Supersymmetric Black Holes

TL;DR

The paper demonstrates a non-renormalization property of the $d=4$, $N=2$ supergravity effective action in backgrounds with super-covariantly constant spinors, including the Robinson-Bertotti solution and multi–extreme Reissner-Nordström black holes. By formulating manifestly supersymmetric black holes as flatness conditions ${\cal D}^2=0$ in a shortened superspace, it shows that all on-shell quantum corrections vanish in these configurations. The results rely on the special structure of the backgrounds (conformal flatness, constant superfields, and preserved supersymmetries) and a flat superspace description that eliminates possible higher-order invariants. This work highlights a deep link between supersymmetry and exact black-hole solutions, offering non-perturbative insights with potential implications for quantum gravity and string theory.

Abstract

The effective action of $N=2$, $d=4$ supergravity is shown to acquire no quantum corrections in background metrics admitting super-covariantly constant spinors. In particular, these metrics include the Robinson-Bertotti metric (product of two 2-dimensional spaces of constant curvature) with all 8 supersymmetries unbroken. Another example is a set of arbitrary number of extreme Reissner-Nordström black holes. These black holes break 4 of 8 supersymmetries, leaving the other 4 unbroken. We have found manifestly supersymmetric black holes, which are non-trivial solutions of the flatness condition $\cd^{2} = 0$ of the corresponding (shortened) superspace. Their bosonic part describes a set of extreme Reissner-Nordström black holes. The super black hole solutions are exact even when all quantum supergravity corrections are taken into account.