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Chern-Weil homomorphism

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

For GG a Lie group with Lie algebra 𝔤\mathfrak{g}, a GG-principal bundle PXP \to X on a smooth manifold XX induces a collection of classes in the de Rham cohomology of XX: the classes of the curvature characteristic forms

F AF AΩ closed 2n(X) \langle F_A \wedge \cdots F_A \rangle \in \Omega^{2n}_{closed}(X)

of the curvature 2-form F AΩ 2(P,𝔤)F_A \in \Omega^2(P, \mathfrak{g}) of any connection on PP, and for each invariant polynomial \langle -\rangle of arity nn on 𝔤\mathfrak{g}.

This is a map from the first nonabelian cohomology of XX with coefficients in GG to the de Rham cohomology of XX

char:H 1(X,G) n iH dR 2n i(X) char : H^1(X,G) \to \prod_{n_i} H_{dR}^{2 n_i}(X)

where ii 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).

Definition

For XX a smooth manifold, PXP \to X a smoth GG-principal bundle with smooth classifying map f:XBGf : X \to B G and connection \nabla. Write CS(,f * univ)CS(\nabla, f^* \nabla_{univ}) for the Chern-Simons form for the interpolation between \nabla and the pullback of the universal connection along ff.

Then defined the cocycle in ordinary differential cohomology given by the function complex

c^:=(f *c,f *h+CS(,f * univ),w(F t))(c,h,w)C k(BG,)×C k1(BG,)×Ω cl k(X). \hat \mathbf{c} := (f^* c , f^* h + CS(\nabla, f^* \nabla_{univ}), w(F_{\nabla_t})) \in (c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times \Omega_{cl}^k(X) \,.
Proposition

The above construction constitutes a map

c^:GBund (X) H diff k(X) \hat \mathbf{c} : G Bund_\nabla(X)_\sim \to H_{diff}^k(X)

from equivalence classes of GG-principal bundles with connection to degree kk ordinary differential cohomology.

Examples

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References

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 (82.172.178.200)