AI 中文总结
该研究针对平面$\boldsymbol{\textit{N}=4}$超杨-米尔斯理论的手征应力张量形状因子，建立簇代数描述，提出$\boldsymbol{n}$点形状因子字母表的周期框架，发现对跖对偶性及相关自对偶箭图，匹配共线边界代数。
AI 中文摘要
我们针对平面$\boldsymbol{\textit{N}=4}$超杨-米尔斯理论中的手征应力张量形状因子的符号字母表，建立了簇代数描述。受三点$\boldsymbol{C_2}$箭图展开的启发，我们基于带有两个冻结节点周期的$\boldsymbol{\textrm{Gr}(4,2n)}$箭图折叠，提出了一个用于$\boldsymbol{n}$点形状因子字母表的周期框架。在四点情况下，该折叠导出了具有无穷多个簇变量的$\boldsymbol{\boldsymbol{\textrm{VI}}}$簇代数。热带截断选取了有限的有理坐标集和四条物理极限射线，趋近这些射线的$\boldsymbol{A_1^{(1)}}$突变序列生成了四个平方根及对应的代数字母空间。在所得的有限有理候选集中，对跖闭包恰好排除了八个额外字母。在宇称守恒面上，对跖映射经进一步折叠后，由两个可交换突变及重标记实现。此外，我们在同一突变类中发现了另一个箭图，其在宇称折叠后成为自对跖的，镜像了四点MHV形状因子的对跖自对偶性；其双共线与三共线边界分别重现了三点形状因子的$\boldsymbol{C_2}$代数与六点振幅的$\boldsymbol{A_3}$代数，将后者折叠为$\boldsymbol{C_2}$后，对跖映射交换了两个$\boldsymbol{C_2}$边界。
英文摘要
We develop a cluster-algebraic description of symbol alphabets for chiral stress-tensor form factors in planar $\mathcal N=4$ super-Yang--Mills theory. Motivated by unfolding the three-point $C_2$ quiver, we propose a periodic framework for $n$-point form-factor alphabets based on folding a $\mathrm{Gr}(4,2n)$ quiver with two periods of frozen nodes. At four points, the folding leads to the $\widetilde{\mathrm{VI}}$ cluster algebra, which has infinitely many cluster variables. Tropical truncation selects a finite set of rational coordinates and four physical limit rays. The $A_1^{(1)}$ mutation sequences approaching these rays generate the four square roots and the corresponding algebraic-letter spaces. Within the resulting finite set of rational candidates, antipodal closure excludes precisely eight additional letters. On the parity-preserving surface, the antipodal map is realized, after a further folding, by two commuting mutations followed by a relabelling. Furthermore, we find an alternative quiver within the same mutation class that becomes self-antipodal after parity folding, mirroring the antipodal self-duality of the four-point MHV form factor. Its double- and triple-collinear boundaries reproduce, respectively, the $C_2$ algebra of the three-point form factor and the $A_3$ algebra of the six-point amplitude; after folding the latter to $C_2$, the antipodal map exchanges the two $C_2$ boundaries.
Comments29 pages, 12 figures, and 3 tables
