In this note, we explore computational algebraic geometry techniques to compute $14^{ th}$ Helicity Trace Index of 4-charge, $\frac{ 1}{ 8}$-BPS, $\mathcal{ N}=8$ pure D-brane configurations dual to D1-D5-P-KK monopole dyonic black holes. We extend the analysis of our previous work to higher values of charges and address subtleties involving compatible gauge choices for $(1,1,1,N)$ charge configurations. For explicit SUSY state counting, we use a parametric monodromy method for the 4-charge $(1,1,1,5)$ and $(1,1,1,6)$ configurations and find that the results match the U-dual picture. By a different choice of the R-symmetry representations, it is possible to explicitly break all supersymmetry and study related (non-)abelian versions as 4-charge non-BPS pure D-brane systems. Using analytical Gr\"{ o}bner bases, we show that the potential has no zero energy configuration. The higher end of the spectrum asymptotes towards the Coulomb branch local minima submanifold representing unbounded D-brane configurations, and the Mixed branch local minima represent bound states at parametrically lower values of the potential. In between, there are several marginally bound stabilizer submanifolds representing partially bounded D-brane configurations. We developed physics-inspired computational techniques to deform the potentials and lift flat directions, thereby cataloging the energy landscape and counting the low-energy stable states with degeneracy.
Categories: hep-th