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 $f\in\mathbb K(\mathfrak g^*)^{Ad(G)}. \,$

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. If G is compact, then under the homomorphism, the cohomology ring of the classifying space for G-bundles B^G is isomorphic to the algebra $\mathbb K(\mathfrak g^*)^{Ad(G)}$ of invariant polynomials:

$H^*(B^G, \mathbb{K}) \cong \mathbb K(\mathfrak g^*)^{Ad(G)}.$

For non-compact groups like SL(n,R), there may be cohomology classes that are not represented by invariant polynomials.

Definition of the homomorphism

Choose any connection form w in P, and let Ω be the associated curvature 2-form. If $f\in\mathbb K(\mathfrak g^*)^{Ad(G)}$ is a homogeneous polynomial of degree k, let f(Ω) be the 2k-form on P given by

$f(\Omega)(X_1,\dots,X_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in\mathfrak S_{2k}}\epsilon_\sigma f(\Omega(X_{\sigma(1)},X_{\sigma(2)}),\dots,\Omega(X_{\sigma(2k-1)}, X_{\sigma(2k)}))$

where $\epsilon_\sigma$ is the sign of the permutation σ in the symmetric group on 2k numbers $\mathfrak S_{2k}$.

(see Pfaffian).

One can then show that

f(Ω)

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

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

References

• Bott, R. (1973), "On the Chern–Weil homomorphism and the continuous cohomology of Lie groups", Advances in Math 11: 289–303, doi:10.1016/0001-8708(73)90012-1 .
• Chern, S.-S. (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes .
• Chern, S.-S.; Simons, J (1974), "Characteristic forms and geometric invariants", The Annals of Mathematics, Second Series 99 (1): 48–69, JSTOR 1971013 .
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Vol. 2, Wiley-Interscience (published new ed. 2004) .
• Narasimhan, M.; Ramanan, S. (1961), "Existence of universal connections", Amer. J. Math. 83: 563–572, doi:10.2307/2372896, JSTOR 2372896 .