Motion Groupoids and Mapping Class Groupoids

A central role in the description of topological phases of matter in (2 + 1)-dimensions is played by the braid groups. Motions exchanging non-abelian anyons in the disk induce unitary representations of the braid groups. It is of interest to generalise these ideas to different manifolds, to particles which are not point like (loop or string excitations in a 3ball, for example) and to motions of particles that do not necessarily start and end in the same configuration. In this paper we construct a categorical framework for studying such generalisations of anyons. Mathematically, there are a number of ways to define the braid groups but the one that arises from the physical picture of particles moving in space is the definition as (equivalence classes of) motions of points in a disk. In this paper we construct for each manifold M its motion groupoid MotM . (One recovers the classical definitions of motion groups associated to a pair of a manifold M and a subset by considering the automorphisms of the corresponding object.) Braid groups can be equivalently defined as the boundary fixing mapping class groups of points in a disk. We also give a construction of a mapping class groupoid MCGM associated to a manifold. It is not the case that for all manifolds the mapping class groupoid MCGM and motion groupoids MotM are isomorphic. We give conditions under which we have an isomorphism (for M = [0,1] for any n ∈ N for example). We will investigate the constructions of motion groupoids and mapping class groupoids, and their relations, in the topological category. We expect that our constructions and results can be formulated in the smooth category. Acknowledgement: JFM and PM are partially funded by the Leverhulme trust research project grant “RPG-2018-029: Emergent Physics From Lattice Models of Higher Gauge Theory”. FT is funded by a University of Leeds PhD Scholarship. FT thanks Carol Whitton; and PM thanks Paula Martin for useful conversations. FT also thanks Simona Pauli, without her invitation to give a lecture series, this work would not have existed. We all thank Celeste Damiani for useful discussions. ∗f.m.torzewska@leeds.ac.uk †j.fariamartins@leeds.ac.uk ‡p.p.martin@leeds.ac.uk 1 ar X iv :2 10 3. 10 37 7v 1 [ m at hph ] 1 8 M ar 2 02 1

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