Compactifications of String/M-Theory and the Swampland: A Study of the AdS$_\text{4}$ Mass Spectrum of Eleven-Dimensional Supergravity on the Squashed Seven-Sphere

The landscape of possible four-dimensional low-energy effective theories arising from compactifications of string/M-theory seems vast. This might lead one to believe that any consistent-looking effective field theory coupled to gravity can be obtained as a low-energy limit of string theory. However, a set of "swampland" conjectures suggests that this is not true and that, in fact, there is an even larger set of effective field theories that cannot be obtained in this way. In particular, the AdS instability swampland conjecture asserts that nonsupersymmetric anti-de Sitter vacua are unstable. These swampland criteria can have implications for, for instance, low-energy physics and cosmology. M-theory is a nonperturbative unification of all superstring theories. Its low-energy limit, eleven-dimensional supergravity, admits two compactifications on the squashed seven-sphere. One of the solutions has one unbroken supersymmetry ($\mathcal{N} = 1$) while the other has none ($\mathcal{N} = 0$). Due to the AdS instability swampland conjecture, the latter should be unstable. However, this has not been demonstrated explicitly. To study the stability of the $\mathcal{N} = 0$ vacuum, we investigate the mass spectrum of the theory. The main advancement compared to previous attempts is the realisation that all mass operators in the Freund-Rubin compactification are related to a universal Laplacian, allowing us to relate Weyl tensor terms to group invariants. One limitation of the group-theoretical method we employ is that it can lead to false roots. This is remedied, at least in part, by demanding that the fields form supermultiplets in the $\mathcal{N} = 1$ case. Although we arrive at an eigenvalue spectrum for all operators of interest, there is a hint that the results may be incomplete. Thus, we do not reach a decisive conclusion regarding the investigated type of instability.