Counterexamples in Levin-Wen string-net models, group categories, and Turaev unimodality

TL;DR

This paper examines the limitations of the Levin-Wen string-net models in representing Turaev-Viro topological quantum field theories, highlighting the role of unimodality in group categories and providing counterexamples and classifications.

Contribution

It identifies where the Levin-Wen model fails to capture Turaev-Viro theory due to unimodality issues and offers a classification of group categories based on unimodality.

Findings

The simplest example of a group category fails to be unimodal.

Unimodality can be straightforwardly computed for group categories.

Counterexamples show where the Levin-Wen model needs extra structure.

Abstract

We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may be needed in the Levin-Wen model for this to hold (however we believe that some form of the correspondence is true). In order to be accessible to the condensed matter community we provide a brief and gentle introduction to the relevant concepts in category theory (relying heavily on analogy with ordinary group representation theory). Likewise, some physical ideas are briefly surveyed for the benefit of the more mathematical reader. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.