Spin, statistics, orientations, unitarity

TL;DRAbstract

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 [math] –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 [math] , but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over [math] . 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 [math] . We interpret Deligne’s “existence of super fiber functors”

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