Pure D-brane black holes: BPS counting and non-BPS vacua

A

bstract

In this paper, we present a unified computational framework to analyze the microscopic vacuum structure of 4-charge extremal black holes in Type IIA string theory, applying techniques from computational algebraic geometry and numerical topology to their pure D-brane effective quantum mechanics. We apply this approach to two physically distinct configurations. First, in the supersymmetric sector, we compute the 14 ^th helicity trace index of

-BPS,

$$ \mathcal{N} $$ N

= 8, D2-D2-D2-D6 configurations dual to D1-D5-P-KK monopole dyonic black holes. Extending previous work to higher charges, we employ a parametric monodromy method to explicitly resolve the vacua for the (1, 1, 1, 5) and (1, 1, 1, 6) configurations, reproducing the degeneracies predicted by the U-dual picture [ JHEP 05 (2024) 091]. Second, we apply complementary techniques to a configuration where supersymmetry is explicitly broken at the level of the effective action. The corresponding 4-charge non-BPS extremal pure D-brane system is obtained by replacing the D6-brane with an anti-D6-brane and assigning incompatible R-symmetry rotations to different brane triplets [ Phys. Rev. D 113 (2026) 026010]. Analyzing the associated scalar potential using analytical Gröbner bases, we demonstrate the absence of zero-energy classical ground states. To handle the continuous flat directions populating the non-BPS landscape, we implement specific topological regularizations, namely Morse-Bott deformations and gated soft-trapping. These methods allow us to systematically characterize the classical energy landscape, identifying a non-compact Coulomb branch, marginally bound stabilizer submanifolds, and an isolated collection of doubly degenerate low-energy stable states.