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Relative anomalies in (2+1)D symmetry enriched topological states

TitleRelative anomalies in (2+1)D symmetry enriched topological states
Publication TypeJournal Article
Year of Publication2020
AuthorsM. Barkeshli, and M. Cheng
JournalSciPost Phys.
Date Publishedfeb

Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for Z(2)(T) space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for U(1) x Z(2)(T) and U (1) (sic) Z(2)(T) symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as Z(4)(T) and mixed anomalies for Z(2) x Z(2)(T) symmetry, and unitary Z(2) x Z(2) symmetry with non-trivial anyon permutations.