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On the Classification of Topological Orders

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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 results by L. Kong, X.G. Wen, and their collaborators.

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  1. Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Theo Johnson-Freyd

  2. Department of Mathematics, Dalhousie University, Halifax, Canada

    Theo Johnson-Freyd

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I thank D. Freed, D. Gaiotto, L. Kong, D. Reutter, and M. Yu, and in particular the anonymous referee, for discussion and comments. I thank C. Scheimbauer for Remark II.7, and D. Nykshych for the end of Remark V.2. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.

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Johnson-Freyd, T. On the Classification of Topological Orders. Commun. Math. Phys. 393, 989–1033 (2022). https://doi.org/10.1007/s00220-022-04380-3

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