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

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.