Abelianization of the $\operatorname{SL}_2$ Hitchin connection at level four

TL;DR

This paper demonstrates that the $ ext{SL}_2$ Hitchin connection at level four can be described via abelian theta functions on Prym torsors, leading to the conclusion that its monodromy is finite, extending previous curve-specific results.

Contribution

It provides a new understanding of the Hitchin connection at level four through abelian theta functions and shows its monodromy is finite, using equivariant embeddings and duality.

Findings

Hitchin connection at level four relates to Mumford-Welters connections

Monodromy of the connection is finite

Extension of previous curve-specific results to a broader setting

Abstract

We prove that the Hitchin connection for $\operatorname{SL}_2$ at level four can be understood in terms of the Mumford-Welters connections on bundles of abelian theta functions for Prym torsors of all unramified double covers, and use this to show that its monodromy is finite. This builds on earlier works, for individual curves, of the last named author with Oxbury and Ramanan. The key ingredients in making this work on the level of connections are equivariant conformal embeddings, and anti-invariant level-rank duality.