Algebraic aspects of boundaries in the Kitaev quantum double model

Abstract

We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a *-subalgebra Ξ ⊆ D(G) and we explicate its structure as a quasi-Hopf *-algebra dependent on a choice of transversal R. We provide decomposition formulae for irreducible representations of D(G) pulled back to Ξ. As an application of our treatment, we study patches with boundaries based on K = G horizontally and K = {e} vertically and show how these could be used in a quantum computer using the technique of lattice surgery. More abstractly, we also provide explicitly the monoidal equivalence of the category of Ξ-modules and the category of G-graded K-bimodules and use this to prove that different choices of R are related by Drinfeld cochain twists. Examples include Sn−1 ⊂ Sn and an example related to the octonions where Ξ is also a Hopf quasigroup.