Perturbing AdS$_6 \times_w S^4$: linearised equations and spin-2 spectrum

We initiate the analysis of the Kaluza--Klein mass spectrum of massive IIA supergravity on the warped AdS$_6 \times_w S^4$ background, by deriving the linearised equations of motion of bosonic and fermionic fluctuations, and determining the mass spectrum of those of spin-2. The spin-2 modes are given in terms of hypergeometric functions and a careful analysis of their boundary conditions uncovers the existence of two branches of mass spectra, bounded from below. The modes that saturate the bounds belong to short multiplets which we identify in the representation theory of the $\mathfrak{f}(4)$ symmetry superalgebra of the AdS$_6 \times_w S^4$ solution.