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 adapt two complementary evaluation methods to the BMN index: 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≤ 9. In the equal-fugacity expansion, the coefficients near charges j∼ N2 exhibit substantial cancellations between vacuum sectors, which tend to reduce the effective N2 coefficient of the total index entropy. The finite-N data also reveal a nontrivial sectoral organization: near j=N2, the sector giving the largest contribution changes with N, from single-partition sectors at small rank to double-partition sectors starting at N=6. 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 in the controlled type-IIA regime a possible D2 dressed black-hole interpretation in which D0 black-hole sectors are accompanied by macroscopic spherical D2-brane degrees of freedom, analogous to dual dressed black holes in AdS5× S5.
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BibTeX
@misc{chang2026finiten,
title = {{Finite-$N$ BMN index across all vacuum sectors}},
author = {Chi-Ming Chang and Sarthak Duary and Kangning Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.25560},
eprint = {2605.25560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25560},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Finite-N BMN index across all vacuum sectors
AU - Chi-Ming Chang
AU - Sarthak Duary
AU - Kangning Liu
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.25560
UR - https://arxiv.org/abs/2605.25560
AB - 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 adapt two complementary evaluation methods to the BMN index: 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≤ 9. In the equal-fugacity expansion, the coefficients near charges j∼ N^2 exhibit substantial cancellations between vacuum sectors, which tend to reduce the effective N^2 coefficient of the total index entropy. 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=6. 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 in the controlled type-IIA regime a possible D2 dressed black-hole interpretation in which D0 black-hole sectors are accompanied by macroscopic spherical D2-brane degrees of freedom, analogous to dual dressed black holes in AdS_5× S^5.
ER -