Squashed 7-spheres, octonions and the swampland

Abstract We give a brief account of how to derive the entire eigenvalue spectrum of the operators on the squashed S 7 that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in AdS 4 and are important for the corresponding 𝒩 = 1 supermultiplet structure. By an orientation-flip on the squashed S 7 we can also determine the spectrum of the corresponding non-supersymmetric theory, and, e.g., its spectrum of marginal operators on the boundary of AdS 4 which has some relevance for the AdS stability conjecture in the swampland program. Here we review recent work in [1, 2, 3] which is a continuation of the work in [4] where the complete spectrum of irreducible isometry representations of the fields in AdS 4 was derived for this compactification. Details are here given primarily for 2-forms while comments are also made on the key role of G 2 and octonions for the structure of the operator equations and mode functions on the squashed S 7 . Key features of these improved methods were obtained in Joel Karlsson’s 2021 MSc thesis [5]. This is a write-up of a talk given at ISQS28, Prague, Czech Republic, July 4, 2024.