Spin, statistics, orientations, unitarity

Understand

A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation.

• A field theory satisfies spin-statistics if it is both spin and super, and $360^\circ$-rotation of the spin structure agrees with the operation of flipping the signs of all fermions.
• We set up a framework in which these two notions are precisely analogous.
• In this framework, field theories are defined over $\mathrm{Vect}_{\mathbb R}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $\mathrm{Spec}(\mathbb R)$.