Cyclic Framed Little Disks Algebras, Grothendieck–Verdier Duality And Handlebody Group Representations. (19th July 2022)

• Record Type: Journal Article
• Authors: Müller, Lukas; Woike, Lukas
• Is Part Of: Quarterly journal of mathematics. Volume 74:Part 1(2023)
• Page Start: 163
• Page End: 245
• Publication Date: 2022-07-19
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/qmath/haac015
• Languages: English
• ISSNs: 0033-5606

Abstract

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, that is 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.