模空间上的平坦联络I：DHS联络的局部(1,0)-延拓
Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection
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中文总结 AI 辅助
本文作为系列研究的第一篇，研究DHS联络的局部(1,0)-延拓，分离并求解对应Teichmüller空间全纯方向的联络分量的局部显式解，相关整体延拓工作留待后续。
中文摘要 AI 辅助
arXiv:2602.01461中构造的平坦DHS联络$\boldsymbol{\textit{J}}_{\text{DHS}}$在亏格为$h$的任意固定紧黎曼曲面$\boldsymbol{\textit{Σ}}$上的$n$个点的构型空间上是光滑的，取值于无限维李代数$\boldsymbol{\textit{\hat{\mathfrak{t}}}}_{h,n}$，且在模群$\text{Sp}(2h,\boldsymbol{\textit{Z}})$下不变。本文是一系列研究的第一篇，该系列的目标是将联络$\boldsymbol{\textit{J}}_{\text{DHS}}$延拓到取值于$\boldsymbol{\textit{\hat{\mathfrak{t}}}}_{h,n}$导子李代数的Teichmüller空间$\boldsymbol{\textit{\mathcal{T}}}_{h,n}$上的整体平坦联络。在选取适配映射$\boldsymbol{\textit{\mathcal{T}}}_{h,n}\to\boldsymbol{\textit{\mathcal{T}}}_{h,n}$的局部坐标后，该联络可拆分为三部分：$\boldsymbol{\textit{J}}_{\text{DHS}}$、对应$\boldsymbol{\textit{\mathcal{T}}}_{h}$全纯方向的部分$\boldsymbol{\textit{L}}$，以及对应$\boldsymbol{\textit{\mathcal{T}}}_{h}$反全纯方向的第三部分。本文分离出$\boldsymbol{\textit{L}}$满足的方程组，并在局部得到其显式解。将$\boldsymbol{\textit{J}}_{\text{DHS}}$整体延拓到$\boldsymbol{\textit{\mathcal{T}}}_{h,n}$，以及将arXiv:1112.0864的亚纯联络延拓到$\boldsymbol{\textit{\mathcal{T}}}_{h,n}$的工作，将留待该系列后续论文完成。
英文摘要
The flat DHS connection $\mathcal J_{\mathrm{DHS}}$ constructed in arXiv:2602.01461 is smooth on the configuration space of $n$ points on a fixed compact Riemann surface $Σ$ of arbitrary genus $h$, takes values in an infinite-dimensional Lie algebra $\hat{\mathfrak t}_{h,n}$ and is invariant under the modular group $\mathrm{Sp}(2h,\mathbb Z)$. This paper is the first in a series for a program whose goal is to extend the connection $\mathcal J_{\mathrm{DHS}}$ to a global flat connection on the Teichmüller space $\mathcal T_{h,n}$ valued in the Lie algebra of derivations of $\hat{\mathfrak t}_{h,n}$. Upon the choice of local coordinates adapted to the map $\mathcal T_{h,n}\to \mathcal T_h$, such a connection splits into three pieces: $\mathcal J_{\mathrm{DHS}}$, a piece $\mathcal L$ corresponding to holomorphic directions of $\mathcal T_h$ and a third piece corresponding to anti-holomorphic directions in $\mathcal T_h$. In this paper, we isolate the system of equations satisfied by $\mathcal L$ and obtain its solution locally and explicitly. The construction of a global extension of $\mathcal J_{\mathrm{DHS}}$ to $\mathcal T_{h,n}$ and the extension of the meromorphic connection of arXiv:1112.0864 to $\mathcal T_{h,n}$, are relegated to future publications in this series.
发表机构
- IRMA and Université de Strasbourg（斯特拉斯堡大学）
- Department of Physics and Astronomy, Department of Mathematics, Centre for Geometry and Physics, Uppsala University（乌普萨拉大学）
- Departamento de Matemáticas Fundamentales, UNED（西班牙国立远程教育大学）
机构由 AI 辅助整理，请以论文原文为准。
