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

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