多变量单值平坦连接在任意黎曼面上的构造
Single-valued flat connections in several variables on arbitrary Riemann surfaces
AI总结：
本文提出了一种在任意黎曼面上构造多变量单值平坦连接的方法，并将其与多重值Enriquez连接联系起来，同时探讨了其与交换和Fay恒等式的关系。
摘要：
在黎曼面上构造多项对数函数可能通过高效的平坦连接来实现，这些连接可以享有各种代数和分析性质。本文中，我们提出了一个单值且模不变的连接J_DHS，在任意数n个点的配置空间Cf_n(Σ)上，其中Σ是一个任意紧致黎曼面，可能有或没有奇点。该连接J_DHS扩展了之前单变量的构造，并基于相同的积分核。我们证明J_DHS在Cf_n(Σ)上是平坦的。对于无奇点的情况，我们通过规范变换和李代数的自同构的组合，将其与meromorphic多重值Enriquez连接K_E在Σ的universal cover~Σ上联系起来。在 companion paper 中，我们将建立这些连接的平坦性与相应的交换和Fay恒等式之间的等价关系，适用于任意紧致黎曼面。
英文摘要：
Polylogarithms on Riemann surfaces may be constructed efficiently in terms of flat connections that can enjoy various algebraic and analytic properties. In this paper, we present a single-valued and modular invariant connection ${\cal J}_\text{DHS}$ on the configuration space $\text{Cf}_n(Σ)$ of an arbitrary number $n$ of points on an arbitrary compact Riemann surface $Σ$ with or without punctures. The connection ${\cal J}_\text{DHS}$ generalizes an earlier construction for a single variable and is built out of the same integration kernels. We show that ${\cal J}_\text{DHS}$ is flat on $\text{Cf}_n(Σ)$. For the case without punctures, we relate it to the meromorphic multiple-valued Enriquez connection ${\cal K}_\text{E}$ in $n$ variables on the universal cover $\tilde Σ$ of $Σ$ by the composition of a gauge transformation and an automorphism of the Lie algebra in which ${\cal J}_\text{DHS}$ and ${\cal K}_\text{E}$ take values. In a companion paper, we shall establish the equivalence between the flatness of these connections and the corresponding interchange and Fay identities, for arbitrary compact Riemann surfaces.
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