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PreprintCC BY 4.0arXiv

Unistochastic reduction of multi-matrix coherent-state kernels

  • Shannon Wang
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Abstract

We study finite-N coherent-state overlap kernels in multi-matrix BPS sectors of 𝒩=4 super Yang--Mills theory. For commuting coherent-state parameters, the kernel is a Laplace transform of the Haar-pushforward measure on the unistochastic set. After quotienting row- and column-shift redundancies, the nontrivial source is an (N−1)×(N−1) matrix D. On the physical m-matrix locus, D is a Gram matrix with rankD≤ m, and the ordinary HCIZ sector is precisely the rank-at-most-one stratum. For N=3 we obtain an exact Bessel-integral representation of the first genuinely rank-two kernel. Along the rank-one ray D=rE11, FN=1F1(1;N;r) for arbitrary finite N. Using KN=log FN as the reduced Kähler potential, we study a controlled two-matrix embedding of the HCIZ sector and determine its transverse metric, second fundamental form, and Gauss--Codazzi geometry. The embedding is non-totally-geodesic at finite N. In the scaling r=Nρ, we find NhN→min(ρ,1), |B|2→2(1−ρ)+.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026unistochastic,
  title = {{Unistochastic reduction of multi-matrix coherent-state kernels}},
  author = {Shannon Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.24483},
  eprint = {2609.24483},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24483},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unistochastic reduction of multi-matrix coherent-state kernels
AU  - Shannon Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.24483
UR  - https://arxiv.org/abs/2609.24483
AB  - We study finite-N coherent-state overlap kernels in multi-matrix BPS sectors of 𝒩=4 super Yang--Mills theory. For commuting coherent-state parameters, the kernel is a Laplace transform of the Haar-pushforward measure on the unistochastic set. After quotienting row- and column-shift redundancies, the nontrivial source is an (N−1)×(N−1) matrix D. On the physical m-matrix locus, D is a Gram matrix with rankD≤ m, and the ordinary HCIZ sector is precisely the rank-at-most-one stratum. For N=3 we obtain an exact Bessel-integral representation of the first genuinely rank-two kernel. Along the rank-one ray D=rE_11, F_N=_1F_1(1;N;r) for arbitrary finite N. Using K_N=log F_N as the reduced Kähler potential, we study a controlled two-matrix embedding of the HCIZ sector and determine its transverse metric, second fundamental form, and Gauss--Codazzi geometry. The embedding is non-totally-geodesic at finite N. In the scaling r=Nρ, we find Nh_N→min(ρ,1), |B|^2→2(1−ρ)_+.
ER  -