Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations

TL;DR

This paper characterizes cyclic algebras over certain operads in symmetric monoidal bicategories, linking them to Grothendieck-Verdier categories, and applies these results to quantum topology by constructing handlebody group representations and dualities.

Contribution

It provides a new characterization of cyclic algebras in bicategories and connects them to Grothendieck-Verdier categories, with applications to quantum topology and mapping class group representations.

Findings

Cyclic associative and framed little 2-disks algebras are equivalent to pivotal and ribbon Grothendieck-Verdier categories.

Constructed handlebody group representations from ribbon Grothendieck-Verdier categories.

Established Grothendieck-Verdier duality for categories from modular functors without semisimplicity assumptions.

Abstract

We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and ribbon Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov.