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 generalized particles in topological phases, loop particles in 3-dimensions for example, is to generalize (some of) the
several different realizations 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 quotient 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 discuss several examples throughout.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Topological Quantum Computation.
