发表机构
International Centre for Theoretical Sciences-TIFR; Centre for Theoretical Physics, Department of Physics and Astronomy, Queen Mary University of London; School of Physics and Shing-Tung Yau Centre of Southeast University（国际理论科学中心-TIFR; 伦敦大学玛丽女王学院物理与天文系理论物理中心; 东南大学物理学院及施一公中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限N下N=4超杨-米尔斯理论su(2)区的一圈膨胀算子，建立其特征多项式的整除关系，提出SU(N)不变多项式空间的同构猜想及对应膨胀算子，计算谱与本征空间分解预测一致。
AI 中文摘要
我们研究了次数为L的规范不变双矩阵多项式，对应于具有规范群U(N)或SU(N)的N=4超杨-米尔斯理论su(2)区的态，其中N为有限值。对于U(N)理论，其态空间具有代数A_N(L)的结构，该代数是置换中心化子代数的直和，即对称群S_L的群代数ℂ[S_L]的子代数。当N≥L时，A_N(L)=A(L)与N无关，且一圈膨胀算子以N为参数作用于A(L)；当N<L时，A_N(L)是A(L)的子空间，通过对具有L个框的杨图施加有限N截断来定义。我们在一圈膨胀算子的本征空间上得到两个结果：第一个主要结果是，作用于A_N(L)的一圈膨胀算子的特征多项式，可整除延拓至N<L时A(L)的特征多项式；第二个结果是，SU(N)理论中一圈膨胀算子的每个本征态，通过单迹的乘法会在U(N)理论中生成无限多个本征态，这些本征态构成由迹的数量标记的U(N)理论的本征空间分解。此外，我们提出猜想：有限N下两个无迹矩阵的SU(N)不变多项式空间，同构于A_N(L)与由错位排列张成的ℂ[S_L]子空间的交集，该交集猜想得到了维数计数的支持。最后，我们在该交集上提出了一个一圈膨胀算子，为有限N下SU(N)谱问题提供了直接的置换框架，基于该提议计算的谱与U(N)理论中本征空间分解的预测一致。
英文摘要
We study gauge-invariant two-matrix polynomials of degree $L$, corresponding to the states in the $su(2)$ sector of $\mathcal{N}=4$ super Yang-Mills theory with gauge group $U(N)$ or $SU(N)$ at finite $N$. For the $U(N)$ theory, the state space has the structure of an algebra $\mathcal{A}_N(L)$, which is a direct sum of permutation centraliser algebras, i.e. sub-algebras of the group algebra $\mathbb{C}[S_L]$ of $S_L$. For $N \ge L$, $\mathcal{A}_N (L) = \mathcal{A} (L)$ is independent of $N$, and the one-loop dilatation operator acts on $\mathcal{A}(L)$ with $N$ as a parameter. For $N < L$, $\mathcal{A}_N (L)$ is a subspace of $\mathcal{A}(L)$ defined using a finite-$N$ cutoff on Young diagrams with $L$ boxes. We establish two results on the eigenspaces of the one-loop dilatation operator. Our first main result is that the characteristic polynomial of the one-loop dilatation operator on $\mathcal{A}_N(L)$ divides the characteristic polynomial from $\mathcal{A}(L)$, continued to $N < L$. Our second result is that each eigenstate of the one-loop dilatation operator in the $SU(N)$ theory generates an infinite tower of eigenstates in the $U(N)$ theory, through the multiplication of single-letter traces. These towers define an eigenspace factorisation of the $U(N)$ theory labelled by the number of such traces. Further, we conjecture that the space of $SU(N)$-invariant polynomials in two traceless matrices at finite $N$ is isomorphic to the intersection of $\mathcal{A}_N(L)$ with the subspace of $\mathbb{C}[S_L]$ spanned by derangements. This intersection conjecture is supported by dimension counting. Finally, we propose a one-loop dilatation operator on the intersection, thus providing a direct permutation framework for the $SU(N)$ spectral problem at finite $N$. The computed spectra based on the proposal agree with the prediction from the eigenspace factorisation in the $U(N)$ theory.
Comments47 pages + 28 pages of appendices
