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In a recent paper, Teo and Kane proposed a 3D model in which the defectssupport Majorana fermion zero modes. They argued that exchanging and twistingthese defects would implement a set R of unitary transformations on the zeromode Hilbert space which is a 'ghostly' recollection of the action of the braidgroup on Ising anyons in 2D. In this paper, we find the group T_{2n} whichgoverns the statistics of these defects by analyzing the topology of the spaceK_{2n} of configurations of 2n defects in a slowly spatially-varying gappedfree fermion Hamiltonian: T_{2n}\equiv {\pi_1}(K_{2n})$. We find that the groupT_{2n}= Z \times T^r_{2n}, where the 'ribbon permutation group' T^r_{2n} is amild enhancement of the permutation group S_{2n}: T^r_{2n} \equiv \Z_2 \timesE((Z_2)^{2n}\rtimes S_{2n}). Here, E((Z_2)^{2n}\rtimes S_{2n}) is the 'evenpart' of (Z_2)^{2n} \rtimes S_{2n}, namely those elements for which the totalparity of the element in (Z_2)^{2n} added to the parity of the permutation iseven. Surprisingly, R is only a projective representation of T_{2n}, apossibility proposed by Wilczek. Thus, Teo and Kane's defects realize`Projective Ribbon Permutation Statistics', which we show to be consistent withlocality. We extend this phenomenon to other dimensions, co-dimensions, andsymmetry classes. Since it is an essential input for our calculation, we reviewthe topological classification of gapped free fermion systems and its relationto Bott periodicity.

Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions