Cyclic Framed Little Disks Algebras, Grothendieck–Verdier Duality And Handlebody Group Representations
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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 ⋆-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.
|The Quarterly Journal of Mathematics
|Number of pages
|Published - 4 Apr 2023