Embedded cobordisms, motion groupoids and topological quantum field theories

Topological phases of matter are a particular class of phases of matter which are potentially of interest in the construction of quantum computers. Examples are given by fractional quantum Hall states. Topological quantum field theories (TQFTs), and generalisations of TQFTs, are mathematical constructions that axiomatise the properties of topological phases. In this thesis we are motivated by the aim of understanding possible statistics of generalised quasiparticles (loops or strings in 3-dimensions, for example), in topological phases of arbitrary dimension. In 2-dimensional topological phases, the worldlines of monotonic evolutions of point particles, which start and end in the same configuration, can be modelled by the braid groups. The braid group has several different topological realisations, each with possible generalisations. In particular it has realisations as a mapping class group and as a motion group. In Chapter 4 we construct for each manifold M its motion groupoid MotM, whose objects are the power set of M, and a mapping class groupoid MCGM with the same object class. These generalise the classical definition of a motion group and mapping class group associated to a pair of a manifold and a subset. The classical definitions can be recovered by considering the automorphisms of the corresponding object. Our motivating aim is to frame questions that inform the modelling of the worldlines of particles in topological phases. These include questions about the skeletons of these categories, and about monoidal structures. But our constructions also frame technical questions that we answer here, such as the following. For a chosen manifold M we explicitly 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 we have an isomorphism in the physically important case M = [0, 1] n with fixed boundary, for any n ∈ N.
In Chapter 5 we are motivated by the construction of embedded TQFTs. These are functors from some choice of embedded cobordism category, which models the worldlines of particles in topological phases, into Vect. We construct a category HomCob, and a family of functors ZG∶HomCob → Vect, one for each finite group G. The category HomCob has equivalence classes of cospans of topological spaces as morphisms. This is a very general construction, making it possible to later fix a choice of a categorical model of particle worldlines, and obtain a TQFT by precompsing ZG with a functor into HomCob. Roughly, such a functor can be realised by taking the complement of the particle worldlines in the ambient space. Notice we do not require that the complement be modelled as a manifold. We also give an interpretation of the functor ZG showing that it is explicitly calculable. The construction is a generalisation of an untwisted version of Dijkgraaf-Witten.