is $\mathcal{C}$ under conditions such as a mild non-degeneracy assumption on $\mathcal{C}$, an assumption that every simple object of $\mathcal{C}_A$ is local, and that induction from $\mathcal{C}$ to $\mathcal{C}_A$ commutes with duality. These conditions are motivated by free field-like VOA extensions $V\subseteq A$ where $A$ is often an indecomposable $V$-module, so our result will make it more feasible to prove rigidity for many vertex algebraic monoidal categories. In a follow-up work, our result will be used to prove rigidity of the category of weight modules for the simple affine VOA of $\mathfrak{sl}_2$ at any admissible level.

Theorems & Definitions (125)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3