Braid Group Representations from Braiding Gapped Boundaries of Dijkgraaf-Witten Theories

2017/07/12 by Nicolás Escobar-Velásquez, Escobar-Velásquez, Nicolás, César Galíndo +3
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Quantum many-body systems #Topological Materials and Phenomena

paper · pdf · doi:10.48550/arxiv.1707.03884

openalex publication_date 2017/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. 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.

Citations

• Fault-tolerant quantum computation by anyons

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