Mapping cone Thom forms

TL;DR

This paper explicitly constructs the mapping cone Thom form for a closed 2-form using the Mathai-Quillen formalism, employing Berezin integrals to demonstrate its key properties.

Contribution

It provides an explicit formula for the mapping cone Thom form in the context of de Rham cohomology with a closed 2-form, incorporating the mapping cone covariant derivative.

Findings

Thom form is closed under the mapping cone differential

Its fiber integration equals 1

It satisfies the transgression formula

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.