Higher Complex Structures and Flat Connections
Abstract
In the physics literature, Bilal--Fock--Kogan 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 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.