In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifoldM in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra, and let denote the algebra of -valued polynomials on Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in, Given a principal G-bundleP on M, there is an associated homomorphism of -algebras, called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles,, is isomorphic to the algebra of invariant polynomials:
Definition of the homomorphism
Choose any connection form ω in P, and let Ω be the associated curvature form; i.e.,, the exterior covariant derivative of ω. If is a homogeneous polynomial function of degree k; i.e., for any complex numbera and x in, then, viewing f as a symmetric multilinear functional on , let be the 2k-form on P given by where vi are tangent vectors to P, is the sign of the permutation in the symmetric group on 2k numbers . If, moreover, f is invariant; i.e.,, then one can show that is a closed form, it descends to a unique form on M and that the de Rham cohomology class of the form is independent of. First, that is a closed form follows from the next two lemmas: Indeed, Bianchi's second identity says and, since D is a graded derivation, Finally, Lemma 1 says satisfies the hypothesis of Lemma 2. To see Lemma 2, let be the projection and h be the projection of onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that As for Lemma 1, first note which is because and f is invariant. Thus, one can define by the formula: where are any lifts of :. Next, we show that the de Rham cohomology class of on M is independent of a choice of connection. Let be arbitrary connection forms on P and let be the projection. Put where t is a smooth function on given by. Let be the curvature forms of. Let be the inclusions. Then is homotopic to. Thus, and belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology. Finally, by naturality and by uniqueness of descending, and the same for. Hence, belong to the same cohomology class. The construction thus gives the linear map: In fact, one can check that the map thus obtained: is an algebra homomorphism.
Let and its Lie algebra. For each x in, we can consider its characteristic polynomial in t: where i is the square root of -1. Then are invariant polynomials on, since the left-hand side of the equation is. The k-th Chern class of a smooth complex-vector bundle E of rank n on a manifold M: is given as the image of under the Chern–Weil homomorphism defined by E. If t = 1, then is an invariant polynomial. The total Chern class of E is the image of this polynomial; that is, Directly from the definition, one can show that and c given above satisfy the axioms of Chern classes. For example, for the Whitney sum formula, we consider where we wrote for the curvature 2-form on M of the vector bundleE. The Chern–Weil homomorphism is the same if one uses this. Now, suppose E is a direct sum of vector bundles 's and the curvature form of so that, in the matrix term, is the block diagonal matrix with ΩI's on the diagonal. Then, since, we have: where on the right the multiplication is that of a cohomology ring: cup product. For the normalization property, one computes the first Chern class of the complex projective line; see. Since, we also have: Finally, the Chern character of E is given by where is the curvature form of some connection on E Then ch is a ring homomorphism: Now suppose, in some ring R containing the cohomology ring, there is the factorization of the polynomial in t: where are in R Then.
Example: Pontrjagin classes
If E is a smooth real vector bundle on a manifold M, then the k-th Pontrjagin class of E is given as: where we wrote for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial on given by:
Let E be a holomorphic vector bundle on a complex manifoldM. The curvature form of E, with respect to some hermitian metric, is not just a 2-form, but is in fact a -form. Hence, the Chern–Weil homomorphism assumes the form: with,
