Kitaev lattice models as a Hopf algebra gauge theory

          @misc{ pirsa_PIRSA:17080002,
            doi = {10.48660/17080002},
            url = {https://pirsa.org/17080002},
            author = {Meusburger, Catherine},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Kitaev lattice models as a Hopf algebra gauge theory},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {aug},
            note = {PIRSA:17080002 see, \url{https://pirsa.org}}
          }
          

Abstract

We show that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). As a result, Kitaev models are a special case of combinatorial quantization of Chern-Simons theory by Alekseev, Grosse and Schomerus. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories.