Codimension-2 defects and higher symmetries in (3+1)D topological phases

Aug 15, 2022
70 pages
Published in:
  • SciPost Phys. 14 (2023) 4, 065,
  • SciPost Phys. 14 (2023) 065
  • Published: Apr 11, 2023
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Abstract: (SciPost Fundation)
(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 Z2\mathbb{Z}_2 gauge theory with fermionic charges, in Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral (DnD_n) and alternating (A6A_6) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H4H^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 A6A_6 gauge theory.
Note:
  • 70 pages, 42 figures
  • string: twist
  • phase: topological
  • defect: topological
  • string: interaction
  • fermion: charge
  • defect: condensation
  • gauge field theory: discrete
  • flux: nonabelian
  • symmetry: topological
  • abelian