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

Abstract

We extend the $κ$-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.

Figures (5)

• Figure 1: A system comprised by a particle/anti-particle pair can be BPS if the total generalized angular momentum satisfies $|\boldsymbol{J}_{\rm tot}| = |\boldsymbol{J}_1| + |\boldsymbol{J}_2|$. $\textbf{(a)}$ Static configuration with the probes located at antipodal points on $\mathbf{S}^2$. $\textbf{(b)}$ Stationary case with the particles rotating in opposite directions.
• Figure 2: Penrose diagram of 2d Anti-de Sitter space in Poincaré and global (strip) coordinates. The triangular region corresponds to a single Poincaré patch, whereas global AdS contains an infinite sequence of such consecutive slices.
• Figure 3: Effective potential $V(\chi)$ controlling the radial dynamics in global AdS$_2 \times \mathbf{S}^2$, cf. eq. \ref{['eq:effectiveradialpoteglobal']}. The dashed vertical line denotes the 'center' of Anti-de Sitter space at $\chi=0$. The qualitative features of the potential depend on whether $\textbf{(a)}$$q_e^2 <\tilde{m}^2 + \ell^2$ (subextremal), $\textbf{(b)}$$q_e^2 >\tilde{m}^2+ \ell^2$ (superextremal), or $\textbf{(c)}$$q_e^2 =\tilde{m}^2 +\ell^2$ (extremal). We show the corresponding effective potential for both the particle ($E q_e>0$, yellow) and its CPT conjugate ($E q_e<0$, blue).
• Figure 4: Depiction of the spacetime trajectories associated to charged subextremal particles in 4d $\mathcal{N}= 2$ AdS$_2 \times \mathbf{S}^2$ geometries. The dynamics along the sphere (right) is controlled by the generalized angular momentum vector $\boldsymbol{J}$, around which the particle precesses, whereas in AdS$_2$ (left) particles are confined within some finite distance from the conformal boundaries and exhibit periodic motion.
• Figure 5: Whenever the global energy of the BPS probe reaches certain minimum value, i.e., for $E=\sqrt{j^2}$, the on-shell trajectory becomes stationary in AdS$_2\times\mathbf{S}^2$, such that the particle stays at a constant radial distance from the boundary determined by its conserved charges. The resulting effective potential therefore exhibits a minimum at $\chi_{\rm min}=\text{sinh}^{-1} (q_e/|j|)$ that verifies $V(\chi_{\min})=0$. The yellow (blue) line indicates a charged particle with $p_\tau \, q_e <0$ ($p_\tau \, q_e >0$).