Nondegeneracy of the annular pairing beyond finite rigidity

Fix a vertex operator algebra $V$ and a cyclic category $\mathcal{T}$ of its graded modules in the sense of Müller–Woike, §2.2, Definition 2.1: $\mathcal{T}$ is a full subcategory equipped with their nondegenerate pairing and coevaluation, associative unital promonoidal structure, braiding, balancing, and cyclicity isomorphism. Set $\mathrm{Rep}(V)=\mathrm{Lin}({\mathcal{T}}^{\mathrm{o}\mathrm{p}},\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t})$, the source cocompletion of $\mathcal{T}$. Write ${\int }_{S^1}\mathrm{Rep}(V)$ for the source's shorthand for factorization homology of the framed $E_2$-category $\mathrm{Rep}(V)$ on the annular cylinder $S^1\times [0,1]$, defined as the homotopy colimit over oriented embeddings of finite disjoint unions of $2$-disks into that cylinder; the source identifies it with categorical Hochschild homology. Theorem 5.4 constructs an open-closed modular functor with singularities from this data. Evaluating it on an annulus with two parametrized boundary circles gives the symmetric pairing $\beta :({\int }_{S^1}\mathrm{Rep}(V))⊠({\int }_{S^1}\mathrm{Rep}(V))⟶\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{.}$

Put $\mathcal{A}={\int }_{S^1}\mathrm{Rep}(V)$. The ambient symmetric monoidal bicategory is ${\mathrm{P}\mathrm{r}}_{\mathbb{C}}^L$ of locally presentable complex-linear categories, complex-linear functors preserving all small colimits, and natural transformations, as in the source's §2.1. Here locally presentable means cocomplete and generated under $\lambda$-filtered colimits by a set of $\lambda$-presentable objects for some regular cardinal $\lambda$; an object is $\lambda$-presentable when its hom functor preserves those colimits. The tensor product $⊠$ is the Deligne–Kelly product, characterized by functors out of it preserving colimits corresponding to bilinear functors preserving colimits separately; its unit $\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}$ is the category of all complex vector spaces. The pairing $\beta$ is nondegenerate precisely when there exists a colimit-preserving complex-linear coevaluation $\delta :\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\to \mathcal{A}⊠\mathcal{A}$ with natural isomorphisms $({\mathrm{i}\mathrm{d}}_{\mathcal{A}}⊠\beta )\circ (\delta ⊠{\mathrm{i}\mathrm{d}}_{\mathcal{A}})\cong {\mathrm{i}\mathrm{d}}_{\mathcal{A}},(\beta ⊠{\mathrm{i}\mathrm{d}}_{\mathcal{A}})\circ ({\mathrm{i}\mathrm{d}}_{\mathcal{A}}⊠\delta )\cong {\mathrm{i}\mathrm{d}}_{\mathcal{A}}\mathrm{.}$ Use the canonical tensor-unit and associativity identifications in these composites. Thus $\beta$ must exhibit $\mathcal{A}$ as its own dual; the existence of an unrelated copairing is insufficient.

In the source's finite-rigid case, $V\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is the category of strongly graded $V$-modules and is a finite tensor category: a finite $\mathbb{C}$-linear abelian category with finite-dimensional morphism spaces, finitely many simple isomorphism classes, enough projectives, finite length, a rigid monoidal product, and simple unit. The chosen rigid dual agrees with the VOA contragredient operation $X\mapsto X^{\mathrm{\prime }}$. Is $\beta$ nondegenerate when these finite-rigid hypotheses do not hold?

As of 2026-10-26, Müller–Woike, Remark 5.8, state that the annular pairing is nondegenerate in the finite-rigid case and that its nondegeneracy beyond that case is not known. Their Theorem 5.4 constructs the modular functor with singularities under its stated cyclic-category hypothesis. The construction supplies a symmetric copairing ${\mathrm{\Delta }}^{\circ }$ for the modular functor, but Remark 5.8 cautions that it is generally not a copairing for $\beta$, even when $\beta$ is nondegenerate; its existence therefore does not settle the question. No answer is claimed here, and this item is not a premise of another result.