On the Classification of Topological Orders

By Theo Johnson-Freyd. Communications in Mathematical Physics, 2020

147 citations recorded by the source corpus.

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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n{-}1)$$\end{document}-categories. We then discuss the classification in low spacetime dimension, proving in particular that all (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1{+}1)$$\end{document}- and (3+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3{+}1)$$\end{document}-dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and results by L. Kong, X.G. Wen, and their collaborators.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0e8f89cf67e483980e6347955bc9b05877c18c25

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

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Citations in the corpus

Orbifold groupoids (2020, background): “In the language of [59] (see also [60]), a massive Majorana fermion with m > 0 is isomorphic to the massive Majorana fermion with m < 0 as “anomalous field theories.”