平坦连接与任意黎曼面上的Fay恒等式等价性
Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces
AI总结：
本文证明了单值和模形式DHS连接的平坦条件等价于所有交换和Fay恒等式的联合，并展示了多变量Enriquez连接的平坦条件与Enriquez核的恒等式之间的关系。
摘要：
在具有无限维李代数值的黎曼面上，平坦连接提供了一种系统且有效的方法，通过迭代积分生成无限族的多项对数。最近的文献提供了在紧致黎曼面（有或无孔）上的一元或多元连接，且在解析或单值类别中。在本文中，我们证明了在多个变量中单值和模形式DHS连接的平坦条件，该连接在arXiv:2602.01461中引入，等价于arXiv:2407.11476中证明的所有交换和Fay恒等式之间的联合。基于相同的组合技术，多变量Enriquez连接的平坦条件被证明暗示了Enriquez核的所有交换和Fay恒等式的联合。
英文摘要：
A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.
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