Contents

group theory

# Contents

## Idea

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).

## References

Original articles:

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

in 4d BF-theory:

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

• Liang Kong, Yin Tian, Zhi-Hao Zhang, Section 2.2 of: Defects in the 3-dimensional toric code model form a braided fusion 2-category, J. High Energ. Phys. 2020, 78 (2020) (arXiv:2009.06564, doi:10.1007/JHEP12(2020)078)

• Zhen Bi, Yi-Zhuang You, Cenke Xu, Anyon and loop braiding statistics in field theories with a topological $\Theta$ term, Phys. Rev. B 90 (2014) 081110(R) (doi:10.1103/PhysRevB.90.081110)

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 October 25, 2022 at 05:04:58. See the history of this page for a list of all contributions to it.