Monadic reconstruction of unitary Drinfeld centers and Factorization Homology
Lucas Hataishi
TL;DR
The paper addresses how the unitary Drinfeld center of a unitary tensor category can be realized monadically as the category of unitary bimodules over a canonical W*-algebra object, enabling a C*-algebraic formulation of factorization homology. It develops a monadic reconstruction of the center via centralizer endofunctors and free half-braidings, and identifies the center with bimodules for the canonical algebra object $\mathcal{S}$, thereby connecting to Yetter-Drinfeld and internal algebra objects. Factorization homology is then formulated with coefficients in $\mathcal{Z}\mathrm{Hilb}(\mathcal{C})$ and realized through symmetric enveloping algebras, leading to extensions $(A^{\mathrm{op}}\otimes A) \underset{F}{\rtimes} \mathcal{Y}_\Sigma$ and actions of modular groups on these algebras. In the complex quantum group case, specifically $\mathcal{C}=\mathrm{Rep}_{fd}(K_q)$, the center corresponds to representations of the Drinfeld double $D K_q$, and one obtains a concrete description of factorization homology as a category of equivariant Hilbert modules over a $D K_q$-C*-algebra, providing a pathway to quantum gauge-theoretic observables in 2D TQFTs.
Abstract
We prove that the unitary Drinfeld center of a unitary tensor category is equivalente to the category of unitary bimodules for the canonical W*-algebra object, generalizing Müger's result to the non-fusion case. This is then used to express factorization homology in terms of C*-algebraic extensions of symmetric enveloping algebras and actions of Drinfeld dobules of compact quantum groups.