Finite-$N$ BMN index across all vacuum sectors

Abstract

We compute the finite-$N$ Witten index of BMN matrix quantum mechanics after summing over all partition-labeled supersymmetric vacuum sectors. Starting from the unitary-matrix integral for each sector, we develop two complementary evaluation methods: a symmetric-group character expansion, which reduces each fixed fugacity order to a finite combinatorial sum, and a residue expansion in which the contributing poles are organized by rooted trees, with a colored-tree generalization for multi-partition sectors. Where practical, direct integration and extraction of the constant term in the expanded integrand give independent coefficient-by-coefficient checks. We evaluate every vacuum sector for $N\leq 9$. In the equal-fugacity expansion, the coefficients near charges $j\sim N^2$ show entropy growth of order $N^2$, and, in this range, the sector sum does not cancel this growth. The finite-$N$ data also reveal a nontrivial sectoral organization: near $j=N^2$, the sector giving the largest contribution changes with $N$, from single-partition sectors at small rank to double-partition sectors starting at $N=5$. We call this phenomenon dominance switching. These results provide quantitative finite-$N$ input for using the BMN index as a diagnostic of protected plane-wave black-hole sectors and suggest a D2 dressed black-hole interpretation in the controlled type-IIA regime, where D0 black-hole sectors are accompanied by macroscopic spherical D2-brane degrees of freedom, analogous to dual dressed black holes in $AdS_5\times S^5$.