Chern-Weil forms and abstract homotopy theory

TL;DR

This paper characterizes Chern-Weil forms as the unique natural differential forms from connections on principal bundles, using homotopy theory of simplicial sheaves and classical invariant theory.

Contribution

It introduces a novel homotopy-theoretic framework to prove the uniqueness of Chern-Weil forms and provides new interpretations of the Weil algebra within this context.

Findings

Chern-Weil forms are the only natural forms associated to principal G-bundle connections.

The Weil algebra is identified as the de Rham complex of a specific simplicial sheaf.

A new interpretation of the Weil model in equivariant de Rham theory is provided.

Abstract

We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors. This paper is dedicated to the memory of Dan Quillen.