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

DOI: 10.2140/pjm.2019.300.1

Abstract

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.

Keywords

topological quantum computation, modular categories, braid group

Mathematical Subject Classification 2010

Primary: 18D10, 20C35, 20F36

Milestones

Received: 12 July 2017

Revised: 21 August 2018

Accepted: 17 September 2018

Published: 20 July 2019