Vol. 300, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Braid group representations from braiding gapped boundaries of Dijkgraaf–Witten theories

Nicolás Escobar-Velásquez, César Galindo and Zhenghan Wang

Vol. 300 (2019), No. 1, 1–16

We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf–Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the braid representations associated to Lagrangian algebras are all monomial with respect to some specific bases. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf–Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.

topological quantum computation, modular categories, braid group
Mathematical Subject Classification 2010
Primary: 18D10, 20C35, 20F36
Received: 12 July 2017
Revised: 21 August 2018
Accepted: 17 September 2018
Published: 20 July 2019
Nicolás Escobar-Velásquez
Departamento de Matemáticas,
Universidad de los Andes
César Galindo
Departamento de Matemáticas
Universidad de Los Andes
Zhenghan Wang
Microsoft Research-Station Q and Department of Mathematics
University of California at Santa Barbara
Santa Barbara, CA
United States