nLab loop braid group



Group Theory


topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



Where the braid group is the group of continuous “motions” of sets of distinct points in the Euclidean plane, the loop braid group (Dahm 62, Goldsmith 81) is the group of continuous motions of disjoint unknotted loops in 3d Euclidean space.

Accordingly, the role that the braid group plays in 3d TQFTs such as Chern-Simons theory/Reshetikhin-Turaev theory with point-like defects (anyons), the loop braid group plays in 4d TQFT with line defects (codimension-2 defect branes).

graphics from Baez, Wise, Crans 06


Original articles on motion groups:

  • D. Dahm, A generalization of braid theory, PhD thesis, Princeton University, 1962.

  • Deborah L. Goldsmith, The theory of motion groups, Michigan Math. J. 28(1): 3-17 (1981) (doi:10.1307/mmj/1029002454)

Specifically on the loop braid group:

See also:

Discussion via 4d TQFT and higher gauge theory/higher parallel transport:

in 4d BF-theory:

and in the 3d toric code-4d TQFT:

and in 3d symmetry protected topological phases:

  • Chao-Ming Jian, Xiao-Liang Qi, Layer Construction of 3D Topological States and String Braiding Statistics, Phys. Rev. X 4 (2014) 041043 [[doi:10.1103/PhysRevX.4.041043]]

  • Shenghan Jiang, Andrej Mesaros, and Ying Ran, Generalized Modular Transformations in (3+1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding, Phys. Rev. X 4 031048 (doi:10.1103/PhysRevX.4.031048)

and similar models in condensed matter theory:

Relation to braided tensor categories:

Relation to integrable systems:

  • Pramod Padmanabhan, Abhishek Chowdhury, Loop braid groups and integrable models [arXiv:2210.12932]

Last revised on December 13, 2023 at 08:55:25. See the history of this page for a list of all contributions to it.