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
–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
,
but rather than being defined in terms of a single tangential structure,
they are defined in terms of a bundle of tangential structures over
.
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
. 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. 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.
Keywords
TQFT, spin, super, categorification, torsors, Galois theory
