Symplectically Flat Connections and Their Functionals

TL;DR

This paper extends the concept of symplectically flat connections to a broader class called -flat connections on smooth manifolds, introduces related functionals and geometric flows, and explores their characteristic classes.

Contribution

It generalizes symplectically flat connections to -flat connections, introduces new functionals and geometric flows, and studies their characteristic classes.

Findings

Introduction of -flat connections on smooth manifolds.

Development of functionals with symplectically flat connections as zeroes.

Description of characteristic classes of -flat bundles.

Abstract

We continue our study of symplectically flat bundles. We broaden the notion of symplectically flat connections on symplectic manifolds to $\zeta$-flat connections on smooth manifolds. These connections on principal bundles can be represented by maps from an extension of the base's fundamental group to the structure group. We introduce functionals with zeroes being symplectically flat connections and study their critical points. Such functionals lead to novel geometric flows. We also describe some characteristic classes of the $\zeta$-flat bundles.