Generically, the supersymmetry and extremality conditions do not coincide. Adopting the nomenclature of [2], we denote by “BPS” the extremal supersymmetric solutions.

There is yet another distinct way to obtain dyonic supersymmetric black holes, by considering C-metric solutions describing accelerating black holes with magnetic charge related to the acceleration parameter [28, 29, 30, 31, 32]. In the present paper we restrict our attention to dyonic twisted non-accelerating solutions.

Though a different limiting procedure to define the on-shell action of the BPS black hole by studying perturbations to the supersymmetric solution has been proposed in the literature [36, 37, 38].

See [17, 40] for a review of the cases of three- and four-dimensional SCFTs.

There also exists a family of static solutions with unconstrained charges [47], and rotating extensions thereof [47, 48]. Euclidean solutions generalizing (23) have been presented in [49]. However, for the purposes of this paper it is appropriate to work with this solution.

For $\mathrm{\Delta }=0$ and $p_1=p_2$, $q_1=q_2$ the parameter $a_1$ is unconstrained by the equations of motion, and can be identified with the parameter $\eta$ in (7).

Rewriting the action in terms of ${\phi }_a$, $P_a$ is ambiguous. We choose the formula to be consistent with the large-$N$ limit of the partition function of mass-deformed ABJM [53].

On the other hand, there also exist saddles of the path integral that are intrinsically Euclidean, such as the ones found in [49], that do not have a Lorentzian black hole counterpart and whose on-shell action can matched to a field theory computation.

If ${\mathrm{\Sigma }}_g\cong S^2$, there is an additional isometry, the states in ${\mathcal{H}}_{S^2}$ are labelled by the angular momentum as well, and it is possible to further refine the index [33]. The bulk dual states would be dyonic rotating solutions with spherical horizons such as [55].