标题： 超共形杨-米尔斯理论中的格点路径组合学 显示英文标题

标题： Lattice path combinatorics in superconformal Yang-Mills theories

摘要： 我们研究四维超共形Yang-Mills理论中的一类可观测量，这些可观测量在平面极限下有限't Hooft耦合时，可以表示为由贝塞尔函数构建的半无限矩阵的行列式。这种行列式表示指向了一种底层的可积结构，我们通过展示这些可观测量满足非线性微分差分方程来明确这一点。我们认为该方程的解可以展开为均匀超越权的迭代Chen积分。值得注意的是，此展开中的系数是普遍的正整数，与特定的可观测量无关，表明存在隐藏的组合起源。基于这一观察，我们证明所得表达式在枚举组合学中具有自然的解释：它们与受限于非平凡区域的格点路径系综的划分函数（或生成函数）相符。这种对应关系扩展并推广了经典的Dyck路径，适用于规范理论中有意义的更丰富的路径系综。 显示英文摘要

摘要： We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This determinant representation points to an underlying integrable structure, which we make explicit by showing that the observables satisfy a nonlinear differential-difference equation. We argue that the solution to this equation admits an expansion in terms of iterated Chen integrals of uniform transcendental weight. Remarkably, the coefficients in this expansion are universal positive integers, independent of the particular observable, suggesting a hidden combinatorial origin. Building on this observation, we show that the resulting expressions possess a natural interpretation in enumerative combinatorics: they coincide with the partition function (or generating function) of an ensemble of lattice paths constrained to a nontrivial domain. This correspondence extends and generalizes the classical Dyck paths to a richer family of path ensembles relevant in gauge theory.