Twisted Quantum Double Model of Topological Orders with Boundaries
  We generalize the twisted quantum double model of topological orders in two
dimensions to the case with boundaries by systematically constructing the
boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group $G$
and a three-cocycle in the third cohomology group of $G$ over $U(1)$, a
boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a two-cochain
in the second cochain group of $K$ over $U(1)$. The consistency between the
bulk and boundary Hamiltonians is dictated by what we call the Frobenius
condition that constrains the two-cochain given the three-cocyle. We offer a
closed-form formula computing the ground state degeneracy of the model on a
cylinder in terms of the input data only, which can be naturally generalized to
surfaces with more boundaries. We also explicitly write down the ground-state
wavefunction of the model on a disk also in terms of the input data only.
