Paravortices: loop braid representations with both generators involutive

2025/12/19 by Paul Martin, Martin, Paul P., Torzewska, Fiona +1
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2512.17830

openalex publication_date 2025/12/19 · openalex created_date 2025/12/23 · openalex updated_date 2026/07/28

Abstract

We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory. For reasons that will become clear, we call this quotient the mixed doubles category, MD. Then our main result is a theorem classifying all mixed doubles representations in rank-2. Each representation yields a mixed doubles group representation for every loop braid group LB_n, and we are able to analyse the unified linear representation theory of many of these sequences of representations, using a mixture of very classical, classical, and new techniques. In particular this is a motivating example for the `glue' generalisation of charge-conserving representation theory (a form of rigid higher non-semisimplicity) introduced recently.

Citations

• A Categorical Perspective on Braid Representations
• Classification of charge-conserving loop braid representations
• Classification of spin-chain braid representations
• On a canonical lift of Artin's representation to loop braid groups
• Mathematics of Topological Quantum Computing
• A journey through loop braid groups
• Unrestricted virtual braids, fused links and other quotients of virtual braid groups
• Local representations of the loop braid group
• Multipermutation solutions of the Yang--Baxter equation
• Non-Abelian anyons and topological quantum computation
• Extending representations of braid groups to the automorphism groups of free groups
• The virtual and universal braids
• Topological Quantum Computation
• AN ALGEBRAIC APPROACH TO VERTEX MODELS AND TRANSFER MATRIX SPECTRA

Related

• The ribbon category framework for topological quantum computing
• Loop braid groups and integrable models
• Local representations of the loop braid group
• Classification of spin-chain braid representations
• Introduction to representations of braid groups