The role of singletons in S⁷ compactifications

We derive the isometry irrep content of squashed seven-sphere compactifications of eleven-dimensional supergravity, i.e., the left-squashed (LS⁷) with $N$ = 1 and right-squashed (RS⁷) with $N$ = 0 supersymmetry, in a manner completely independent of the round sphere. Then we compare this result with the spectrum obtained by Higgsing the round sphere spectrum. This way we discover features of the spectra which makes it possible to argue that the only way the round spectrum can be related by a Higgs mechanism to the one of LS⁷ is if the singletons are included in the round sphere spectrum. For this to work also in the RS⁷ case it seems that the gravitino of the LS⁷ spectrum must be replaced by a fermionic singleton present in the RS⁷ spectrum.