On the classification of topological orders

Abstract

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion $n$-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such $n$-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion $n$-categories (i.e. multifusion $n$-categories with indecomposable unit) with centreless braided fusion $(n{-}1)$-categories. We then discuss the classification in low spacetime dimension, proving in particular that all $(1{+}1)$- and $(3{+}1)$-dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and proposals by X.G. Wen et al.