规范不变单重态扇区中U(N)矩阵模型的精确算法
An exact algorithm for U(N) matrix models in the gauge-invariant singlet sector
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中文总结 AI 辅助
研究规范不变单重态扇区中U(N)矩阵模型，提出精确算法，利用舒尔多项式等张成扇区，经群论约化计算可观测量矩阵元，验证单矩阵实现，概述多矩阵扩展，为全息矩阵模型非平面区域提供新计算方法。
中文摘要 AI 辅助
矩阵模型是M理论和D膜动力学的基本描述，通过规范/引力对偶，其规范不变或单重态扇区描述了全息对偶中的纯引力自由度。本文提出一种新的精确算法，用于计算规范不变单重态扇区中玻色子U(N)矩阵模型的可观测量。该扇区由舒尔多项式（单矩阵）和受限舒尔多项式（多矩阵）的正交基张成，可对角化自由哈密顿量并按激发数截断希尔伯特空间。通过群论约化到对称群合适子群的陪集和双陪集以及对称群上的特征和来计算相互作用哈密顿量或任何规范不变可观测量的矩阵元。结果条目是规范群秩N的封闭形式多项式，由预先计算一次的数据组装而成，可用于任何N和耦合常数。通过与N个非相互作用费米子的精确映射验证了单矩阵实现，展示了低能谱与截断的快速收敛。概述了多矩阵扩展，其主要瓶颈是对称群受限特征的计算，目前尚无与穆尔纳根 - 中山法则可比的算法。该框架可直接访问规范不变态的有限N、有限耦合动力学，并为全息矩阵模型的非平面区域打开了新的计算窗口。
英文摘要
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.
发表机构
- Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究所（DESY））
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