Topological orders and factorization homology

Abstract

In the study of 2d (the space dimension) topological orders, it is well-known that bulk excitations are classified by unitary modular tensor categories. But these categories only describe the local observables on an open 2-disk in the long wave length limit. For example, the notion of braiding only makes sense locally. It is natural to ask how to obtain global observables on a closed surface. The answer is provided by the theory of factorization homology. We compute the factorization homology of a closed surface $\Sigma$ with the coefficient given by a unitary modular tensor category, and show that the result is given by a pair $(\mathbf{H}, u_\Sigma)$, where $\mathbf{H}$ is the category of finite-dimensional Hilbert spaces and $u_\Sigma\in \mathbf{H}$ is a distinguished object that coincides precisely with the Hilbert space assigned to the surface $\Sigma$ in Reshetikhin-Turaev TQFT. We also generalize this result to a closed stratified surface decorated by anomaly-free topological defects of codimension 0,1,2. This amounts to compute the factorization homology of a stratified surface with a coefficient system satisfying an anomaly-free condition.