Morita triviality beyond the finite rigid case

Determine whether factorization homology of a closed surface with coefficients in Rep(V) is Morita-trivial, or otherwise generalize the Morita triviality of skein algebras, for arbitrary vertex operator algebras and presentable module categories arising from cyclic subcategories.

Background

In the finite rigid case, the factorization homology of a closed surface is equivalent to the category of modules over a skein algebra, and that module category is generated by a single simple object associated with a handlebody. The paper identifies this as Morita triviality of skein algebras.

For an arbitrary vertex operator algebra and its presentable module category Rep(V), the corresponding factorization homology is not understood. The unresolved issue is whether an analogous one-generator or Morita-trivial structure persists without finiteness and rigidity assumptions.

References

For arbitrary V and its presentable module category Rep(V ) based on some cyclic subcategory ofV -modules, no such results are known, and the best course of action would be to understand ∫Σ Rep(V ). Is there, in that case, any generalization of the Morita triviality of skein algebras?

— Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness  (2608.28579 - Müller et al., 28 Aug 2026) in Question 5.15, Section 5.5

Question 6.9. Denote by D ∈ N0 ∪ {∞} the order of the operator d∗ on ΩV (A; −, −). In the finite rigid case from Section 3.3 in which Rep(V ) is the ind completion of a modular category V −mod,D = |θ| ,where θ is the balancing of V −mod. Indeed, by the comparison from Corollary 5.11 we know that D is the order of the balancing θ−1 ⊠ θ of V −mod ⊠ V −mod, which is |θ|. On closed surfaces, the order of the operator associated to a Dehn twist about a non-separating simple closed curve is also |θ| [MW25b, Theorem 4.1]. How can these results be generalized beyond the finite rigid case?

— Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness  (2608.28579 - Müller et al., 28 Aug 2026) in Question 6.9, Section 6.4