Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
For a Lie group with Lie algebra , a -principal bundle on a smooth manifold induces a collection of classes in the de Rham cohomology of : the classes of the curvature characteristic forms
of the curvature 2-form of any connection on , and for each invariant polynomial of arity on .
This is a map from the first nonabelian cohomology of with coefficients in to the de Rham cohomology of
where runs over a set of generators of the invariant polynomials. This is the analogy in nonabelian differential cohomology of the generalized Chern character map in generalized Eilenberg-Steenrod-differential cohomology.
Refined Chern-Weil homomorphism
We describe the refined Chern-Weil homomorphism (which associates a class in ordinary differential cohomology to a principal bundle with connection) in terms of the universal connection on the universal principal bundle. We follow (HopkinsSinger, section 3.3).
Let be a compact Lie group
with Lie algebra ;
and write for the dg-algebra of invariant polynomials on (which has trivial differential).
Write for the smooth level classifying space
and for the colimit, a smooth model of the classifying space of .
Write for the universal connection on .
Let be a characteristic class
and choose a refinement in ordinary differential cohomology represented by a differential function
For a smooth manifold, a smoth -principal bundle with smooth classifying map and connection . Write for the Chern-Simons form for the interpolation between and the pullback of the universal connection along .
Then defined the cocycle in ordinary differential cohomology given by the function complex
A classical textbook reference is
The description of the refined Chern-Weil homomorphism in terms of differential function complexes is in section 3.3. of
For more references see Chern-Weil theory.
Revised on March 28, 2012 09:33:18
by Urs Schreiber