How to (Non-)Perturb a BPS Black Hole

Abstract

We relate the structure of non-perturbative corrections to BPS black hole observables in flat-spacetime theories with certain properties of probe charged particles in the near-horizon geometry. Concretely, we consider 4d $\mathcal{N} = 2$ supergravity with an infinite tower of F-terms and probe branes in $\text{AdS}_2\times \mathbf{S}^2$ backgrounds threaded by constant electric-magnetic fields. The higher dimensional operators we pick are computed by Type II topological string theory, and we approximate them via the constant map contribution, which is valid at large volume and can be interpreted as arising from D0-branes integrated out in M-theory on a Calabi-Yau threefold times a circle. We analyze the resulting force conditions on massive particles carrying $(q_A, p^A)$ charges, their classical trajectories, and the 1-loop effective action they produce. A simple semiclassical analysis allows us to understand qualitatively the structure of the non-perturbative corrections. The exact path integral assessment then reproduces the Gopakumar--Vafa integral of the flat-spacetime theory, now evaluated in the black hole attractor geometry. Thus, we make explicit how the physics of the fully backreacted black hole solution is controlled by the behaviour of the light D-brane states which generate the relevant set of higher derivative corrections.

Figures (4)

• Figure 1: Integration contour in the complex s-plane employed to evaluate the loop integral \ref{['contourIntgral']}. The non-perturbative singularities lie along $\mathbb{R}e^{i(\pi/2 - \theta_\alpha)}$, with $\theta_\alpha$ the complex phase of $\alpha$, whereas the perturbative ones fall onto the real axis. In the limit $\text{Re}\, \alpha \rightarrow 0$, all the poles become real.
• Figure 2: Classical paths of charged particles in the Poincaré patch of AdS$_2$ (shaded grey region). The solid green (red) line represents the D0-brane (resp., anti-D0-brane) trajectory parametrized by real proper time, while the dashed blue curves correspond to those in imaginary time. $\textbf{(a)}$ For $\bar{m} < |q_e|$, charged particles with $q_e p_t < 0$ can undergo pair production in the bulk. $\textbf{(b)}$ In contrast, configurations with $q_e p_t > 0$ do not allow for such effects. $\textbf{(c)}$ For $\bar{m} > |q_e|$, the classical trajectories remain confined at a finite radial distance from the boundary, signaling vacuum stability. This corresponds to BPS particles with non-zero magnetic charge, i.e., $q_m \neq 0$. $\textbf{(d)}$ Finally, when $\bar{m} = |q_e|$, the anti-D0-brane trajectory asymptotically approaches the AdS boundary and disappears, while the one of the D0-brane becomes tangent to the latter. This special case corresponds to a BPS particle with $q_m = 0$.
• Figure 3: 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 4: To connect \ref{['eq:susyeffaction2']} with the Gopakumar-Vafa integral and make manifest the perturbative and non-perturbative structure of the black hole free energy one separates the latter into two and deforms the contours toward the preferred ray at $\theta=\tan^{-1} (|q_m|/|q_e|)$. As a result, one gets a line integral $\mathcal{I}_{\rm line}$ plus some residues.