Chern–Weil homomorphism

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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).

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

when BG = \varinjlim B_jG and B_jG 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[edit]

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 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 vi 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[edit]

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 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.

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 λj are in R (they are sometimes called Chern roots.) Then \operatorname{ch}(E) = e^{\lambda_j}.

Example: Pontrjagin classes[edit]

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[edit]

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[edit]

  1. ^ Kobayashi-Nomizu 1969, Ch. XII.
  2. ^ The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing [1]. Kobayashi-Nomizu, the main reference, gives a more concrete argument.
  3. ^ 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.
  4. ^ 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[edit]

  • 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", The 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", A.M.S monograph 201 .