Gram--Wishart--Stiefel formulation of the $N=2$, large--$d$ gauge theory in 1D

Abstract

We develop in this paper the Gram/Wishart/Stiefel formulation of the \(N=2\), large--\(d\) planar endpoint theory of the BFSS/BMN matrix quantum mechanics on the lattice, obtained in our previous work. In this formulation, the endpoint degrees of freedom are reorganized into rank--two Wishart eigenvalues and relative Stiefel angular variables. This allows the holonomy invariants \(A\), \(B\), and \(R^2=A^2+B^2\) to be analyzed directly in terms of radial and angular Gram data. A central point is the large-\(R\) aligned asymptotics of the holonomy potential. Its universal linear contribution \(-A\) is absorbed into the Gaussian sector, producing the shifted mass parameter \((α_Λ)_{\rm eff}=α_Λ-1/2\). In the Gram/Wishart/Stiefel variables, the exact \(O(2)\) angular integral encodes this shifted sector in a rank--two Bessel kernel. The pure \(-A\) theory, which is exactly solvable in Cartesian variables, then fixes the leading Bessel/HCIZ structure: its exponential part selects the aligned configuration, while its prefactor removes the spurious doubled Wishart entropy. We then apply this structure to the transverse \(B\)-type expansion and its non-polynomial toy completion. Finite polynomial truncations lead to an apparent large--\(d\) perturbativity bound incompatible with the continuum limit, but this bound is shown to be an artifact of truncation. After summing the local transverse completion and balancing the compensating \(+A\) term, the Wishart saddle is recovered with the physical shifted mass. The resulting continuum behavior reproduces the universal \(-2d\) contribution of the \(D_Λ\)-channel, while the genuinely anisotropic \(β_Λ\)-channel lies outside the scope of a pure transverse \(B\)-type description.