Drinfel’d double symmetry of the 4d Kitaev model

Abstract

Following the general theory of categorified quantum groups developed by the author previously, we construct the Drinfel’d double 2-bialgebra associated to a finite group N = G0. For N = ℤ2, we explicitly compute the braided 2-categories of 2-representations of certain version of this Drinfel’d double 2-bialgebra, and prove that they characterize precisely the 4d toric code and its spin-ℤ2 variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and displays an instance of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying Drinfel’d double 2-bialgebra is responsible for much of the higher-structural properties that arise in 4d topological orders.