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Transgression in the primitive cohomology

Hao Zhuang

TL;DR

The paper develops a primitive Chern–Weil theory on symplectic manifolds by formulating a primitive superconnection $\mathbb{A}$ acting on a superbundle and proving a primitive Bianchi identity $\partial\,\mathrm{Str}(\mathbb{A}^{2k})=(0,0)$. It then derives a transgression formula for a smooth family of primitive superconnections, establishing an explicit $\partial$-exact relation for the derivative of $\mathrm{Str}(\mathbb{A}_t^{2k+2})$, which underpins the independence of primitive cohomology classes from the chosen connection. Using these tools, it defines primitive characteristic classes such as the primitive Chern character $\mathrm{ch}(\mathbb{A},\omega)$ and a primitive $\hat{A}$-genus, and discusses their behavior in the integral $\omega$-case where connections relate to circle bundles and Chern–Simons forms. The work suggests further links between primitive invariants and geometric structures (e.g., symplectic flatness, Kähler–Einstein conditions) and raises questions about the geometric meaning of these primitive 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.

Transgression in the primitive cohomology

TL;DR

The paper develops a primitive Chern–Weil theory on symplectic manifolds by formulating a primitive superconnection acting on a superbundle and proving a primitive Bianchi identity . It then derives a transgression formula for a smooth family of primitive superconnections, establishing an explicit -exact relation for the derivative of , which underpins the independence of primitive cohomology classes from the chosen connection. Using these tools, it defines primitive characteristic classes such as the primitive Chern character and a primitive -genus, and discusses their behavior in the integral -case where connections relate to circle bundles and Chern–Simons forms. The work suggests further links between primitive invariants and geometric structures (e.g., symplectic flatness, Kähler–Einstein conditions) and raises questions about the geometric meaning of these primitive 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.
Paper Structure (4 sections, 5 theorems, 83 equations)

This paper contains 4 sections, 5 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Primitive Bianchi identity
  3. Transgression formula
  4. Characteristic classes involving $\omega$

Key Result

Theorem 1.6

For a smooth family of primitive superconnections $\mathbb{A}_t$$(t\in\mathbb{R})$ and any $k\in\mathbb{N}$, we identify maps $\mathbb{A}_t^{2k}$ and $\dfrac{d\mathbb{A}_t}{dt}\mathbb{A}_t^{2k-2}$ with unique elements and respectively according to (A2k expression) and (dA_tA_t^2k expression). Then, for any polynomial $f\in\mathbb{C}[z]$, we have

Theorems & Definitions (18)

  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 8 more