- Chern-Weil homomorphism
In
mathematics , the Chern-Weil homomorphism is a basic construction in theChern-Weil theory , relating for asmooth manifold "M" thecurvature of "M" to thede Rham cohomology groups of "M", i.e., geometry to topology. This theory ofShiing-Shen Chern andAndré Weil from the 1940s was an important step in the theory ofcharacteristic class es. It is a generalization of theChern-Gauss-Bonnet theorem .Denote by mathbb K either the
real field orcomplex field . Let G be a real or complexLie group withLie algebra mathfrak g; and let:mathbb K(mathfrak g^*)
denote the algebra of mathbb K-valued
polynomial s 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 associatedcurvature 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).
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