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 × S 7 and AdS 5 × S 5 by modes that are not part of the 𝒩 = 8 supermultiplet. As an application, we compute the full Kaluza-Klein spectrum of the 𝒩 = 1 and 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.