The distinguished invertible object as ribbon dualizing object in the Drinfeld center.

We prove that the Drinfeld center Z (C) of a pivotal finite tensor category C comes with the structure of a ribbon Grothendieck–Verdier category in the sense of Boyarchenko–Drinfeld. Phrased operadically, this makes Z (C) into a cyclic algebra over the framed E 2 -operad. The underlying object of the dualizing object is the distinguished invertible object of C appearing in the well-known Radford isomorphism of Etingof–Nikshych–Ostrik. Up to equivalence, this is the unique ribbon Grothendieck–Verdier structure on Z (C) extending the canonical balanced braided structure that Z (C) already comes equipped with. The duality functor of this ribbon Grothendieck–Verdier structure coincides with the rigid duality if and only if C is spherical in the sense of Douglas–Schommer-Pries–Snyder. The main topological consequence of our algebraic result is that Z (C) gives rise to an ansular functor, in fact even a modular functor regardless of whether C is spherical or not. In order to prove the aforementioned uniqueness statement for the ribbon Grothendieck–Verdier structure, we derive a seven-term exact sequence characterizing the space of ribbon Grothendieck–Verdier structures on a balanced braided category. This sequence features the Picard group of the balanced version of the Müger center of the balanced braided category. [ABSTRACT FROM AUTHOR]