Monadicity of the band tensor adjunction

Determine for which vertex operator algebras and cyclic categories of modules the adjunction between the band tensor functor and its right adjoint is monadic, and construct non-rigid examples yielding a monadic adjunction.

Background

For a cyclic category of modules over a vertex operator algebra V, the paper considers the band tensor functor L from Rep(V) to its factorization homology over the circle, together with its right adjoint R. Monadicity asks whether the factorization-homology category can be reconstructed as the Eilenberg–Moore category of algebras over the monad RL.

Monadicity is known in the possibly non-finite rigid situation through reconstruction results, but the status for general cyclic module categories, especially non-rigid ones, is unresolved.