An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector
- 1Deutsches Elektronen-Synchrotron DESY
Abstract
A bstract : 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.
Versions
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BibTeX
@article{brehm2026exact,
title = {{An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector}},
author = {Enrico Brehm and Jean Cazalis},
journal = {Journal of High Energy Physics},
year = {2026},
month = oct,
doi = {10.1007/jhep10(2026)048},
url = {https://link.springer.com/10.1007/JHEP10(2026)048},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector
AU - Enrico Brehm
AU - Jean Cazalis
PY - 2026
DA - 2026/10/05
JO - Journal of High Energy Physics
DO - 10.1007/jhep10(2026)048
UR - https://link.springer.com/10.1007/JHEP10(2026)048
AB - A bstract : 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.
SN - 1029-8479
ER -