String-Net Models with $Z_N$ Fusion Algebra

Understand

We study the Levin-Wen string-net model with a $Z_N$ type fusion algebra.

• Solutions of the local constraints of this model correspond to $Z_N$ gauge theory and double Chern-simons theories with quantum groups.
• For the first time, we explicitly construct a spin-$(N-1)/2$ model with $Z_N$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a $Z_N$ type fusion algebra on a honeycomb lattice.
• This exact duality exists only when the spins are coupled to a $Z_N$ gauge field living on the links of the triangular lattice.