Chern-Weil forms and abstract homotopy theory
Daniel S. Freed, Michael J. Hopkins
TL;DR
The paper addresses the problem of identifying differential forms naturally attached to a G-connection on principal G-bundles. It develops a universal framework using simplicial presheaves to encode connections and their gauge symmetries, introducing the universal bundle E_{ abla}G and its classifying space B_{ abla}G, and showing that the Weil algebra arises as the de Rham complex of E_{ abla}G. The main result proves that Chern-Weil forms exhaust all natural differential forms associated to a G-connection, with the de Rham complex of B_{ abla}G identified as the invariant polynomial algebra with zero differential, and a Weil-model interpretation for equivariant de Rham theory. The work combines abstract homotopy theory, Verdier hypercovering, and invariant theory to provide a cohesive, generalizable approach to differential-geometric characteristic classes, and includes an appendix on the polynomial-functor transformations that underlie key technical steps.
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.