Transgression in the primitive cohomology

TL;DR

This paper develops a transgression formula for primitive cohomology in symplectic manifolds, extending Chern-Weil theory and introducing primitive characteristic classes for new geometric insights.

Contribution

It proves a primitive version of the Bianchi identity and establishes a transgression formula, advancing the understanding of primitive cohomology in symplectic geometry.

Findings

Proved a primitive Bianchi identity.

Derived a transgression formula for primitive cohomology.

Introduced primitive characteristic classes.

Abstract

We study the Chern-Weil theory for the primitive cohomology of a symplectic manifold. First, given a symplectic manifold, we review the superbundle-valued forms on this manifold and prove a primitive version of the Bianchi identity. Second, as the main result, we prove a transgression formula associated with the boundary map of the primitive cohomology. Third, as an application of the main result, we introduce the concept of primitive characteristic classes and point out a further direction.