Near-AdS$_2$ Spectroscopy: classifying the spectrum of operators and interactions in $\mathcal{N} = 2$ 4D supergravity

We describe holographic properties of near-AdS$_2$ spacetimes that arise within spherically symmetric configurations of ${\cal N}=2$ 4D $U(1)^4$ supergravity, for both gauged and ungauged theories. These theories pose a rich space of AdS$_2\times S^2$ backgrounds, and their responses in the near-AdS$_2$ region are not universal. In particular, we show that the spectrum of operators dual to the matter fields, and their cubic interactions, are sensitive to properties of the background and the theory it is embedded in. The properties that have the most striking effect are whether the background is supersymmetric or not, and if the theory is gauged or ungauged. Interesting effects are due to the appearance of operators with $\Delta < 2$, which depending on the background can lead to, for instance, instabilities or extremal correlators. The resulting differences will have an imprint on the quantum nature of the microstates of near-extremal black holes, reflecting that not all extremal black holes respond equally when kicked away from extremality.