Admissible Skein Modules and Ansular Functors: A Comparison

TL;DR

This paper demonstrates an isomorphism between admissible skein modules and ansular functors derived from finite ribbon categories, revealing their equivalence up to a boundary component, with implications for handlebody group actions.

Contribution

It establishes a formal equivalence between two different constructions of handlebody invariants, connecting skein modules and modular functors through a universal property approach.

Findings

Proves the isomorphism between skein modules and ansular functors.

Shows the constructions differ by a boundary component labeled by a distinguished invertible object.

Includes the handlebody group action and skein algebra action in the comparison.

Abstract

Given a finite ribbon category, which is a particular case of a cyclic algebra over the operad of genus zero surfaces, there are two possibilities for an extension defined on all three-dimensional handlebodies: On the one hand, one can use the admissible skein module construction of Costantino-Geer-Patureau-Mirand. On other hand, by a construction of the authors using Costello's modular envelope, one can build a so-called ansular functor, a handlebody version of the notion of a modular functor. Unlike the admissible skein modules with their construction through the Reshetikhin-Turaev graphical calculus, the ansular functor is defined purely through a universal property. In this note, we prove the widely held expectation that these constructions are related by giving an isomorphism between them, with the somewhat surprising subtlety that we need to include consistently on one of the sides an additional boundary component labeled by the distinguished invertible object of Etingof-Nikshych-Ostrik. In other words, the constructions agree on handlebodies up to a `background charge' that becomes trivial in the unimodular case. Our comparison result includes the handlebody group action as well as the skein algebra action.