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.

Let G be a real or complex Lie group with Lie algebra $\mathfrak g$ ; and let $\mathbb{C}[\mathfrak g]$ denote the algebra of $\mathbb{C}$ -valued polynomials on $\mathfrak g$ (exactly the same argument works if we used $\mathbb{R}$ instead of $\mathbb{C}$ .) Let $\mathbb{C}[\mathfrak g]^G$ be the subalgebra of fixed points in $\mathbb C[\mathfrak g]$ under the adjoint action of G; that is, it consists of all polynomials f such that for any g in G and x in $\mathfrak{g}$ , $f(\operatorname{Ad}_g x) = f(x).$

Given principal G-bundle P on M, there is an associated homomorphism of $\mathbb C$ -algebras

\mathbb C[\mathfrak g]^{G} \to H^*(M,\mathbb C)

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 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_jG) \to \Omega^{k+1}(B_jG))/\operatorname{im} d.

when $BG = \varinjlim B_jG$ and $B_jG$ are manifolds.)

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., $f(a x) = a^k f(x)$ 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)}))

where v_i are tangent vectors to P, $\epsilon_\sigma$ is the sign of the permutation $\sigma$ in the symmetric group on 2k numbers $\mathfrak S_{2k}$ (see Lie algebra-valued forms#Operations as well as Pfaffian).

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:^[1]

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.

Indeed, Bianchi's second identity says $D \Omega = 0$ and, since D is a graded derivation, $D f(\Omega) = 0.$ Finally, Lemma 1 says $f(\Omega)$ satisfies the hypothesis of Lemma 2.

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 $d \pi(h v) = d \pi(v)$ (the kernel of $d \pi$ is precisely the vertical subspace.) As for Lemma 1, first note

f(\Omega)(d R_g(v_1), \dots, d R_g(v_{2k})) = f(\Omega)(v_1, \dots, v_{2k}), \, R_g(u) = ug;

which is because $R_g^* \Omega = \operatorname{Ad}_{g^{-1}} \Omega$ and f is invariant. Thus, one can define $\overline{f}(\Omega)$ by the formula:

\overline{f}(\Omega)(\overline{v_1}, \dots, \overline{v_{2k}}) = f(\Omega)(v_1, \dots, v_{2k})

where $v_i$ are any lifts of $\overline{v_i}$ : $d \pi(v_i) = \overline{v}_i$ .

Next, we show that the de Rham cohomology class of $\overline{f}(\Omega)$ on M is independent of a choice of connection.^[2] 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

where t is a smooth function on $P \times \mathbb{R}$ given by $(x, s) \mapsto s$ . Let $\Omega', \Omega_0, \Omega_1$ be the curvature forms of $\omega', \omega_0, \omega_1$ . Let $i_s: M \to M \times \mathbb{R}, \, x \mapsto (x, s)$ be the inclusions. Then $i_0$ is homotopic to $i_1$ . Thus, $i_0^* \overline{f}(\Omega')$ and $i_1^* \overline{f}(\Omega')$ belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology. Finally, by naturality and by uniqueness of descending,

i_0^* \overline{f}(\Omega') = \overline{f}(\Omega_0)

and the same for $\Omega_1$ . Hence, $\overline{f}(\Omega_0), \overline{f}(\Omega_1)$ belong to the same cohomology class.

The construction thus gives the linear map: (cf. Lemma 1)

\mathbb C[\mathfrak g]^{G}_k \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

Let $G = GL_n(\mathbb{C})$ and $\mathfrak{g} = \mathfrak{gl}_n(\mathbb{C})$ its Lie algebra. For each x in $\mathfrak{g}$ , we can consider its characteristic polynomial in t:

\det \left( I - t{x \over 2 \pi i} \right) = \sum_{k=0}^n f_k(x) t^k,

^[3]

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 f_k 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 c_j, 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 E_i's and Ω_i the curvature form of E_i so that, in the matrix term, Ω is the block diagonal matrix with Ω_I'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.

Since $\Omega_{E \otimes E'} = \Omega_E \otimes I_{E'} + I_{E} \otimes \Omega_{E'}$ ,^[4] we also have:

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 λ_j 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})

where we wrote $E \otimes \mathbb{C}$ for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial $g_{2k}$ on $\mathfrak{gl}_n(\mathbb{R})$ given by:

\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)].

Notes

↑ Kobayashi-Nomizu 1969, Ch. XII.
↑ The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing . Kobayashi-Nomizu, the main reference, gives a more concrete argument.
↑ Editorial note: This definition is consistent with the reference except we have t, which is t ⁻¹ there. Our choice seems more standard and is consistent with our "Chern class" article.
↑ Proof: By definition, $\nabla^{E \otimes E'}(s \otimes s') = \nabla^{E} s \otimes s' + s \otimes\nabla^{E'} s'$ . Now compute the square of $\nabla^{E \otimes E'}$ using Leibniz's rule.

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 .
Shiing-Shen Chern, Complex Manifolds Without Potential Theory (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.
The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
Chern, S.-S.; Simons, J (1974), "Characteristic forms and geometric invariants", Annals of Mathematics. Second Series 99 (1): 48–69, JSTOR 1971013 .
Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Vol. 2 (new ed.), Wiley-Interscience (published 2004) .
Narasimhan, M.; Ramanan, S. (1961), "Existence of universal connections", Amer. J. Math. 83: 563–572, doi:10.2307/2372896, JSTOR 2372896 .
Morita, Shigeyuki (2000), "Geometry of Differential Forms", Translations of Mathematical Monographs 201 .