The Arf-Brown TQFT of Pin${}^{-}$ Surfaces

The Arf-Brown invariant $AB(\mathrm{\Sigma })$ is an 8th root of unity associated to a surface $\mathrm{\Sigma }$ equipped with a pin${}^{-}$ structure. In this note we investigate a certain fully extended, invertible, topological quantum field theory (TQFT) whose partition function is the Arf-Brown invariant. Our motivation comes from the recent work of Freed-Hopkins on the classification of topological phases, of which the Arf-Brown TQFT provides a nice example of the general theory; physically, it can be thought of as the low energy effective theory of the Majorana chain, or as the anomaly theory of a free fermion in 1 dimension.