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Spin, statistics, orientations, unitarity

Theo Johnson-Freyd

Algebraic & Geometric Topology 17 (2017) 917–956

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–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  Vect, 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(). 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 π1 ét(Spec()) = π1 BO(). We interpret Deligne’s “existence of super fiber functors” theorem as implying that π2 ét(Spec()) = π2 BO() 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.

TQFT, spin, super, categorification, torsors, Galois theory
Mathematical Subject Classification 2010
Primary: 14A22, 57R56, 81T50
Received: 28 October 2015
Revised: 6 June 2016
Accepted: 24 June 2016
Published: 14 March 2017
Theo Johnson-Freyd
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
United States