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

TL;DR

The paper proves that the Drinfeld center $Z(C)$ of a pivotal finite tensor category is always a ribbon Grothendieck-Verdier category, with the dualizing object given by the distinguished invertible $\alpha$ in $C$ via the Radford isomorphism $-^{\vee\vee\vee\vee} \cong \alpha \otimes - \otimes \alpha^{-1}$. This GV-ribbon structure is unique up to equivalence among extensions of the canonical balanced braided structure; sphericality of $C$ exactly corresponds to the GV-duality agreeing with rigid duality. The authors show that $Z(C)$ yields an ansular (hence modular) functor, providing mapping class group representations independent of sphericality, and derive a seven-term exact sequence describing all GV-ribbon structures in terms of the balanced Müger center $Z_2^{bal}(A)$. They further illustrate the framework with explicit examples and connect the algebraic data to topological invariants, offering a robust bridge between categorical structures and low-dimensional topology.

Abstract

We prove that the Drinfeld center $Z(\mathcal{C})$ of a pivotal finite tensor category $\mathcal{C}$ comes with the structure of a ribbon Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. Phrased operadically, this makes $Z(\mathcal{C})$ into a cyclic algebra over the framed $E_2$-operad. The underlying object of the dualizing object is the distinguished invertible object of $\mathcal{C}$ appearing in the well-known Radford isomorphism of Etingof-Nikshych-Ostrik. Up to equivalence, this is the unique ribbon Grothendieck-Verdier structure on $Z(\mathcal{C})$ extending the canonical balanced braided structure that $Z(\mathcal{C})$ already comes equipped with. The duality functor of this ribbon Grothendieck-Verdier structure coincides with the rigid duality if and only if $\mathcal{C}$ is spherical in the sense of Douglas-Schommer-Pries-Snyder. The main topological consequence of our algebraic result is that $Z(\mathcal{C})$ gives rise to an ansular functor, in fact even a modular functor regardless of whether $\mathcal{C}$ 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.

Theorems & Definitions (41)

• Lemma 2.1
• proof
• Example 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• Remark 2.6
• proof : Proof of Proposition \ref{['proppivotal']}