On Supersymmetric D-brane probes in 4d $\mathcal{N}=2$ $\text{AdS}_2\times\mathbf{S}^2$ Attractors

TL;DR

This paper extends the analysis of supersymmetric D-brane probes in $ ext{AdS}_2 imes ext{S}^2$ to include stationary, non-static trajectories with angular momentum, revealing new BPS states and enriching the spectrum of supersymmetric configurations relevant for black hole microstates.

Contribution

It introduces stationary D-brane probe solutions with angular momentum in $ ext{AdS}_2 imes ext{S}^2$, expanding the known supersymmetric configurations and their role in holography and microstate counting.

Findings

Identified new $rac{1}{2}$-BPS trajectories with fixed radius and velocity.

Demonstrated these paths saturate a lower energy bound related to angular momentum.

Revealed a spectrum of supersymmetric states organized by $SU(2)$ charges.

Abstract

We extend the $\kappa$-symmetry analysis of supersymmetric D-brane probes in the $\mathrm{AdS}_2 \times \mathbf{S}^2$ attractor geometry, originally performed by Simons, Strominger, Thompson, and Yin, to also include stationary -- but non-static -- worldlines carrying angular momentum along the 2-sphere. We demonstrate that certain special trajectories, with fixed radius and orbital velocity, solve the equations of motion and moreover satisfy a supersymmetry preserving condition, thus defining new $\frac12$-BPS configurations. Furthermore, these classical paths are shown to saturate a lower bound for the Hamiltonian generating global time translations, with the corresponding minimal energy depending on a generalized angular momentum vector $\boldsymbol{J}$. The direction of the latter, in turn, determines exactly which supercharges remain unbroken. Our results reveal a richer spectrum of (multi-particle) supersymmetric states in $\mathrm{AdS}_2 \times \mathbf{S}^2$, which can be organized into distinct selection sectors labeled by the conserved $SU(2)$ charges. This construction has direct applications in black hole microstate counting, the analysis of probe dynamics and $\text{AdS}_2/\text{CFT}_1$ holography.