Squashed 7-spheres, octonions and the swampland

The entire eigenvalue spectrum of the operators on the squashed $S^7$ that appear in the Freund-Rubin compactification of eleven-dimensional supergravity was recently derived in [1 - 4]. Here we give a brief account of this work which started with [1] where the complete spectrum of irreducible isometry representations of the fields in $AdS_4$ was derived for the squashed $S^7$ compactification. The operator spectra determine the mass spectrum of the fields in $AdS_4$ and are important for the corresponding $\mathcal{N} =1$ supermultiplet structure which appears in two versions depending on the choice of boundary conditions. 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 may have some relevance for the $AdS$ stability conjecture in the swampland program. The role of singletons is discussed and a possible new Higgsing phenomenon turning them into bulk fields is suggested. Details are here given primarily for 2-forms and comments are made on the key role of $G_2$ and octonions for the structure of the operator equations and mode functions on the squashed $S^7$. Some important features of these improved methods were obtained in Joel Karlsson's 2021 MSc thesis [5]. This is an extended version of the author's contribution to the proceedings of the conference ISQS28, Prague, Czech Republic, July 1 - 5, 2024.