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Paravortices: loop braid representations with both generators involutive

Paul P. Martin, Eric C. Rowell, Fiona Torzewska

TL;DR

This work introduces the mixed doubles category ${\mathsf{MD}}$, a quotient of the loop braid category that encodes paravortex loop statistics by enforcing $R^2=I$ and yields two copies of the symmetric group ${\Sigma_n}$. It develops a BaBeDa framework via a generalised wreath construction to model ${\mathsf{MD}}_n$ and provides a complete rank-2 classification of strict monoidal functors ${\mathsf{MD}} \to {\mathsf{Mat}}$, including a detailed analysis of the resulting mixed-doubles representations. The study reveals rich, non-semisimple representation theory with indecomposable structures, Wangian cases, and glue-type families (f-glue, a-glue, antislash, etc.), and it characterises when these representations are finite, semisimple, or unitary. The results illuminate how higher-paragraphure symmetry constraints shape quantum-resource representations in 3D loop-like systems and connect to broader non-semisimple higher representation theory, while suggesting avenues for extending the framework to higher rank and alternative quotient relations.

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.

Paravortices: loop braid representations with both generators involutive

TL;DR

This work introduces the mixed doubles category , a quotient of the loop braid category that encodes paravortex loop statistics by enforcing and yields two copies of the symmetric group . It develops a BaBeDa framework via a generalised wreath construction to model and provides a complete rank-2 classification of strict monoidal functors , including a detailed analysis of the resulting mixed-doubles representations. The study reveals rich, non-semisimple representation theory with indecomposable structures, Wangian cases, and glue-type families (f-glue, a-glue, antislash, etc.), and it characterises when these representations are finite, semisimple, or unitary. The results illuminate how higher-paragraphure symmetry constraints shape quantum-resource representations in 3D loop-like systems and connect to broader non-semisimple higher representation theory, while suggesting avenues for extending the framework to higher rank and alternative quotient relations.

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, . 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 , 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.

Paper Structure

This paper contains 18 sections, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalised Wreaths and the BaBeDa isomorphism
  4. Warm-up: wreaths, their conjugacy classes, and their representations
  5. Generalised wreaths
  6. The BaBeDa isomorphism
  7. Mixed doubles representations
  8. Notions of equivalence of representations/strict monoidal functors
  9. On non-local but unentangled equivalences
  10. Towards classification of solutions in rank 2: classification of functors $F:\Sigma\rightarrow\mathsf{Mat}^2$
  11. Manji calculus
  12. Rank-2 classification of ${\mathsf{MD}}$ representations: Theorem and proof
  13. Analysis of MD representations in the classification
  14. Structure of representations: some preliminaries
  15. The MD representation in the f-glue case
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Passing from a diagram $d$ for an element of $G\wr \Sigma_n$ to the diagram-trace graph of the diagram, which shows its conjugacy class. Here we have coloured the strands the same colour if they are in the same cycle in the perm in $d$ - the black cycle is the 3-cycle $(145)$ and so on. The final identity slides $G$ elements around their cycle until they are together - note that conjugating by, say $d \mapsto \tau_4^{-1} d \tau_4$, cancels the $\tau_4$ from the top and moves it to the bottom (diagrammatically this is as if the $\tau_4$ is slid around the trace loop). Then from the bottom, straightening in the sense of (\ref{['eq:straighten-eg']}) then moves this factor to the first strand in the 3-cycle.

Theorems & Definitions (1)

  • proof