Unitary fermionic topological field theory

Abstract

Atiyah's axioms are one way to define rigorously what is a topological quantum field theory. He defined it as a symmetric monoidal functor from a bordism category to the category of vector spaces. To include fermions in this definition, we take instead a functor from a spin bordism category to the category of super vector spaces. In quantum field theory, we can ask whether there is a connection between the spin (integer or half-integer) and its statistics (fermionic or bosonic). The spin-statistics theorem says that every unitary quantum field theory has a spin-statistics connection. In fermionic topological quantum field theory, a spin-statistics connection is a relationship between the 360 degree rotation in spacetime and the fermion parity operator on state space. Not every such field theory has a spin-statistics connection. We construct a symmetric monoidal dagger category of spin bordisms and define a unitary fermionic topological quantum field theory to be a symmetric monoidal dagger functor from this category to the symmetric monoidal category of super Hilbert spaces. The main goal of the thesis is to show that every unitary fermionic topological quantum field theory has a spin-statistics connection.

To achieve this, we develop a theory on how to construct dagger categories from categories with anti-involutions equipped with Hermitian pairings. This construction is analogous to the construction of the dagger category of Hilbert spaces out of the category of vector spaces. In case the category we started with admits duals, we study lifts of the dual functor to a symmetric monoidal dagger functor. As observed by Dave Penneys, dagger dual functors are subtle. We shed some light on their existence and uniqueness. The dagger categories studied in this thesis additionally come equipped with a unitary BZ/2-action. This action is provided by the 360 degree rotation in the bordism category and the parity operator in the category of Hilbert spaces. The essential categorical property of these dagger categories is that they admit dagger dual functors that are `twisted by the BZ/2-action'. Given a bordism dagger category with this property, a dagger functor to super Hilbert spaces will have a spin-statistics connection. We prove that our construction of the spin bordism category satisfies this property and the spin-statistics theorem for topological quantum field theory follows.