Schreiber Chern-Weil theory in a smooth (∞,1)-topos

under construction

Ordinary Chern-Weil theory provides for GG a Lie group and for any connection \nabla on a smooth GG-principal bundle PXP \to X a morphism of dg-algebras

Ω (X)inv(𝔤):P(), \Omega^\bullet(X) \leftarrow inv(\mathfrak{g}) : P(\nabla) \,,

where inv(𝔤)inv(\mathfrak{g}) is the dg-algebra of invariant polynomials on the Lie algebra 𝔤\mathfrak{g} of GG: the image of a given invariant polynomial is the corresponding curvature? characteristic class of the bundle.

More generally, in a smooth (∞,1)-topos we have the notion of Ehresmann ∞-connection on a AA-principal ∞-bundle. For XX 0-truncated this comes with a morphism

Ω (X)inv(𝔞):P() \Omega^\bullet(X) \leftarrow inv(\mathfrak{a}) : P(\nabla)

from the invariant polynomial on the ∞-Lie algebroid 𝔞\mathfrak{a} of AA.

Created on February 22, 2010 at 17:26:25. See the history of this page for a list of all contributions to it.