Motion Groupoids

          @misc{ pirsa_PIRSA:24030087,
            doi = {10.48660/24030087},
            url = {https://pirsa.org/24030087},
            author = {Torzewska, Fiona},
            keywords = {Condensed Matter},
            language = {en},
            title = {Motion Groupoids},
            publisher = {Perimeter Institute},
            year = {2024},
            month = {mar},
            note = {PIRSA:24030087 see, \url{https://pirsa.org}}
          }
          

Abstract

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group. In this talk I will construct for each manifold M its motion groupoid $Mot_M$, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the quotient used in the construction $Mot_M$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold M with the same object class. For each manifold M I will construct a functor $F \colon Mot_M \to MCG_M$, and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of M is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in\mathbb{N}$. I will discuss several examples throughout.