Topological orders and factorization homology

TL;DR

This paper computes the factorization homology of closed surfaces with coefficients from unitary modular tensor categories, linking local topological order data to global surface observables via TQFT, and extends to stratified surfaces with defects.

Contribution

It provides a computation of factorization homology for closed surfaces with modular tensor category coefficients, connecting local and global topological order descriptions, and generalizes to stratified surfaces with defects.

Findings

Factorization homology yields the surface Hilbert space in TQFT.

Global observables are characterized by a pair involving Hilbert spaces.

Extension to stratified surfaces with anomaly-free defects.

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.