Motion groupoids and mapping class groupoids
Moduli and Friends seminar
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Monday 8 May 2023

Fiona Torzewska (University of Vienna)Motion groupoids and mapping class groupoids

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 MotM, 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 congruence relation used in the construction MotM 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 MCGM associated to a manifold M with the same object class. For each manifold M I will construct a functor F : MotM → MCGM, and prove that this is an isomorphism if π0 and π1 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∈N.

I will give several examples throughout.

* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.