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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