An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector

• URL: http://arxiv.org/abs/2607.13725v2
• Date: Mon, 20 Jul 2026 20:18:33 GMT
• Title: An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector
• Abstract summary: We present a new exact observable for computing observables of boson(N) matrix models in the gauge-invariant singlet sector.<n>We validate the one-matrix implementation against the exact mapping to N non-interacting fermions, demonstrating rapid convergence of the low-lying spectrum with the cutoff.<n>The framework gives direct access to finite-N, finite-coupling dynamics of a gauge-invariant states and a new computational window on the non-planar regime of matrix models.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Matrix models appear as fundamental descriptions of M-theory and D-brane dynamics, and via the gauge/gravity duality their gauge-invariant, or singlet, sector describes the purely gravitational degrees of freedom in the holographic dual. We present a new exact algorithm for computing observables of bosonic U(N) matrix models in the gauge-invariant singlet sector. This sector is spanned by an orthogonal basis of Schur polynomials (for a single matrix) and restricted Schur polynomials (for multiple matrices), which diagonalizes the free Hamiltonian and provides a natural truncation of the Hilbert space by excitation number. Matrix elements of the interaction Hamiltonian, or any gauge-invariant observable, are evaluated through a group-theoretic reduction to cosets and double cosets of suitable subgroups of the symmetric group, together with character sums on the symmetric group. The resulting entries are closed-form polynomials in the gauge-group rank N, assembled from group-theoretic data that are precomputed once and can be reused for any N and any coupling constants. We validate the one-matrix implementation against the exact mapping to N non-interacting fermions, demonstrating rapid convergence of the low-lying spectrum with the cutoff. The multi-matrix extension is outlined; its main bottleneck is the computation of restricted characters of the symmetric group, for which no algorithm comparable to the Murnaghan--Nakayama rule is currently known. The framework gives direct access to finite-N, finite-coupling dynamics of gauge-invariant states and opens a new computational window on the non-planar regime of holographic matrix models.

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