Hopf algebra gauge theory from lattice models

Kitaev lattice models play an important role in topological quantum computing and are closely related to 3d TQFTs of Reshetikhin-Turaev and of Turaev-Viro type. The Kitaev model for a finite-dimensional semisimple Hopf algebra H describes the Hilbert space a 3d Turaev-Viro TQFT assigns to an oriented surface, and the corresponding model with excitations is related to TQFTs of Reshetikhin-Turaev type for the Drinfeld double D(H). However, this relation is rather implicit, and many questions remain open.

In this talk, we show how a Kitaev lattice model for a finite-dimensional semisimple Hopf algebra H gives rise to a Hopf algebra lattice gauge theory for the Drinfeld double D(H). This explicitly relates Kitaev models to a known Hamiltonian quantisation formalism for Chern-Simons gauge theory, namely combinatorial quantisation and can be viewed as an algebraic counterpart of the correspondence between Turaev-Viro TQFTs for H and Reshetikhin-Turaev TQFTs for D(H). It also allows one to formulate a semiclassical limit of these models in terms of Poisson-Lie structures and to relate them to the symplectic structures on moduli spaces of flat connections on the underlying surface.