Chern-weil Homomorphism

In mathematics, the Chern-Weil homomorphism is a basic construction in the Chern-Weil theory, relating for a smooth manifold M curvature to its de Rham cohomology groups, i.e. geometry to topology. This theory of Chern and 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_n)
for all f\in\mathbb 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 f\in\mathbb 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_{\sigma\in\mathfrak S_{2k}}\epsilon_\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 closed df(\Omega)=0, and that the 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)}\rightarrow H^*(M,\mathbb K).

 

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