Wilson Line Picture of Levin-Wen Partition Functions

TL;DR

This paper connects Levin-Wen lattice models of doubled Chern-Simons theories to Wilson loop link invariants in 2+1D spacetime, providing a geometric interpretation of their topological properties and chiral sectors.

Contribution

It introduces a geometric Chain-Mail construction linking Levin-Wen partition functions to Wilson loop expectations, clarifying topological invariance and chiral sectors.

Findings

Partition function expressed as Wilson loop expectation

Topological invariance of the model established

Chiral sectors explained through geometric construction

Abstract

Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice Hamiltonian description of doubled Chern-Simons theories. We relate the partition function of these theories to an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as Chain-Mail. This geometric construction gives physical interpretation of the Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining, and how two opposite chirality sectors of the doubled theory arise.