Unistochastic reduction of multi-matrix coherent-state kernels

Abstract

We study finite-$N$ coherent-state overlap kernels in multi-matrix BPS sectors of $\mathcal{N}=4$ 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 $(N-1)\times (N-1)$ matrix $D$. On the physical $m$-matrix locus, $D$ is a Gram matrix with $\mathrm{rank}D\le m$, 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 $D=r{E}_{11}$, $F_N={}_1{F}_1(1;N;r)$ 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 $r=N\rho$, we find \begin{equation} Nh_N\to\min(ρ,1), \qquad \|B\|^2\to2(1-ρ)_+. \end{equation}

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