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.

To appear in Algebraic and Geometric Topology

Cited by in corpus (8)

• Reflection positivity and invertible topological phases
• On the classification of topological orders
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• Topological Orders in (4+1)-Dimensions
• Snowmass White Paper: The Quest to Define QFT
• Reflection Structures and Spin Statistics in Low Dimensions
• The Smith Fiber Sequence and Invertible Field Theories
• A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories