Cyclic Framed Little Disks Algebras, Grothendieck–Verdier Duality And Handlebody Group Representations

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt


  • Fulltext

    Forlagets udgivne version, 1,91 MB, PDF-dokument

  • Lukas Müller
  • Lukas Woike
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.
TidsskriftThe Quarterly Journal of Mathematics
Udgave nummer1
Sider (fra-til)163-245
Antal sider83
StatusUdgivet - 4 apr. 2023

Antal downloads er baseret på statistik fra Google Scholar og

Ingen data tilgængelig

ID: 344726086