# Chern–Weil homomorphism

In mathematics, the Chern–Weil homomorphism is a basic construction in the Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M 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.

The Chern–Weil homomorphism is a homomorphism of $\mathbb {C}$ -algebras

$\mathbb {C} [{\mathfrak {g}}]^{G}\to H^{*}(M,\mathbb {C} )$ where on the right cohomology is de Rham cohomology. 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 BG is isomorphic to the algebra $\mathbb {C} [{\mathfrak {g}}]^{G}$ of invariant polynomials:

$H^{*}(BG,\mathbb {C} )\cong \mathbb {C} [{\mathfrak {g}}]^{G}.$ (The cohomology ring of BG can still be given in the de Rham sense:

$H^{k}(BG,\mathbb {C} )=\varinjlim \operatorname {ker} (d:\Omega ^{k}(B_{j}G)\to \Omega ^{k+1}(B_{j}G))/\operatorname {im} d.$ when $BG=\varinjlim B_{j}G$ and $B_{j}G$ are manifolds.) 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 ω in P, and let Ω be the associated curvature 2-form; i.e., Ω = Dω, the exterior covariant derivative of ω. If $f\in \mathbb {C} [{\mathfrak {g}}]^{G}$ is a homogeneous polynomial function of degree k; i.e., $f(ax)=a^{k}x$ for any complex number a and x in ${\mathfrak {g}}$ , then, viewing f as a symmetric multilinear functional on $\prod _{1}^{k}{\mathfrak {g}}$ (see the ring of polynomial functions), let

$f(\Omega )$ be the (scalar-valued) 2k-form on P given by

$f(\Omega )(v_{1},\dots ,v_{2k})={\frac {1}{(2k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\epsilon _{\sigma }f(\Omega (v_{\sigma (1)},v_{\sigma (2)}),\dots ,\Omega (v_{\sigma (2k-1)},v_{\sigma (2k)}))$ If, moreover, f is invariant; i.e., $f(\operatorname {Ad} _{g}x)=f(x)$ , then one can show that $f(\Omega )$ 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 ω. First, that $f(\Omega )$ is a closed form follows from the next two lemmas:

Lemma 1: The form $f(\Omega )$ on P descends to a (unique) form ${\overline {f}}(\Omega )$ on M; i.e., there is a form on M that pulls-back to $f(\Omega )$ .
Lemma 2: If a form φ on P descends to a form on M, then dφ = Dφ.

To see Lemma 2, let $\pi :P\to M$ be the projection and h be the projection of $T_{u}P$ onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that $d\pi (hv)=d\pi (v)$ (the kernel of $d\pi$ is precisely the vertical subspace.) As for Lemma 1, first note

$f(\Omega )(dR_{g}(v_{1}),\dots ,dR_{g}(v_{2k}))=f(\Omega )(v_{1},\dots ,v_{2k}),\,R_{g}(u)=ug;$ ${\overline {f}}(\Omega )({\overline {v_{1}}},\dots ,{\overline {v_{2k}}})=f(\Omega )(v_{1},\dots ,v_{2k})$ Next, we show that the de Rham cohomology class of ${\overline {f}}(\Omega )$ on M is independent of a choice of connection. Let $\omega _{0},\omega _{1}$ be arbitrary connection forms on P and let $p:P\times \mathbb {R} \to P$ be the projection. Put

$\omega '=t\,p^{*}\omega _{1}+(1-t)\,p^{*}\omega _{0}$ $i_{0}^{*}{\overline {f}}(\Omega ')={\overline {f}}(\Omega _{0})$ The construction thus gives the linear map: (cf. Lemma 1)

$\mathbb {C} [{\mathfrak {g}}]_{k}^{G}\rightarrow H^{2k}(M,\mathbb {C} ),\,f\mapsto \left[{\overline {f}}(\Omega )\right].$ In fact, one can check that the map thus obtained:

$\mathbb {C} [{\mathfrak {g}}]^{G}\rightarrow H^{*}(M,\mathbb {C} )$ is an algebra homomorphism.

## Example: Chern classes and Chern character

$\det \left(I-t{x \over 2\pi i}\right)=\sum _{k=0}^{n}f_{k}(x)t^{k},$ where i is the square root of -1. Then $f_{k}$ are invariant polynomials on ${\mathfrak {g}}$ , 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:

$c_{k}(E)\in H^{2k}(M,\mathbb {Z} )$ is given as the image of fk under the Chern–Weil homomorphism defined by E (or more precisely the frame bundle of E). If t = 1, then $\det \left(I-{x \over 2\pi i}\right)=1+f_{1}(x)+\cdots +f_{n}(x)$ is an invariant polynomial. The total Chern class of E is the image of this polynomial; that is,

$c(E)=1+c_{1}(E)+\cdots +c_{n}(E).$ Directly from the definition, one can show cj, c given above satisfy the axioms of Chern classes. For example, for the Whitney sum formula, we consider

$c_{t}(E)=[\det \left(I-t{\Omega /2\pi i}\right)]$ where we wrote Ω for the curvature 2-form on M of the vector bundle E (so it is the descendent of the curvature form on the frame bundle of E). The Chern–Weil homomorphism is the same if one uses this Ω. Now, suppose E is a direct sum of vector bundles Ei's and Ωi the curvature form of Ei so that, in the matrix term, Ω is the block diagonal matrix with ΩI's on the diagonal. Then, since $\det(I-t\Omega /2\pi i)=\det(I-t\Omega _{1}/2\pi i)\wedge \dots \wedge \det(I-t\Omega _{m}/2\pi i)$ , we have:

$c_{t}(E)=c_{t}(E_{1})\cdots c_{t}(E_{m})$ 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 Chern class#Example: the complex tangent bundle of the Riemann sphere.

$c_{1}(E\otimes E')=c_{1}(E)\operatorname {rk} (E')+\operatorname {rk} (E)c_{1}(E').$ Finally, the Chern character of E is given by

$\operatorname {ch} (E)=[\operatorname {tr} (e^{-\Omega /2\pi i})]\in H^{*}(M,\mathbb {Q} )$ where Ω is the curvature form of some connection on E (since Ω is nilpotent, it is a polynomial in Ω.) Then ch is a ring homomorphism:

$\operatorname {ch} (E\oplus F)=\operatorname {ch} (E)+\operatorname {ch} (F),\,\operatorname {ch} (E\otimes F)=\operatorname {ch} (E)\operatorname {ch} (F).$ Now suppose, in some ring R containing the cohomology ring H(M, C), there is the factorization of the polynomial in t:

$c_{t}(E)=\prod _{j=0}^{n}(1+\lambda _{j}t)$ where λj are in R (they are sometimes called Chern roots.) Then $\operatorname {ch} (E)=e^{\lambda _{j}}$ .

## 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:

$p_{k}(E)=(-1)^{k}c_{2k}(E\otimes \mathbb {C} )\in H^{4k}(M,\mathbb {Z} )$ $\operatorname {det} \left(I-t{x \over 2\pi }\right)=\sum _{k=0}^{n}g_{k}(x)t^{k}.$ ## The homomorphism for holomorphic vector bundles

Let E be a holomorphic (complex-)vector bundle on a complex manifold M. The curvature form Ω of E, with respect to some hermitian metric, is not just a 2-form, but is in fact a (1, 1)-form (see holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle). Hence, the Chern–Weil homomorphism assumes the form: with $G=GL_{n}(\mathbb {C} )$ ,

$\mathbb {C} [{\mathfrak {g}}]_{k}\to H^{k,k}(M,\mathbb {C} ),f\mapsto [f(\Omega )].$ 