# Definition of Topological Order in terms of categories

I have a question regarding the definition of topological order as defined in Wen's review article http://www.hindawi.com/journals/isrn/2013/198710/.

Is the distinction between boundary-gapped topological orders in 2+1 dimensions in terms of LU transformations or in terms of spherical fusion categories equivalent?

If so, is there a similar correspondence between categories for the general case?

## 2 Answers

The definition in terms of LU transformations is more fundamental. In the general case, we believe that topological orders are described by modular tensor categories (MTC), and the equivalence of topological orders translates into the equivalence of MTCs. In the special case of boundary-gapped topological orders, they are all realized by quantum double (also known as Drinfeld center, if you are more mathematically complicated) of unitary fusion categories. So the equivalence on the fusion categories is just that they need to give the same quantum double as modular tensor categories. This is called the Morita equivalence for fusion categories.

• That answers my questions, thank you very much! One more question related to this: Is there also a formalism in terms of categories that includes symmetries, hence an equivalence to symmetric LU transformations? – Soliton Jun 5 '15 at 16:19
• Yes, of course there is. It is the subject of symmetry-protected phases and symmetry-enriched topological phases. – Meng Cheng Jun 6 '15 at 18:22

My paper with Liang Kong, arXiv:1405.5858, presents the following conjecture: Bosonic topological orders in $n$-space-time dimensions (after quotient out the invertible topological orders) are described/classified by modular unitary $n$-categories with one object.

The reason that modular unitary $n$-categories fail to classify topological orders is because modular unitary $n$-categories with one object describe/classify topological excitations. The invertible topological orders has no non-trivial topological excitations. Thus all invertible topological orders correspond to the same trivial unitary $n$-category with one object. But after we quotient out the invertible topological orders, modular unitary $n$-categories with one object classify bosonic topological orders in $n$-space-time dimensions.

• Perhaps you could expand this out a little more? As it stands, it is perilously close to a link-only answer. – Jon Custer Aug 5 '15 at 23:14