Kitaev's quantum double model, bulk and boundary, in graphical calculus
Lucy Liuxuan Zhang
Kitaev's quantum double (QD) model is a Hamiltonian model defined on a ribbon graph of qudits lying in a physical surface. The famous toric code is a special case of the QD models. We study QD models parameterized by finite-dimensional semisimple Hopf algebras, which determine the Hilbert space and the Hamiltonian. Throughout the talk, we make reference to the parallel study of TQFTs. For a QD model parameterized by Hopf algebra H, both the Turaev-Viro TQFT based on the spherical fusion category Rep(H) and the Reshetikhin-Turaev TQFT based on the modular category Z(Rep(H))=Rep(D(H)) play a major role.
In this talk, we re-express this relatively general version of Kitaev's quantum double model by writing the Hamiltonian operator in terms of the PROP of finite-dimensional semisimple Hopf algebras. In doing so, we obtain a tensor diagram living in the 3-manifold of the physical surface x [0,1], which we compare with the tensor diagram construction used for the Kuperberg invariant for closed 3-manifolds with involutory Hopf algebra input (basically equivalent to the Turaev-Viro TQFT based on Rep(H)).
If time permits, we present work-in-progress on the boundary of the QD model. The study of boundaries and domain walls is more or less the study of defect TQFTs. Some prior analyses on this topic exist, but our work differs in its level of generality (finite-dimensional semisimple Hopf algebras instead of group-theoretic) and in terms of concreteness (working with algebras rather than categories).
This is joint work with Tobias Fritz.