Spin Structures and AdS$_4$ Holography

We study the importance of spin structures as defining data for 11d supergravity backgrounds of the form AdS$_4\times S^7/\mathbf{Z}_k$ with a free orbifold action. For a generic choice of the orbifold action, there is only one spin structure that preserves invariant Killing spinors. There are, however, orbifold actions that allow for more than one spin structure with preserved Killing spinors. In these cases the two different spin structures should lead to distinct holographically dual 3d SCFTs with different amounts of supersymmetry. We illustrate this phenomenon by studying the KK spectrum of 11d supergravity on AdS$_4\times S^7/\mathbf{Z}_4$. For one choice of spin structure this background is dual to the 3d $\mathcal{N}=6$ ABJM theory at level $k=4$. For the alternative choice of spin structure our results suggest that the holographic dual is a different 3d $\mathcal{N}=2$ SCFT and we discuss some of its properties.