The spin-statistics theorem for topological quantum field theories

Abstract

We establish the spin-statistics theorem for topological quantum field theories (TQFTs) in the framework of Atiyah. We incorporate spin via spin structures on bordisms, and represent statistics using super vector spaces. Unitarity is implemented using dagger categories, in a manner that is equivalent to the approach of Freed-Hopkins, who employed $\mathbb{Z}/2$-equivariant functors to address reflection-positivity. A key contribution of our work is the introduction of the notion of fermionically dagger compact categories, which extends the well-established concept of dagger compact categories. We show that both the spin bordism category and the category of super Hilbert spaces are examples of fermionically dagger compact categories. The spin-statistics theorem for TQFTs emerges as a specific case of a more general result concerning symmetric monoidal dagger functors between fermionically dagger compact categories.