The complete Kaluza-Klein spectra of $\mathcal{N} = 1$ and $\mathcal{N} = 0$ M-theory on $AdS_4 \times (\text{squashed } S^7)$

The squashed seven-sphere operator spectrum is completed by deriving the spectrum of the spin-3/2 operator. The implications of the results for the $AdS_4$ $\mathcal{N} = 1$ supermultiplets obtained from compactification of eleven-dimensional supergravity are analysed. The weak $G_2$ holonomy plays an important role when solving the eigenvalue equations on the squashed sphere. Here, a novel and more universal algebraic approach to the whole eigenvalue problem on coset manifolds is provided. Having obtained full control of all the operator spectra, we can finally determine the irreps $D(E_0, s)$ for all supermultiplets in the left-squashed vacuum. This includes an analysis of possible boundary conditions. By performing an orientation flip on the seven-sphere, we also obtain the full spectrum for the non-supersymmetric right-squashed compactification which is of interest in the swampland context and in particular for the $AdS$ swampland conjecture. Here, a number of boundary condition ambiguities arise making the analysis of dual marginal operators somewhat involved. This work is a direct continuation of [1] and [2].