Levin-Wen is a Gauge Theory: Entanglement from Topology

Abstract

We show that the Levin-Wen model of a unitary fusion category \({\mathcal {C}}\) is a gauge theory with gauge symmetry given by the tube algebra \({\text {Tube}}({\mathcal {C}})\). In particular, we define a model corresponding to a \({\text {Tube}}({\mathcal {C}})\) symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case \({\mathcal {C}}=\textsf{Hilb}(G,\omega )\), we show how our procedure reduces to the twisted gauging of a trival G-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.