# nLab braid representation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

A braid representation is a linear representation of a braid group.

In topological quantum computation on anyons, braid representations serve as quantum gates.

## References

### Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

Review:

in relation to modular tensor categories:

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:

and understood in terms of anyon statistics:

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

• R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

Introduction and review:

Realization of Fibonacci anyons on quasicrystal-states:

• Marcelo Amaral, David Chester, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 $[$arXiv:2207.08928, doi:10.3390/sym14091780$]$

Realization on supersymmetric spin chains:

• Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains $[$arXiv:2209.03822$]$

Last revised on December 26, 2022 at 14:54:43. See the history of this page for a list of all contributions to it.