Spectral flow for skew-adjoint Fredholm operators

An analytic definition of a $\mathbb Z_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through 0 along the path. The $\mathbb Z_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $\mathbb Z_2$-index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the $\mathbb Z_2$-polarization in these models.