Kaluza-Klein Spectrometry beyond Consistent Truncations: The Squashed $S^7$

We show how to use Exceptional Field Theory to compute the full Kaluza-Klein spectra of 10- and 11-dimensional supergravity around deformations of backgrounds of maximal gauged supergravity by scalar modes that do not form part of the consistent truncation. This includes deformations of AdS${}_4\times S^7$ and AdS${}_5\times S^5$ by modes that are not part of the $\mathcal{N}=8$ supermultiplet. As an application, we compute the full Kaluza-Klein spectrum of the $\mathcal{N}=1$ and $\mathcal{N}=0$ squashed $S^7$. In this example, all conformal dimensions are captured by a universal formula in terms of the Casimir operators and additional quantum numbers which organise the spectrum.