Exact Path Integral Methods in Supersymmetric $\text{AdS}_2\times \mathbf{S}^2$ Backgrounds

We determine the exact functional determinants of charged, massive spin-0 and spin-$\frac12$ particles in $\text{AdS}_2\times \mathbf{S}^2$ backgrounds threaded by constant electric and magnetic fields. This is achieved using Schwinger proper-time formalism, which allows us to derive the full non-perturbative effective action in the 1-loop and constant background field approximations. We then specialize the computation to supersymmetric settings and we obtain the effective action for a 4d $\mathcal{N}=2$ BPS massive hypermultiplet in a supersymmetric $\text{AdS}_2\times \mathbf{S}^2$ spacetime. This setup can be seen to be equivalent to the near-horizon geometry of a BPS black hole which solves the attractor equations of 4d $\mathcal{N}=2$ supergravity. Our results provide a necessary intermediate step for the evaluation of the quantum-corrected black hole partition function. We also comment on the relation with the celebrated Gopakumar-Vafa integral representation.