Citation: Lin, Chien-Hung, Michael Levin, and Fiona J. Burnell. "Generalized string-net models: A thorough exposition." Physical Review B 103.19 (2021): 195155.
Web: https://arxiv.org/abs/2012.14424
Tags: Kitaev-quantum-double
This paper gives a thorough exposition of generalized string-net models, which nowadays typically just go under the name "sting-net models". The idea is as follows. The Drinfeld center gives a mapping of unitary fusion categories to UMTCs. The Levin-Wen model is supposed to take in UFCs and spit out a Hamiltonian describing a topological order described by the center. This is not quite right, however. The original Levin-Wen paper only deals with states that have tetrahedral symmetry. This means that given a ground state, replacing any coloring of spins on tetrahedron-like subgraph by a rotated version of that coloring should give the same amplitude. This is NOT generically the case for a UFC. Algebraically, this is the condition that the F symbol of six indices is equal to the complex conjugate of the F symbol with all dual indices. I.e., acting by particle/antiparticle duality simultaneously on all indicies acts on F symbols by complex conjugation. This is not always true.
The generalized string-net model applies to all UFCs, regardless of whether or not they have tetrahedral symmetry. This changes the model slightly, but not all that much. This paper gives a nice discussion of exactly what you need to know about moving to this generalized setting.