Higher Complex Structures and Flat Connections

In the physics literature, Bilal--Fock--Kogan \cite{BFK} introduced the idea of parabolic reduced flat connections on a surface to give a geometric origin to $W$-algebras. In this paper, we combine these ideas with higher complex structures, geometric structures defined by Fock and the author in \cite{FockThomas}. A semiclassical analysis of the parabolic reduction establishes a direct link between flat connections and higher complex structures. In particular, we study a certain class of connections on a bundle equipped with a line subbundle $L$, which we call $L$-parabolic. The curvature of these connections is of rank at most 1. We describe a certain family of $L$-parabolic connections with vanishing curvature, giving the data of a higher complex structure and a cotangent variation. Infinitesimal higher diffeomorphisms, the natural class of transformations on higher complex structures, are realized by the infinitesimal gauge transformation induced by changing $L$. Constructing flat families of connections of this kind is linked to Toda integrable systems.