Chern–Weil homomorphism explained

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 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 and let

\Complex[ ak g] denote the algebra of \Complex -valued polynomials on ak g (exactly the same argument works if we used \R instead of Let \Complex[ ak g] ^{G} be the subalgebra of fixed points in \Complex[ ak g] under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that f(\operatorname{Ad} _{g} x) = f(x) , for all g in G and x in ak{g} ,

Given a principal G-bundle P on M, there is an associated homomorphism of

\Complex -algebras,\Complex[ ak g] ^{G} \to H ^{*(M;} \Complex) ,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 \Complex[ ak g] ^{G} of invariant polynomials:H ^{*(BG;} \Complex) \cong \Complex[ ak g] ^{G} . (The cohomology ring of BG can still be given in the de Rham sense:H ^{k(BG;} \Complex) = \varinjlim \operatorname{ker} (d\colon \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 form; i.e., the exterior covariant derivative of ω. If

f\in C[ ak 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 then, viewing f as a symmetric multilinear functional on $\backslash prod\_1^k\; \backslash mathfrak$ (see the ring of polynomial functions), letf(\Omega) be the (scalar-valued) 2k-form on P given byf(\Omega)(v _{1,...,v} _{2k} ) = (1)/(( 2 k)!) \sum _{\sigma\in ak S _{2k}} \epsilon _{\sigma} f(\Omega(v _{\sigma(1)} ,v _{\sigma(2)} ), ... ,\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 ak 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 \omega . 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 of

\varphi on P descends to a form on M, then d\varphi = D\varphi .

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\colon 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 notef(\Omega)(d R _{g(v} _{1),} ... , d R _{g(v} _{2k} )) = f(\Omega)(v _{1,} ... , 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},} ... , \overline{v _{2k} }) = f(\Omega)(v_1, \dots, v_),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.^[1] Let \omega _{0,} \omega _{1} be arbitrary connection forms on P and let p\colon P x \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 x 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 x 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)

\Complex[ ak g] ^{G} _{k} \to H ^{2k} (M; \Complex), f \mapsto \left[\overline{f}(\Omega)\right]. In fact, one can check that the map thus obtained:\Complex[ ak g] ^{G} \to H ^{*(M;} \Complex) is an algebra homomorphism.

Example: Chern classes and Chern character

Let

G = \operatorname{GL} _{n(\Complex)} and ak{g} = ak{gl} _{n(\Complex)} its Lie algebra. For each x in ak{g} , we can consider its characteristic polynomial in t:^[2] \det \left( I - t{x \over 2 \pi i} \right) = \sum ^{n} _{k =0} f _{k(x)} t ^{k,} where i is the square root of -1. Then f _{k} are invariant polynomials on ak{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, \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)} + … + 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)} + … + c _{n(E).}

Directly from the definition, one can show that

c _{j} and c given above satisfy the axioms of Chern classes. For example, for the Whitney sum formula, we considerc _{t(E)} = [\det \left( I - t {\Omega / 2 \pi i} \right)], where we wrote \Omega 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 \Omega . Now, suppose E is a direct sum of vector bundles E _{i} 's and \Omega _{i} the curvature form of E _{i} so that, in the matrix term, \Omega is the block diagonal matrix with Ω_I's on the diagonal. Then, since we have:c _{t(E)} = c _{t(E} _{1)} … 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 .

Since

\Omega _{E} = \Omega _{E} ⊗ I _{E'} + I _{E} ⊗ \Omega _{E'} ,^[3] we also have:c _{1(E} ⊗ E') = c _{1(E)} \operatorname{rank} (E') + \operatorname{rank}(E) c _{1(E').}

Finally, the Chern character of E is given by

\operatorname{ch}(E) = [\operatorname{tr}(e ^{-\Omega/2\pi} )] \in H ^{*(M,} \Q) where \Omega is the curvature form of some connection on E (since \Omega is nilpotent, it is a polynomial in \Omega .) Then ch is a ring homomorphism:\operatorname{ch}(E ⊕ F) = \operatorname{ch}(E) + \operatorname{ch}(F), \operatorname{ch}(E ⊗ F) = \operatorname{ch}(E) \operatorname{ch}(F). Now suppose, in some ring R containing the cohomology ring H ^{*(M,} \Complex) , there is the factorization of the polynomial in t:c _{t(E)} = \prod ^{n} _{j =0} ( 1 + λ _{j} t) where λ _{j} are in R (they are sometimes called Chern roots.) Then \operatorname{ch}(E) = e ^{λ _{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 ⊗ \Complex) \in H ^{4k} (M; \Z) where we wrote E ⊗ \Complex for the complexification of E. Equivalently, it is the image under the Chern–Weil homomorphism of the invariant polynomial g _{2k} on ak{gl} _{n(\R)} given by:\operatorname{det}\left(I - t {x \over 2 \pi}\right) = \sum ^{n} _{k =0} 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

\Omega 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 = \operatorname{GL} _{n(\Complex)} ,\Complex[ ak{g}] _{k} \to H ^{k,} (M; \Complex), f \mapsto [f(\Omega)].

References

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• , . (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.)
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Further reading

• Chern-Weil forms and abstract homotopy theory. 1301.5959. 2013. Daniel S.. Freed. Dan Freed. Michael J.. Hopkins. Michael J. Hopkins. Bulletin of the American Mathematical Society. (N.S.) . 50 . 3. 431–468. 10.1090/S0273-0979-2013-01415-0. 3049871. 51755613.

Notes and References

1. The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing Web site: Archived copy . 2014-12-11 . dead . https://web.archive.org/web/20141217025038/https://math.berkeley.edu/~amathew/kodaira.pdf . 2014-12-17 . . Kobayashi-Nomizu, the main reference, gives a more concrete argument.
2. 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.
3. Proof: By definition, \nabla ^{E} (s ⊗ s') = \nabla ^{E} s ⊗ s' + s ⊗ \nabla ^{E'} s' . Now compute the square of \nabla ^{E} using Leibniz's rule.