Squashed 7-spheres, octonions and the swampland

Paper in proceeding, 2024

We give a brief account of how to derive the entire eigenvalue spectrum of the operators on the squashed S7 that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in AdS4 and are important for the corresponding N = 1 supermultiplet structure. By an orientation-flip on the squashed S7 we can also determine the spectrum of the corresponding non-supersymmetric theory, and, e.g., its spectrum of marginal operators on the boundary of AdS4 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 AdS4 was derived for this compactification. Details are here given primarily for 2-forms while comments are also made on the key role of G2 and octonions for the structure of the operator equations and mode functions on the squashed S7. 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.