Relative Anomalies in (2+1)D Symmetry Enriched Topological States

Abstract

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 ${\mathbb{Z}}_2^{\mathbf{T}}$ space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for $U(1)\times {\mathbb{Z}}_2^{\mathbf{T}}$ and $U(1)⋊{\mathbb{Z}}_2^{\mathbf{T}}$ 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 ${\mathbb{Z}}_4^{\mathbf{T}}$ and mixed anomalies for ${\mathbb{Z}}_2\times {\mathbb{Z}}_2^{\mathbf{T}}$ symmetry, and unitary ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry with non-trivial anyon permutations.

arXiv comment34 pages. v2: minor corrections

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