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 $\mathrm{Vect}_{\mathbb R}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $\mathrm{Spec}(\mathbb R)$. Bundles of tangential structures may be etale-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 etale-locally equivalent to Orientations. This bundle owes its existence to the fact that $\pi_1^{et}(\mathrm{Spec}(\mathbb R)) = \pi_1{BO}$. We interpret Deligne's "existence of super fiber functors" theorem as implying that in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings, $\pi_2^{et}(\mathrm{Spec}(\mathbb R)) = \pi_2{BO}$. There are eight bundles etale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of that distinguished tangential structure, one arrives at field theories that are both Hermitian and satisfy spin-statistics. Finally, we formulate a notion of "reflection-positivity" and prove that if an etale-locally-oriented field theory is reflection-positive then it is necessarily Hermitian, and if an etale-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.