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

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 ${\mathbb{Z}}_{\ge 0}$-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.

59 pages - final version, added references

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