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.

32 pages, 23 figures, add references, and a lot of minor refinements, comments are welcome

References in corpus (5)

• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• On the Classification of Topological Field Theories
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Tori Detect Invertibility of Topological Field Theories
• Generalized and quasi-localizations of braid group representations

Cited by in corpus (16)

• On the classification of topological orders
• Boundary-bulk relation in topological orders
• Classification of topological phases with finite internal symmetries in all dimensions
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Gapless edges of 2d topological orders and enriched monoidal categories
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Line and surface defects in Reshetikhin-Turaev TQFT
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• One dimensional gapped quantum phases and enriched fusion categories
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• Pointed Drinfeld center functor
• Extended TQFT arising from enriched multi-fusion categories
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• $C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order
• Categorical computation