A Double--Scaling Large--\(d\) Saddle of BFSS/BMN Matrix Quantum Mechanics

Abstract: We study the large--(d) dynamics of the mass--deformed bosonic (\mathrm{BFSS}_{d+1}) matrix quantum mechanics using a Hubbard--Stratonovich localization of the Yang--Mills interaction. After integrating out the matrix coordinates, the theory reduces to a holonomy--dependent effective action for an auxiliary adjoint kernel. We introduce a commuting--symmetric saddle and its maximally symmetric specialization, in which the interaction is encoded in a single dynamically generated mass shift (k_0). The resulting large--(d) description is a gauged matrix harmonic oscillator with self--consistent frequency (s^2=m+k_0), fixed by a gap equation. We analyze the low--temperature (X)-space physics, the holonomy effective action, the Yang--Mills observable, and the associated phase structure. We then identify a correlated double--scaling limit in which (d\to\infty), (m\to\infty), and (κ=m^{3/2}/d) is held fixed. In this limit the Yang--Mills interaction and the explicit mass deformation remain parametrically balanced: the theory interpolates between the commutator--dominated BFSS regime and the mass--dominated Gaussian regime. The double--scaled theory exhibits two complementary large--(d) regimes. At low temperature, the enhanced gap pushes the deconfinement scale upward and opens a parametrically large uniform--holonomy region, where the bulk dynamics behaves as weakly coupled (\mathrm{BFSS}_2)--type gauged harmonic--oscillator sectors. At the same time, the high--temperature branch reveals an overlap window in which the Gaussian description remains self--consistent while the commutator contribution per matrix pair is parametrically suppressed. The resulting dynamics is therefore (\mathrm{BFSS}_2)--like in its enlarged uniform--holonomy sector and IKKT--like in its almost--commuting matrix behavior.