Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection
Abstract
The flat DHS connection 𝒥DHS 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 t̂h,n and is invariant under the modular group Sp(2h,ℤ). This paper is the first in a series for a program whose goal is to extend the connection 𝒥DHS to a global flat connection on the Teichmüller space 𝒯h,n valued in the Lie algebra of derivations of t̂h,n. Upon the choice of local coordinates adapted to the map 𝒯h,n→ 𝒯h, such a connection splits into three pieces: 𝒥DHS, a piece ℒ corresponding to holomorphic directions of 𝒯h and a third piece corresponding to anti-holomorphic directions in 𝒯h. In this paper, we isolate the system of equations satisfied by ℒ and obtain its solution locally and explicitly. The construction of a global extension of 𝒥DHS to 𝒯h,n and the extension of the meromorphic connection of arXiv:1112.0864 to 𝒯h,n, are relegated to future publications in this series.
Related
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- On leading Gopakumar--Vafa invariants of local ℙ2 and leading Betti numbers of moduli spaces of one-dimensional sheaves on ℙ2
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BibTeX
@misc{dhoker2026flat,
title = {{Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection}},
author = {Eric D'Hoker and Benjamin Enriquez and Oliver Schlotterer and Federico Zerbini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29169},
eprint = {2608.29169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29169},
}
RIS
TY - UNPB
TI - Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection
AU - Eric D'Hoker
AU - Benjamin Enriquez
AU - Oliver Schlotterer
AU - Federico Zerbini
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2608.29169
UR - https://arxiv.org/abs/2608.29169
AB - The flat DHS connection 𝒥_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 t̂_h,n and is invariant under the modular group Sp(2h,ℤ). This paper is the first in a series for a program whose goal is to extend the connection 𝒥_DHS to a global flat connection on the Teichmüller space 𝒯_h,n valued in the Lie algebra of derivations of t̂_h,n. Upon the choice of local coordinates adapted to the map 𝒯_h,n→ 𝒯_h, such a connection splits into three pieces: 𝒥_DHS, a piece ℒ corresponding to holomorphic directions of 𝒯_h and a third piece corresponding to anti-holomorphic directions in 𝒯_h. In this paper, we isolate the system of equations satisfied by ℒ and obtain its solution locally and explicitly. The construction of a global extension of 𝒥_DHS to 𝒯_h,n and the extension of the meromorphic connection of arXiv:1112.0864 to 𝒯_h,n, are relegated to future publications in this series.
ER -