Paravortices: loop braid representations with both generators involutive

TL;DR

This paper classifies all rank-2 representations of a quotient of the loop braid category, called the mixed doubles category, with implications for topological quantum computation and higher representation theory.

Contribution

It provides a complete classification of rank-2 mixed doubles representations and analyzes their linear representation theory using novel and classical methods.

Findings

Classified all rank-2 mixed doubles representations.

Connected representations to loop braid groups and quantum computation.

Introduced techniques for analyzing non-semisimple higher representations.

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.