Spin-$2$ operators in AdS${}_2$/CFT${}_1$

We study spin-2 fluctuations around an infinite family of warped backgrounds of the form ${\text{AdS}}_2\times {\text{S}}^2\times {\text{T}}^4\times {\text{S}}^1\times {\mathcal{I}}_{\rho }$ in the type IIB theory, which possesses $\mathcal{N}=4$ supersymmetry. We find that there exists a special set of solutions which is independent of the background data; the minimal universal class. This particular class of solutions corresponds to operators with scaling dimension $\mathrm{\Delta }=\mathrm{\ell }+1$, with $\mathrm{\ell }$ being the quantum number of the angular-momentum on the ${\text{S}}^2$. Using the mode solutions of these spin-2 states, we compute the central charge of the dual field theories. We comment on the relation of our results to the seed ${\text{AdS}}_3\times {\text{S}}^2\times {\text{T}}^4\times {\mathcal{I}}_{\rho }$ backgrounds.