we study the low-energy behavior of a different model, the Majorana chain. The Arf-Brown invariant is a generalization of the more familiar Arf invariant of a spin surface. The Arf invariant admits three quite different-looking descriptions: one using a quadratic refinement of the intersection pairing; one using a mod 2 index of the spin Dirac operator; and one using KO-theory. This paper presents Arf-Brown Topological quantum Field Theories of Pin- Manifolds. We apply the Arf-Brown theory for studying the Majorana chain with its time-reversal symmetry. The phase predicted to be associated to this system is an example of a special class of phases called symmetry-protected topological (SPT) phases, which are conjectured to correspond to invertible TFTs in the low-energy ansatz. Specifically, it is believed that the group of 2d fermionic SPT phases with a time-reversal symmetry squaring to 1 is isomorphic to Z/8, and that the phase of the Majorana chain is a generator. In the low-energy ansatz, this is related to the Z/8 classification of 2d pin− reflection positive invertible TFTs, generated by the Arf-Brown TFT Z_AB. We give a few different constructions of the Arf-Brown invariant, which is the partition function of Z_AB, then construct ZAB. We investigate this by defining the Majorana chain on a pin− 1-manifold with a triangulation, encoding the pin− structure in additional discrete data. We then compute the space of ground states, and prove that these agree with the state spaces of Z_AB .