Categorified Quantum Groups, Braided Monoidal 2-Categories and the 4d Kitaev Model

Abstract

It is well-known since the work of Hopf, Drinfel’d, Majid, Witten, and etc. in the late 20th century that Hopf algebra quantum groups play a signification role in both physics and mathematics. In particular, the category of representations of quantum groups are braided, and hence captures invariants of knots. This talk is based on works with F. Girelli, where we develop a systematic categorification of the theory of quantum groups/bialgebras, including monoidal weakenings, and prove that their 2-representations form a braided monoidal 2-category. We will then apply our general theory to describe the 4D toric code and its spin variant, uniting the 2-categorical and 2-group gauge theory frameworks.