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 $\Delta = \ell + 1$, with $\ell$ 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.