Chern-Weil homomorphism

In mathematics, the Chern-Weil homomorphism is a basic construction in the Chern-Weil theory, relating for a smooth manifold "M" the curvature of "M" to the de Rham cohomology groups of "M", i.e., geometry to topology. This theory of Shiing-Shen Chern and André Weil from the 1940s was an important step in the theory of characteristic classes. It is a generalization of the Chern-Gauss-Bonnet theorem.

Denote by $$mathbb K either the real field or complex field. Let G be a real or complex Lie group with Lie algebra $$mathfrak g; and let

:$$mathbb K(mathfrak g^*)

denote the algebra of $$mathbb K-valued polynomials on $$mathfrak g. Let $$mathbb K(mathfrak g^*)^{Ad(G)} be the subalgebra of fixed points in $$mathbb K(mathfrak g^*) under the adjoint action of G, so that for instance:$$f(t_1,dots,t_k)=f(Ad_g t_1,dots, Ad_g t_k)for all $$finmathbb K(mathfrak g^*)^{Ad(G)}.

The Chern-Weil homomorphism is a homomorphism of $$mathbb K-algebras from $$mathbb K(mathfrak g^*)^{Ad(G)} to the cohomology algebra $$H^*(M,mathbb K). Such a homomorphism exists and is uniquely defined for every principal G-bundle P on M. One can usually think of the bundle P as living inside the K-theory of M, $$ [P] in K_G(M), so that the class of Chern-Weil homomorphisms is parametrized by $$K_G(M).

Definition of the homomorphism

Choose any connection form w in P, and let $$Omega be the associated curvature 2-form. If $$finmathbb K(mathfrak g^*)^{Ad(G)} is a homogeneous polynomial of degree k, let $$f(Omega) be the 2k-form on P given by:$$f(Omega)(X_1,dots,X_{2k})=frac{1}{(2k)!}sum_{sigmainmathfrak S_{2kepsilon_sigma f(Omega(X_{sigma(1)},X_{sigma(2)}),dots,Omega(X_{sigma(2k-1),sigma(2k)}))where $$epsilon_sigma is the sign of the permutation $$sigma in the symmetric group on 2k numbers $$mathfrak S_{2k}.

(see Pfaffian).

One can then show that

:$$f(Omega)

is a closed form, so that

:$$df(Omega)=0,

and that the de Rham cohomology class of

:$$f(Omega)

is independent of the choice of connection on "P", so it depends only upon the principal bundle.

Thus letting

:$$phi(f)

be the cohomology class obtained in this way from f, we obtain an algebra homomorphism

:$$phi:mathbb K(mathfrak g^*)^{Ad(G)} ightarrow H^*(M,mathbb K).

References

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