从ADE型Stokes数据出发的tt*-结构
On tt*-structures from $ADE$-type Stokes data
AI总结：
本文通过同调变形和Stokes矩阵的模糊性，研究了tt*-结构的ADE分类，揭示了Stokes现象、Br_n对称性和Cartan矩阵正定性之间的关系。
摘要：
Cecotti和Vafa引入了拓扑反拓扑融合（tt*）方程，其解描述了超对称共形场论的质子变形。本文提供了ADE分类tt*-结构的严格解析公式。在自然的结构假设下，复平面上的tt*-结构可通过具有上单位三角实Stokes矩阵的同调变形描述。两个基本问题出现：Stokes矩阵的模糊性由一个由重新排列操作生成的群Br_n作用决定，以及相关Riemann-Hilbert问题的可解性。本文第一主要结果表明，分类归结为允许的Stokes矩阵模Br_n作用，且Stokes矩阵的Br_n轨道确定复平面上的tt*-结构。第二主要结果表明，对称化后与类型A_n、D_n、E_6、E_7或E_8的Cartan矩阵相等的上单位三角矩阵产生复平面上的tt*-结构。这为ADE分类提供了直接的解析实现，并阐明了Stokes现象、Br_n对称性和Cartan型矩阵正定性之间的相互作用。
英文摘要：
Cecotti and Vafa introduced the topological anti-topological fusion (tt*)-equation, whose solutions describe massive deformations of supersymmetric conformal field theories. We provide a rigorous analytic formulation of the $ADE$ classification of tt*-structures. Under natural structural assumptions, a tt*-structure over $\mathbb{C}^*$ can be described via isomonodromic deformations with upper unitriangular real Stokes matrices. Two fundamental issues arise: the ambiguities of Stokes matrices, governed by an action of a group $\tilde{Br}_n$, which is generated by reordering operations, and the solvability of the associated Riemann-Hilbert problem. Our first main result shows that the classification reduces to admissible Stokes matrices modulo $\tilde{Br}_n$-action, and that the $\tilde{Br}_n$-orbit of a Stokes matrix determines a tt*-structure over $\mathbb{C}^*$. Our second main result establishes that upper unitriangular matrices whose symmetrizations coincide with Cartan matrices of type $A_n, D_n, E_6, E_7,$ or $E_8$ give rise to tt*-structures over $\mathbb{C}^*$. This provides a direct analytic realization of the $ADE$ classification and clarifies the interplay between Stokes phenomena, $\tilde{Br}_n$-symmetry, and positivity of Cartan-type matrices.
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