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Mapping cone Thom forms

Hao Zhuang

Abstract

For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone covariant derivative, carrying the extra information brought by the 2-form. Our main tool is the Berezin integral. As the main result, we show that this Thom form is closed with respect to the mapping cone differentiation, its integration along the fiber is 1, and it satisfies the transgression formula.

Mapping cone Thom forms

Abstract

For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone covariant derivative, carrying the extra information brought by the 2-form. Our main tool is the Berezin integral. As the main result, we show that this Thom form is closed with respect to the mapping cone differentiation, its integration along the fiber is 1, and it satisfies the transgression formula.

Paper Structure

This paper contains 6 sections, 8 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Mapping cone covariant derivative
  3. Skew-adjoint Bianchi identity
  4. Berezin integral of pairs
  5. Thom form for the mapping cone complex
  6. Transgression of the Thom form

Key Result

Theorem 1.2

The pair $\mathcal{U}\in\Omega^n(E)\oplus\Omega^{n-1}(E)$ is $d^{\widetilde{\omega}}$-closed. Also, it satisfies Here, $\int_{E/M}$ means the integration along the fiber.

Theorems & Definitions (18)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Corollary 5.1
  • ...and 8 more