Citation: Wen, Xiao-Gang. "A theory of 2+ 1D bosonic topological orders." National Science Review 3.1 (2016): 68-106.
Web: https://arxiv.org/abs/1506.05768
Tags: Expository, Modular-tensor-categories, MTC-reconstruction:
In this lovely expository theory, Xiao-Gang gives an account of the algebraic theory of (2+1)D bosonic topological order in terms of modular categories. It's a good reference for the claim that "it is widely believed that (2+1)D bosonic topological is classified by UMTCs + ccc".
When talking to people about the classification of topological phases, they often point out that Xiao-Gang Wen already has a paper classifying (2+1)D bosonic topological order - this one. The obvious retort to this is that Xiao-Gang does not provide arguments for many of his assertions and that it's is a completely reasonable logical possibility that he could be wrong - there aren't any first-principle arguments for his results.
The obvious counter-retort is “sure, I guess Xiao-Gang could be wrong, but come on he probably isn't let's be reasonable here”. Interestingly, not only could be be wrong, but he is wrong! His stated classification result is not “topological phases = UMTCs + ccc”. His stated classification result is “topological phases = S-matrix + T-matrix + ccc”.
To quote Xiao-Gang: “We have claimed that (S, T, c) can determine all other topological invariants of a topological order” and “We stress that topological order can be defined/probed by measurable quantities (S, T, c)“. He then goes on to give a little list of axioms that (S,T) matrices would need to satisfy to form good modular data, and he wonders if this is a good alternate foundation for topological order which requires less machinery. Of course, this is wrong. There are non-equivalent modular tensor categories with the same (S,T) matrices and thus give non-equivalent topological order with the same (S,T) matrices:
> Mignard, Michaël, and Peter Schauenburg. "Modular categories are not determined by their modular data. arXiv e-prints, page." arXiv preprint arXiv:1708.02796 (2017).
The point is: you can't just take these sorts of conjectures for granted because they can be wrong.