Weak braiding and actions of non-invertible symmetries on vertex algebras

Ascertain whether and how the structure of a weak-braided monoidal category D determines or constrains actions of non-invertible symmetries on vertex operator algebras, thereby providing a categorical mechanism to define or classify such actions.

Background

One of the motivations for weak braiding is to capture generalized (potentially non-invertible) symmetries arising in conformal field theory and topological quantum field theory. Vertex operator algebras (VOAs) provide algebraic models for chiral symmetries of rational CFTs, and their module categories often carry braided or related structures.

The question asks whether the additional structure encoded by weak braiding can illuminate or directly translate into a notion of non-invertible symmetry action at the level of VOAs, extending known frameworks for invertible (group-like) symmetries.