Spin, statistics, orientations, unitarity

Abstract

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 ${\mathrm{\text{ Vect}}}_{ℝ}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $Spec\left(ℝ\right)$. Bundles of tangential structures may be étale-locally equivalent without being equivalent, and hermitian field theories are nothing but the field theories controlled by the unique nontrivial bundle of tangential structures that is étale-locally equivalent to Orientations. This bundle owes its existence to the fact that ${\pi }_1^{\text{ ét}}\left(Spec\left(ℝ\right)\right)={\pi }_{1}BO\left(\infty \right)$. We interpret Deligne’s “existence of super fiber functors” theorem as implying that ${\pi }_2^{\text{ ét}}\left(Spec\left(ℝ\right)\right)={\pi }_{2}BO\left(\infty \right)$ in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings. One finds that there are eight bundles of tangential structures étale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of a field theory with that distinguished tangential structure, one arrives at a field theory that is both hermitian and satisfies spin-statistics. Finally, we formulate in our framework a notion of reflection-positivity and prove that if an étale-locally-oriented field theory is reflection-positive then it is necessarily hermitian, and if an étale-locally-spin field theory is reflection-positive then it necessarily both satisfies spin-statistics and is hermitian. The latter result is a topological version of the famous spin-statistics theorem.