Gram--Wishart--Stiefel formulation of the N=2, large--d gauge theory in 1D: towards emergent Grassmannian geometry
Abstract
We develop the Gram/Wishart/Stiefel formulation of the N=2, large--d planar endpoint theory of lattice BFSS/BMN matrix quantum mechanics on the lattice. The endpoint variables become rank--two Wishart eigenvalues and relative Stiefel angles, expressing the holonomy invariants A, B, and R2=A2+B2 in radial and angular Gram data. The large-R aligned contribution −A is absorbed into the Gaussian sector, producing the shifted mass (αΛ)eff=αΛ−1/2. The exact O(2) angular integral encodes this sector in a rank--two Bessel kernel. The pure −A theory, exactly solvable in Cartesian variables, fixes the leading Bessel/HCIZ structure: its exponential selects alignment, 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 DΛ-channel, while the genuinely anisotropic βΛ-channel lies outside the scope of a pure transverse B-type description. Finally, we outline the emergent Grassmannian geometry encoded by the two endpoint planes. Their relative angular fluctuations form a 2(d−2)-dimensional sector, whose entropy interpretation is developed in the sequel. For d≥4, the two planes generically span a relational four-dimensional subspace of ℝd.
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BibTeX
@misc{ydri2026gramwishartstiefel,
title = {{Gram--Wishart--Stiefel formulation of the $N=2$, large--$d$ gauge theory in 1D: towards emergent Grassmannian geometry}},
author = {Badis Ydri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08481},
eprint = {2607.08481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08481},
}
RIS
TY - UNPB
TI - Gram--Wishart--Stiefel formulation of the N=2, large--d gauge theory in 1D: towards emergent Grassmannian geometry
AU - Badis Ydri
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2607.08481
UR - https://arxiv.org/abs/2607.08481
AB - We develop the Gram/Wishart/Stiefel formulation of the N=2, large--d planar endpoint theory of lattice BFSS/BMN matrix quantum mechanics on the lattice. The endpoint variables become rank--two Wishart eigenvalues and relative Stiefel angles, expressing the holonomy invariants A, B, and R^2=A^2+B^2 in radial and angular Gram data. The large-R aligned contribution −A is absorbed into the Gaussian sector, producing the shifted mass (α_Λ)_eff=α_Λ−1/2. The exact O(2) angular integral encodes this sector in a rank--two Bessel kernel. The pure −A theory, exactly solvable in Cartesian variables, fixes the leading Bessel/HCIZ structure: its exponential selects alignment, 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. Finally, we outline the emergent Grassmannian geometry encoded by the two endpoint planes. Their relative angular fluctuations form a 2(d−2)-dimensional sector, whose entropy interpretation is developed in the sequel. For d≥4, the two planes generically span a relational four-dimensional subspace of ℝ^d.
ER -