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Certain patterns of symmetry fractionalization in topologicallyordered phases of matter are anomalous, in the sense that they can onlyoccur at the surface of a higher dimensional symmetry-protectedtopological (SPT) state. An important question is to determine how tocompute this anomaly, which means determining which SPT hosts a givensymmetry-enriched topological order at its surface. While special casesare known, a general method to compute the anomaly has so far beenlacking. In this paper we propose a general method to compute relativeanomalies between different symmetry fractionalization classes of agiven (2+1)D topological order. This method applies to all types ofsymmetry actions, including anyon-permuting symmetries and generalspace-time reflection symmetries. We demonstrate compatibility of therelative anomaly formula with previous results for diagnosing anomaliesfor \mathbb{Z}_2^{T} ℤ 2 T space-time reflection symmetry (e.g. where time-reversal squares to theidentity) and mixed anomalies for U(1) \times \mathbb{Z}_2^{T} U ( 1 ) × ℤ 2 T and U(1) \rtimes \mathbb{Z}_2^{T} U ( 1 ) ⋊ ℤ 2 T symmetries. We also study a number of additional examples, includingcases where space-time reflection symmetries are intertwined innon-trivial ways with unitary symmetries, such as \mathbb{Z}_4^{T} ℤ 4 T and mixed anomalies for \mathbb{Z}_2 \times \mathbb{Z}_2^{T} ℤ 2 × ℤ 2 T symmetry, and unitary \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 symmetry with non-trivial anyon permutations.
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Codimension-2 defects and higher symmetries in (3+1)D topological phases
(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory.
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- Award ID(s):
- 1753240
- PAR ID:
- 10438924
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 14
- Issue:
- 4
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.14.4.065
