Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection

Abstract

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 $\Sigma$ 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.

76 + 36 pages; v2: added references and appendix D