# Twisted Quantum Double Model of Topological Orders with Boundaries

## TL;DR

This paper extends the twisted quantum double model of topological orders to include boundaries, providing explicit boundary Hamiltonians, ground state formulas, and ground state degeneracy calculations based on algebraic data.

## Contribution

It introduces a systematic construction of boundary Hamiltonians for twisted quantum double models and derives formulas for ground states and degeneracies.

## Key findings

- Boundary Hamiltonians are explicitly constructed using subgroup and cochain data.
- Ground state degeneracy on a cylinder is computed from input algebraic data.
- Ground-state wavefunctions on a disk are explicitly formulated.

## Abstract

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.