Analysing topological aspects of QFT within the locally covariant algebraic framework

The locally covariant approach to quantum field theory (LCQFT) is a manifestly covariant functorial approach to quantum field theory (QFT) that applies to curved spacetimes and which builds on the local algebraic approach. In this thesis we investigate applications of LCQFT to topological aspects of QFT.

We analyse extensions of quantum field theories defined on contractible globally hyperbolic regions of spacetime, using Fredenhagen's universal algebra construction. This construction involves covering a spacetime by open contractible causally convex subregions, and applying the functor that defines the theory to each of them to get a net of local algebras. The universal algebra is then obtained by taking the colimit of this net. Morphisms between universal algebras can be defined with the result that the mapping between spacetimes and their corresponding universal algebras defines a functor. We prove two main results about this universal construction, which both require considerable geometric apparatus.

First we prove that for a broad class of theories modelled on the free scalar/Dirac field, the functor assigning universal algebras satisfies the Einstein causality axiom. We then restrict attention to Fermionic theories in this class, and analyse the universal theories obtained from the subtheories that assign even subalgebras. We show that for each spacetime $\mathcal{M}$, the universal theory assigns an algebra which decomposes into a product (in the categorical sense) of subalgebras, that are in bijective correspondence with the set $H^{1}(\mathcal{M},\mathbb{Z}_2)$. The latter set counts the number of distinct spin structures the spacetime manifold $\mathcal{M}$ permits. The universal algebra for a Fermionic theory therefore has the geometric information encoded in it necessary to define half integer spin fields.